Kleisli Monoids - Vrije Universiteit Brussel...monoid, commute. In Chapter2we give di erent examples...

Faculteit WetenschappenVakgroep Wiskunde

Kleisli Monoids

Proefschrift ingediend met het oog op het behalen van de graad Master in de Wiskunde

Stijn Toth

Promotor: Prof. Dr. Eva Colebunders

Academiejaar 2013-2014

Abstract

For a Set-monad T we consider (T, 2)-algebras as a generalization of Barr’spresentation of topological spaces in terms of ultrafilter convergence. Theselax algebras require the existence of lax extensions of T. Through these laxalgebras we describe different topological structures via a convergence struc-ture.

By generalizing the description of topological spaces by neighborhoodsystems, we obtain the notion of Kleisli monoids. In general the categoryof Kleisli monoids is a neighborhood-like presentation of categories of laxalgebras. A notable advantage of this approach is that it does not requireexplicitly the lax extension of the associated monad. In this thesis, wepresent, specifically, ordered sets, topological spaces and closure spaces asKleisli monoids and lax algebras.

i

Samenvatting

Voor een Set-monad T beschouwen we (T, 2)-algebra’s als een veralgemeeningvan Barr’s voorstelling van topologische ruimten in termen van ultrafilterconvergentie. Deze lax algebra’s vereisen het bestaan van lax extensies vanT. Via deze lax algebra’s beschrijven we verschillende topologische structurendoor een convergentiestructuur.

Door de beschrijving van topologische ruimten via omgevingensystemente veralgemenen, verkrijgen we de notie van Kleisli monoıden. In het alge-meen is de categorie van Kleisli monoıden een omgevingen-achtige presen-tatie van categorieen van lax algebra’s. Een noemenswaardig voordeel vandeze benadering is dat het bestaan van een lax extensie van de geassocieerdemonad niet expliciet gebruikt wordt. In deze thesis beschrijven we specifiekgeordende verzamelingen, topologische ruimten en closure ruimten als Kleislimonoıden en lax algebra’s.

ii

Contents

Abstract i

Samenvatting ii

Contents iii

Acknowledgments v

Introduction vi

Inleiding x

1 Basic concepts 11.1 The compositional structure of relations . . . . . . . . . . . . 11.2 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Adjunctions for ordered sets . . . . . . . . . . . . . . . . . . . 51.5 Closure operation and interior operation . . . . . . . . . . . . 61.6 Completeness and lattices . . . . . . . . . . . . . . . . . . . . 71.7 Filters, ultrafilters, cliques and ultracliques . . . . . . . . . . . 8

2 Monads 152.1 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Kleisli triples . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Lax extensions 303.1 Lax functors and their transformations . . . . . . . . . . . . . 303.2 Lax extensions of functors . . . . . . . . . . . . . . . . . . . . 313.3 Lax extensions of monads . . . . . . . . . . . . . . . . . . . . 363.4 The Barr extension . . . . . . . . . . . . . . . . . . . . . . . . 373.5 The Beck-Chevalley condition . . . . . . . . . . . . . . . . . . 413.6 The Barr extension of a monad . . . . . . . . . . . . . . . . . 43

iii

CONTENTS iv

4 Algebra structures and unitary relations 464.1 Eilenberg-Moore algebra . . . . . . . . . . . . . . . . . . . . . 464.2 Lax algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 The Kleisli category . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Kleisli convolution . . . . . . . . . . . . . . . . . . . . . . . . 564.5 Unitary (T, 2)-relations . . . . . . . . . . . . . . . . . . . . . . 594.6 Associativity of unitary (T, 2)-relations . . . . . . . . . . . . . 604.7 Fundamental example of a lax algebra . . . . . . . . . . . . . 644.8 �-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.9 Induced order on lax algebras . . . . . . . . . . . . . . . . . . 71

5 Kleisli Monoids 735.1 Topological spaces via neighborhood filters . . . . . . . . . . . 745.2 Power-enriched monads . . . . . . . . . . . . . . . . . . . . . . 775.3 T-monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4 The Kleisli extension . . . . . . . . . . . . . . . . . . . . . . . 875.5 Topological spaces via filter convergence . . . . . . . . . . . . 92

6 Initial extensions 986.1 Algebraic functors . . . . . . . . . . . . . . . . . . . . . . . . . 986.2 Initial extensions . . . . . . . . . . . . . . . . . . . . . . . . . 996.3 Sup-dense and interpolating monad morphisms . . . . . . . . . 1036.4 (S, 2)-categories as Kleisli monoids . . . . . . . . . . . . . . . 1066.5 Compact Hausdorff closure spaces . . . . . . . . . . . . . . . . 112

Bibliography 117

Index 119

Acknowledgments

In the first place, I would like to thank my promotor professor Eva Colebundersfor introducing me to the research of monoidal topology and for the profes-sional guidance she has given me the past year. I also want to thank all theprofessors I have encountered in the past five years for teaching and inspiringme in mathematics.

Next, I want to thank my parents, brothers and partner, for standingby my side and motivating me, although they don’t always understand thetopics I am talking about.

For the support and moments of relaxation they provide, I want to thankall my friends and classmates.

Finally, I wish you take pleasure in reading my thesis.

Stijn TothMay 2014

v

Introduction

Monoidal Topology describes an active research that provides a powerful uni-fying framework and theory for ordered, metric and topological structures.For this framework we use category theory as language. By this theory, wewill describe, inter alia, ordered1 spaces and topological spaces by a conver-gence structure.

Monoids are a simple structure, i.e. a set with an associative multipli-cation and a unit. By viewing these conditions in diagrams we can generalizethis notion. Instead of a set, we use an endofunctor T : Set → Set and themultiplication and unit become natural transformations m : TT → T ande : 1Set → T. We have a monad, when the diagrams for the natural transfor-mations, similar to the diagrams of the multiplication and the unit of amonoid, commute. In Chapter 2 we give different examples of monads.

With a monad T = (T,m, e) Eilenberg and Moore (1965) constructed acategory SetT with objects sets X equipped with a map a : TX → X makingthe two diagrams

TTX Ta //

mX��

TX

a��

XeX //

1X !!

TX

a��

TX a// X X

commutative and with appropriate morphisms. This category will be calledthe Eilenberg-Moore category and its objects are the Eilenberg-Moore alge-bras.

The starting point for monoidal topology is the proof by Manes [10] in1969 that compact Hausdorff spaces are precisely the Eilenberg-Moore alge-bras of the ultrafilter monad � = (β,m, e). In this description, the mapa : βX → X of a compact Hausdorff space X assigns to every ultrafilterits unique point of convergence in X, requiring the two basic axioms of anEilenberg-Moore algebra a(βa(U )) = a(mX(U )) and a(eX(x)) = x for all

1By an order on a space X, we mean a reflexive and transitive relation. An order doesnot need to satisfy the antisymmetric axiom.

vi

CONTENTS vii

U ∈ ββX and x ∈ X. We will rewrite these axioms by using the principalultrafilter on x

eX(x) = x = {A ⊆ X | x ∈ A}

and the Kowalsky sum of U

mX(U ) = ΣU = {A ⊆ X | {U ∈ βX | A ∈ U} ∈ U }.

These ultrafilters, together with the notations V → z instead of a(V) = zand U → V instead of βa(U ) = V , give a more intuitive form of the basicaxioms, i.e.

U → V & V → z =⇒ ΣU → z

andx→ x

for all U ∈ ββX,V ∈ βX and x, z ∈ X.For metric spaces and convergence of sequences, we can describe the first

axiom in the following way. Take a sequence (xn)n∈N that converges to x ∈ X,then for a collections of sequences, where the first sequence converges tox1, the second one to x2, and so on, we take the diagonal sequence, i.e.the first element of the first sequence, the second element of the secondsequence, . . . Then the diagonal sequence converge to x. The Kowalsky sumis a generalization of the diagonal sequence.

In 1970 Barr [1] observed that in an arbitrary topological space it is nolonger guaranteed that every ultrafilter has a point of convergence (compact-ness) and that there is at most one such point (Hausdorffness) and thereforewe will need a relation instead of a map. Such a relation a describing conver-gence has to satisfy a(βa(U )) ≤ a(mX(U )) (transitivity) and x ≤ a(eX(x))(reflexivity) for all U ∈ ββX and x ∈ X. Thus a topology on X is describedby two simple axioms on a convergence relation between ultrafilters andpoints. In the discussion above we glossed over an important point: havingthe Set-functor β, one knows what βa is when a is a map, but not necessar-ily when a is just a relation. To define βa, when a is a relation, there is astraightforward solution.

By changing the ultrafilter monad � = (β,m, e) by any other monadT = (T,m, e), these conditions on a relation a : TX →7 X give rise to thenotion of lax algebras. Together with the appropriate morphisms, they forma category of lax algebras. Again, we glossed over the issue of defining Tawhen a is a relation. While there is for the ultrafilter monad a straight-forward solution, in general we are confronted with the issue of extendinga monad T = (T,m, e) on Set to Rel. For our purposes it suffices that this

CONTENTS viii

extension is lax, not a genuine monad on Rel. The definition and examplesof lax extensions of monads can be found in Chapter 3.

In section 4.8 we demonstrate the connection between the Eilenberg-Moore algebras and the lax algebras for the ultrafilter monad. We will provethe result of Manes [10], but in a different way. We use the result of Barr [1],altough, historically seen, Barr based himself on the paper of Manes.

With the concept of lax algebras developed, one tries to describe uni-formly all structures that seem to matter in topology, inter alia, ordered sets,closure spaces and topological spaces. Ordered sets are simple to describeas lax algebras. Considering the case where T is the identity monad on Setand the extension of the identity monad on Set is the identity monad on Rel,we see that the lax algebra requirements can be denoted by a · a ≤ a and1X ≤ a, with ≤ to be read as set-theoretical inclusion if the relation a is pre-sented as a ⊆ X ×X. These requirements are precisely the transitivity andthe reflexivity of an order on X. The appropriate morphisms for lax algebrasare in this case precisely the order-preserving maps. In Chapter 4 and thefollowing chapters we give more examples of lax algebras.

When we look at topological spaces and their description in terms ofultrafilter convergence, one might ask if it is possible to trade the ultrafilterconvergence for filter convergence. To answer this question, it is useful toaxiomatize topologies on a set X in terms of neighborhood filters of points.Therefore we consider maps ν : X → FX into the set FX of filters on X, tobe thought of as assigning to each point its neighborhood filter. Ordering suchmaps pointwise by reverse inclusion and considering them as morphisms ofthe Kleisli category SetF of the filter monad F = (F,m, e), i.e using the Kleislicomposition ◦, one obtains another monoidal characterization of topologieson a set X by the axioms:

ν ◦ ν ≤ ν and eX ≤ ν.

In this way topological spaces are represented as Kleisli monoids (X, ν), oras F-monoids [13].

In the previous description the filter monad F may be replaced by anymonad T on Set when the sets TX carry a complete-lattice order, suitablycompatible with the monad operations. Such a monad T may be charac-terized via a monad morphism τ : P→ T, with P the powerset monad, and iscalled power-enriched. The equivalence between neighborhood systems andfilter convergence, given by

ν(x) ⊆ F ⇔ F → x

can be generalized at the level of a power-enriched monad T.

CONTENTS ix

With an appropiate lax extension of T to Rel, this correspondence yields apresentation of Kleisli monoids as lax algebras. By applying this presentationto the filter monad, we can see that the description of topological spacesremains valid if we trade ultrafilter convergence for filter convergence. Theresults regarding Kleisli monoids are explained in Chapter 5.

Since we can describe a topological space with two axioms in termsof ultrafilter convergence and with two axioms in terms of filter conver-gence, we have an isomorphism between the category of lax algebras of theultrafilter monad and the category of lax algebras of the filter monad. InChapter 6 we will try to generalize this interaction for an arbitrary monadmorphism α : S→ T. We discover which conditions are sufficient on themonad morphism in order to obtain an isomorphism between the categoryof lax algebras of S and T, respectively.

Closure spaces are spaces with a closure operation, that is a monotonemap c : PX → PX such that c is reflexive (A ⊆ c(A)) and idempotent(c(c(A)) = c(A)). When such a closure operation c is finitely additive andc(∅) = ∅, it defines a topological space. Closure spaces, on their own, arean interesting structure to investigate. In 1940 Birkhoff’s motivation [2] forconsidering closure spaces came from the observation that the collection ofclosed sets of a closure space forms a complete lattice. Closure spaces havenow applications in data analysis, knowledge representation, quantum logicand in the representation theory of physical systems.

Having at hand lax algebraic characterizations of topological spaces (witheither the ultrafilter or filter monad), we wonder whether a similar descriptionis possible for closure spaces. To find such a description, we will introducethe concept of a clique on X, as a collection C of subsets of X such thattwo elements of C have a non-empty intersection if ∅ /∈ C and that C is anup-set. We call a clique C proper if ∅ /∈ C and we call a clique an ultracliqueif it is proper and maximal with respect to the inclusion. By defining theseconcepts we can write closure spaces in the language of monoidal topology,as lax algebras and Kleisli monoids.

As a final result we investigate the Eilenberg-Moore algebras for the ultra-clique monad. We define compactness and Hausdorffness for closure spacesand prove that the category of Eilenberg-Moore algebras of the ultracliquemonad is isomorphic to the category of compact Hausdorff closure spaces,inspired by the result of Manes [10].

Inleiding

Monoidale Topologie beschrijft een actief onderzoek dat een kader en eentheorie biedt, die geordende, metrische en topologische stucturen op eenzelfdemanier beschrijft. Als taal voor dit kader gebruiken we categorie theorie. Aande hand van deze theorie zullen we, o.a., geordende2 ruimten en topologischeruimten door een convergentiestructuur beschrijven.

Monoıden zijn eenvoudige structuren, i.e. een verzameling met een asso-ciative vermenigvuldiging en een eenheid. Door deze voorwaarden in dia-grammen te bekijken, kunnen we dit begrip veralgemenen. In plaats van eenverzameling gebruiken we een endofunctor T : Set → Set en de vermenig-vuldiging en eenheid worden natuurlijke transformaties m : TT → T ene : 1Set → T. We krijgen een monad als de diagrammen voor de natuurlijketransformaties, gelijkaardig aan de diagrammen voor de vermenigvuldigingen de eenheid van een monoıde, commuteren. In Hoofdstuk 2 geven weverschillende voorbeelden van monads.

Met een monad T = (T,m, e) construeerden Eilenberg en Moore (1965)een categorie SetT met als objecten verzamelingen X uitgerust met eenfunctie a : TX → X die de twee diagrammen

TTXTa //

mX��

TX

a��

XeX //

1X !!

TX

a��

TX a// X X

commutatief maakt en met gepaste morfismes. Deze categorie noemen we deEilenberg-Moore categorie en zijn objecten zijn de Eilenberg-Moore algebra’s.

Het beginpunt voor monoidale topologie is het bewijs door Manes [10] in1969 dat compacte Hausdorff ruimte exact de Eilenberg-Moore algebra’s voorde ultrafilter monad � = (β,m, e) zijn. In deze beschrijving wijst de functiea : βX → X van een compacte Hausdorff ruimte X aan elke ultrafilter zijnuniek convergentiepunt toe, die voldoet aan de twee basis axioma’s van een

2Met een orde op een ruimte X bedoelen we een reflexieve en transitieve relatie. Eenorde moet niet voldoen aan het axioma van de anti-symmetrie.

x

CONTENTS xi

Eilenberg-Moore algebra a(βa(U )) = a(mX(U )) en a(eX(x)) = x voor alleU ∈ ββX en x ∈ X. We zullen deze axioma’s herschrijven door gebruik temaken van de principale ultrafilter op x

eX(x) = x = {A ⊆ X | x ∈ A}

en de Kowalsky som van U

mX(U ) = ΣU = {A ⊆ X | {U ∈ βX | A ∈ U} ∈ U }.

Deze ultrafilters, samen met de notatie V → z in plaats van a(V) = z enU → V in plaats van βa(U ) = V , geven een intuıtievere vorm van de basisaxioma’s, i.e.

U → V & V → z =⇒ ΣU → z

enx→ x

voor alle U ∈ ββX,V ∈ βX en x, z ∈ X.Voor metrische ruimten en de convergentie van rijen kunnen we het eerste

axioma op volgende manier beschrijven. Neem een rij (xn)n∈N die naar x ∈ Xconvergeert. Dan nemen we voor een collectie rijen, waarvan de eerste rij naarx1 convergeert, de tweede rij naar x2, etc. , de diagonaalrij, i.e. het eersteelement van de eerste rij, het tweede element van de tweede rij, . . . Danconvergeert de diagonaalrij naar x. De Kowalsky som is de veralgemeningvan de diagonaalrij.

In 1970 observeerde Barr [1] dat in een willekeurige topologische ruimtehet niet langer gegarandeerd is dat elke ultrafilter een convergentiepunt heeft(compactheid) en dat er hoogstens een zo’n punt is (Hausdorff) en we daaromeen relatie nodig zullen hebben in plaats van een functie. Zo’n relatie a datconvergentie beschrijft moet voldoen aan a(βa(U )) ≤ a(mX(U )) (transi-tiviteit) en x ≤ a(eX(x)) (reflexiviteit) voor alle U ∈ ββX en x ∈ X.Dus een topologie op X is beschreven door twee eenvoudige axioma’s opeen convergentierelatie tussen ultrafilters en punten. In de bovenstaandediscussie verdoezelden we een belangrijk punt: wanneer we de Set-functor βkennen, weten we wat βa is, als a een functie is, maar niet noodzakelijk alsa slechts een relatie is. Om βa te bepalen, als a een relatie is, bestaat er eenrechtstreekste oplossing.

Als we de ultrafilter monad � = (β,m, e) vervangen door een anderemonad T = (T,m, e), dan geven deze voorwaarden op een relatie a : TX →7 Xaanleiding tot het begrip van lax algebra’s. Samen met geschikte morfismes,vormen ze de categorie van lax algebra’s. Opnieuw moeten we nog Ta

CONTENTS xii

definieren als a een relatie is. Terwijl er voor de ultrafilter monad een recht-streekse oplossing bestaat, worden we in het algemeen geconfronteerd metde kwestie van een monad T = (T,m, e) op Set uit te breiden naar Rel. Vooronze doeleinden volstaat het dat deze extensie lax is en niet een volwaardigemonad op Rel. De definitie en voorbeelden van lax extensies van monads zijnte vinden in Hoofdstuk 3.

In Sectie 4.8 tonen we de connectie tussen Eilenberg-Moore algebra’s enlax algebra’s voor de ultrafilter monad aan. We zullen het resultaat vanManes [10] bewijzen op een andere manier dan Manes het zelf heeft bewezen.We gebruiken het resultaat van Barr [1], alhoewel, historisch gezien, Barrzich gebaseerd heeft op de paper van Manes.

Met het begrip “lax algebra” ontwikkeld, probeert men alle structuren dieer toe doen in de topologie, zoals, geordende verzamelingen, closure ruimtenen topologische ruimten, uniform te beschrijven. Geordende verzamelingenzijn eenvoudig te beschrijven als lax algebra’s. Beschouwen we het geval waarT de identiteit monad op Set is en de extensie van de identiteit monad op Setde identiteit monad op Rel is, dan kunnen de voorwaarden om een lax algebrate zijn beschreven worden door a · a ≤ a en 1X ≤ a, waarbij ≤ moet gelezenworden als de verzameling-theoretische inclusie als de relatie a voorgesteldwordt als a ⊆ X × X. Deze voorwaarden geven precies de transitiviteit ende reflexiviteit van een orde op X. De gepaste morfismes voor lax algebra’szijn in dit geval precies de orde-bewarende functies. In Hoofdstuk 4 en dedaarop volgende hoofdsukken geven we meer voorbeelden van lax algebra’s.

Wanneer we naar topologische ruimten en hun beschrijving in termen vanultrafilter convergentie kijken, kunnen we ons afvragen of het mogelijk is omde ultrafilter convergentie te veranderen door filter convergentie. Om dezevraag te beantwoorden, is het nuttig om topologieen op een verzamelingX te axiomatiseren in termen van omgevingenfilters van punten. Daarombeschouwen we functies ν : X → FX naar de verzameling FX van allefilters op X, die we kunnen zien als aan elk punt zijn omgevingenfilter toete kennen. Ordenen we zo’n functies puntgewijs door omgekeerde inclusie enbeschouwen we ze als morfismes van de Kleisli categorie SetF van de filtermonad F = (F,m, e), i.e. gebruik maken van de Kleisli samenstelling ◦, danverkrijgen we een andere monoidale karakterisatie van topologieen op eenverzameling X door de axioma’s:

ν ◦ ν ≤ ν and eX ≤ ν.

Op deze manier worden topologische ruimten voorgesteld als Kleisli monoıden(X, ν), of als F-monoıden [13].

In de voorgaande beschrijving mag de filter monad F vervangen wordendoor elke andere monad T op Set als de verzamelingen TX een volledige

CONTENTS xiii

tralie orde dragen dat compatibel is met de monad operaties. Zo’n monadT kan gekarakteriseerd worden via een monad morfisme τ : P → T, met Pde powerset monad en noemen we power-enriched. De equivalentie tussenomgevingensystemen en filter convergentie, gegeven door

ν(x) ⊆ F ⇔ F → x

kan veralgemeend worden voor een power-enriched monad T.Met een geschikte lax extensie van T naar Rel, houdt deze correspon-

dentie een presentatie van Kleisli monoıden als lax algebra’s in. Door dezepresentatie toe te passen op de filter monad, zien we dat de beschrijving vantopologische ruimten geldig blijft als we ultrafilter convergentie ruilen voorfilter convergentie. De resultaten met betrekking tot Kleisli monoıden zijnuitgelegd in Hoofdstuk 5.

Omdat we topologische ruimten kunnen beschrijven met twee axioma’s intermen van ultrafilter convergentie en met twee axioma’s in termen van filterconvergentie, hebben we een isomorfisme tussen de categorie van lax alge-bra’s van de ultrafilter monad en de categorie van lax algebra’s van de filtermonad. In Hoofdstuk 6 proberen we deze interactie te veralgemenen voor eenwillekeurig monad morfisme α : S→ T. We ontdekken welke voorwaarden opde monad morfisme voldoende zijn om een isomorfisme te verkrijgen tussende categorie van lax algebra van respectievelijk S en T.

Closure ruimten zijn ruimten met een closure operatie, i.e. een mono-tone functie c : PX → PX zodat c reflexief is (A ⊆ c(A)) en idempotent is(c(c(A)) ⊆ c(A)). Als zo’n closure operatie c eindig additief is en c(∅) = ∅,dan definieert c een topologische ruimte. Closure operaties, op zichzelf, zijneen interessante structuur om te onderzoeken. In 1940 kwam Birkhoff’smotivatie [2] om closure ruimten te beschouwen van de observatie dat deverzameling van gesloten verzamelingen van een closure ruimte een volledigetralie vormt. Closure ruimten hebben toepassingen in data analyse, knowl-edge representation, quantum logica en in de representatietheorie van fysischesystemen.

Omdat we de lax algebraısche karakterisaties van topologische ruimten(met ofwel de ultrafilter ofwel de filter monad) kunnen beschrijven, vragenwe ons af of er een gelijkaardige beschrijving mogelijk is voor closure ruimten.Om zo’n beschrijving te vinden, zullen we het begrip “clique” opX definieren,als een collectie C van deelverzamelingen van X zodat twee elementen van Ceen niet-lege doorsnede hebben als ∅ /∈ C en dat C een up-set is. We noemeneen clique C “proper” als ∅ /∈ C en we noemen een clique een ultraclique alsze proper is en maximaal ten opzichte van de inclusie. Door deze begrippente definieren, kunnen we closure ruimten schrijven in de taal van monoidaletopologie, namelijk als lax algebra’s en als Kleisli monoıden.

CONTENTS xiv

Als laatste onderzoeken we de Eilenberg-Moore algebra’s voor de ultra-clique monad. We definieren compactheid en Hausdorff voor closure ruimtenen bewijzen dat de categorie van Eilenberg-Moore algebra’s voor de ultra-clique monad isomorf is met de categorie van compact Hausdorff closureruimten, geınspireerd door het resultaat van Manes [10].

Chapter 1

Basic concepts

In the first chapter of this thesis we give a short introduction to conceptsmostly known to a master student. It provides the notations, terminologyand theory used in the rest of the thesis. This chapter is based on the book“Monoidal Topology” [4], specifically on chapter II, written by Gavin J. Sealand Walter Tholen. The results on cliques and ultracliques are elaboratedby the author.

1.1 The compositional structure of relations

A relation r from a set X to a set Y distinguishes those elements x ∈ Xand y ∈ Y that are r-related; we write x r y if x is r-related to y. Hence,depending on whether we display r as a subset, a two-valued function or amulti-valued function via

r ⊆ X × Y, r : X × Y → {false, true} = {0, 1} = 2, r : X → PY

respectively, where PY denotes the powerset of Y , x r y may be equivalentlywritten as

(x, y) ∈ r, r(x, y) = true = 1, y ∈ r(x).

Writing r : X →7 Y when r is a relation from X to Y , we can “multiply” rwith s : Y →7 Z via ordinary relational composition:

x (s · r) z ⇐⇒ ∃y ∈ Y : (x r y and y s z).

Writing r ≤ r′ (with r′ : X →7 Y ) when, equivalently,

r ⊆ r′, ∀x ∈ X ∀y ∈ Y : (r(x, y) |= r′(x, y)), ∀x ∈ X : (r(x) ⊆ r′(x)),

1

CHAPTER 1. BASIC CONCEPTS 2

we see that the multiplication respects ≤, since

r ≤ r′, s ≤ s′ =⇒ s · r ≤ s′ · r′.

Moreover, relational composition is associative, so that

(t · s) · r = t · (s · r),

when t : Z →7 W, and for the identity relation 1X (with x 1X x′ =⇒ x = x′)one has r · 1X = r = 1Y · r. For r : X →7 Y one has the opposite (or dual)relation r◦ : Y →7 X with

y r◦ x ⇐⇒ x r y,

for all x ∈ X and y ∈ Y, which satisfies

(s · r)◦ = r◦ · s◦, (1X)◦ = 1X , (r◦)◦ = r, r ≤ r′ =⇒ r◦ ≤ (r′)◦.

Note that when r is the graph of a map f : X → Y (so that x r y ⇐⇒f(x) = y), then r◦(y) = f−1(y) is simply the fibre of f over y ∈ Y.

For a map f : X → Y we have that

f · f ◦ ≤ 1Y and 1X ≤ f ◦ · f.

The category Rel, with objects sets and morphisms relations, can be seenas an extension of Set, the category with objects sets and morphisms maps,via the functor

(−)◦ : Set→ Rel : X 7→ X, f 7→ f◦,

for every set X and map f : X → Y , with

f◦ : X →7 Y ; f◦(x, y) =

{1 if f(x) = y0 otherwise

Proof. To prove that (−)◦ is a functor, we check the two conditions for afunctor. First we remark that (1X)◦(x, y) = 1 if and only if 1X(x) = y, thusx = y. So we know that

(1X)◦ = 1X .

Next we look at the two relations (g · f)◦ and g◦ · f◦ for f : X → Y andg : Y → Z,

(g · f)◦(x, z) =

{1 if g(f(x)) = z0 otherwise

and

(g◦ · f◦)(x, z) =

{1 if ∃y ∈ Y : f(x) = y and g(y) = z0 otherwise

and thus(g · f)◦ = g◦ · f◦.

�


1.2 Orders

An order on a set X is a relation a : X →7 X that carries a monoid structurewith respect to the compositional structure of relations; that is, a satisfies

a · a ≤ a, 1X ≤ a.

Hence, a is simply a transitive and reflexive relation on X:

(x ≤ y and y ≤ z =⇒ x ≤ z), x ≤ x

for all x, y, z ∈ X, when we write x ≤ y for x a y. The order is separatedif a ∩ a◦ = 1X , so that

x ≤ y and y ≤ x =⇒ x = y.

A map f : X → Y of ordered sets is monotone (or order-preserving)if

f · a ≤ b · f,where a, b denote the orders on X, Y respectively and f is identified with itsgraph. Hence, if we write ≤ for both a and b,

x ≤ y =⇒ f(x) ≤ f(y)

for all x, y ∈ X.The category with objects ordered sets and monotone maps as morphisms

is called Ord.

1.3 Modules

A relation r : X →7 Y between ordered sets is a module if (≤Y ) ·r ·(≤X) ≤ r,that is, if

x′ ≤ x and x r y and y ≤ y′ =⇒ x′ r y′,

for all x, x′ ∈ X, y, y′ ∈ Y. Hence, the relation r is a module if and onlyif the map r : Xop × Y → {true, false} is monotone (where Xop × Y isordered componentwise). Graphically, we indicate modularity of a relationr : X →7 Y by

r : X →◦ Y.

Every monotone map f : X → Y gives rise to the modules

f∗ =≤Y ·f : X →◦ Y and f ∗ = f ◦· ≤Y : Y →◦ X

that is, x f∗ y ⇐⇒ f(x) ≤ y and y f ∗ x ⇐⇒ y ≤ f(x) for all x ∈ X, y ∈ Y.The following rules may be easily verified when g : Y → Z is monotone.


1. 1∗X = (1X)∗ = (≤X),

2. (g · f)∗ = g∗ · f∗ and (g · f)∗ = g∗ · f ∗,

3. 1∗X ≤ f ∗ · f∗ and f∗ · f ∗ ≤ 1∗Y .

Proof.

1. By definition we have 1∗X =≤X ·1X . Thus, for x, y ∈ X, we get

x (≤X ·1X) y ⇐⇒ ∃z ∈ X : x 1X z and z ≤ y

⇐⇒ x ≤ y.

The other statement is proved analogously.

2. For x ∈ X and z ∈ Z we know that x (g ·f)∗ z means that g(f(x)) ≤ z.On the other hand, we have

x (g∗ · f∗) z ⇐⇒ x (≤Z ·g· ≤Y ·f) z

⇐⇒ ∃w ∈ Z, ∃v, y ∈ Y : x f v & v ≤ y & y g w & w ≤ z

⇐⇒ ∃w ∈ Z, ∃v, y ∈ Y : f(x) = v & v ≤ y & g(y) = w & w ≤ z

⇐⇒ f(x) ≤ y and g(y) ≤ z

⇐⇒ g(f(x)) ≤ z,

because g is monotone. The other statement is proved analogously.

3. If for x ∈ X and y ∈ Y we have that x (f ∗ · f∗) y, then we know thatthere exists v, w, z ∈ Y such that x f v, v ≤ w, w ≤ z, z f ◦ y, in otherwords f(x) ≤ f(y). Thus the statement holds if f is monotone, whichis the fact by definition. The other statement is proved analogously.

�

Modularity is closed under relational composition. Indeed, for modulesr : X →◦ Y and s : Y →◦ Z, one has

(≤Z) · (s · r) · (≤X) ≤ (≤Z) · s · (≤Y ) · (≤Y ) · r · (≤X) ≤ s · r,

so that s · r : X →◦ Z is again a module.Mod is the category with ordered sets as objects and modules as maps.


1.4 Adjunctions for ordered sets

For ordered sets X, Y, the set Ord(X, Y ) = {f | f : X → Y monotone} isitself ordered pointwise by

f ≤ f ′ ⇐⇒ ∀x ∈ X : f(x) ≤ f ′(x).

This order is preserved by composition on either side; whenever h : W → Xand k : Y → Z are monotone then

f ≤ f ′ =⇒ k · f · h ≤ k · f ′ · h.

A monotone map g : Y → X is called

1. right adjoint if there is a monotone map f : X → Y with 1X ≤g · f, f · g ≤ 1Y ;

2. an isomorphism if there is a monotone map f : X → Y with 1X =g · f, f · g = 1Y .

If g is right adjoint, the corresponding f is called left adjoint to g and onewrites

f a g.

The terminology becomes more plausible when we consider the followingfact:

Proposition 1.4.1. A map g : Y → X is right adjoint if and only if thereis a map f : X → Y such that f(x) ≤ y ⇐⇒ x ≤ g(y) for all x ∈ X andy ∈ Y (where f is not assumed to be monotone a priori).

Proof. The necessity of the condition is obvious since x ≤ g(y) impliesf(x) ≤ f · g(y) ≤ y, and dually for “ =⇒ ”. For its sufficiency, observe thatf(x) ≤ f(x) implies x ≤ g · f(x) and dually f · g(y) ≤ y. The monotonicityof f (and also of g) follows, since x ≤ x′ ≤ g · f(x′) yields f(x) ≤ f(x′). �

Corollary 1.4.2. A right adjoint map g (with left adjoint f) is fully faithful,i.e.

g(x) ≤ g(y) =⇒ x ≤ y,

if and only if f · g ≤ 1Y and 1Y ≤ f · g.

Proof. We have f · g ≤ 1Y because g and f are adjoint. Since g and f areadjoint, it also follows that 1X ≤ g · f , and we get

g(y) ≤ g · f · g(y).


Since g is fully faithful, we know y ≤ f · g(y) and thus

1Y ≤ f · g.

Conversely assume that g(x) ≤ g(y), then we have

x ≤ f · g(x) ≤ f · g(y) ≤ y.

�

1.5 Closure operation and interior operation

A closure operation is a monotone map c : X → X such that

c · c ≤ c, 1X ≤ c.

We call the first condition idempotency and the second one reflexivity.A closure space is a set X which comes with a closure operation on thepowerset PX, ordered by inclusion. A map f : X → Y is continuous if

f(cX(A)) ⊆ cY (f(A))

for all A ⊆ X. A interior operation is a monotone map d : X → X suchthat

d ≤ d · d, d ≤ 1X .

A interior space is a set X which comes with a interior operation on thepowerset PX, ordered by inclusion. A map f : X → Y is continuous if

f−1(dY (B)) ⊆ dX(f−1(B))

for all B ⊆ Y.Via the order isomorphism (−){ : (PX)op → PX (which maps A ∈ PX

to its complement A{ := X \ A in X), any closure operation c on PXcorresponds to an interior operation d on PX, and vice-versa:

c(A){ = d(A{), for all A ⊆ X.

If we call Cls the category of all closure spaces and continuous functionsand Int the category of interior spaces and continuous functions, we have

Cls ∼= Int.


1.6 Completeness and lattices

Let X be an ordered set. For an element x ∈ X we define the up-set of xin X as follows

↑X x = {y ∈ X | y ≥ x}.The up-closure of A ⊆ X is

↑X A =⋃x∈A

↑X x.

We say that A is up-closed (or an up-set) if

↑X A = A.

With UpX we denote all the up-sets of X. We can define the dual notionsanalogously. The down-set of x is

↓X x = {y ∈ X | y ≤ x}.

The rest of the notions are down-closure, down-closed, down-set, DnX.We say that an ordered set X is complete if and only if the map

↓: X → DnX

is right adjoint; equivalently, if there is a map∨X =

∨: DnX → X which

for every A ∈ DnX satisfies

∀x ∈ X :∨

A ≤ x ⇐⇒ A ⊆↓ x.

We may call∨A the supremum (or least upper bound) of A.

The dual notion of the adjunctions between the down-set and the suprema,can also be defined. But note that, for ↑X to be monotone, UpX is orderedby reverse inclusion and we get the notion of an infimum (or greatest lowerbound):

∀x ∈ X : x ≤∧

A ⇐⇒ A ⊆↑ x.

A monotone map f : X → Y of ordered sets preserves the supremum∨A of A ⊆ X if f(

∨A) is a supremum of {f(x) | x ∈ A} in Y . Moreover,

f is a sup-map if it preserves every existing supremum in X. The dualnotions are: preserves an infimum, inf-map.

A lattice is a separated ordered set X with finite infima and finitesuprema (for evere finite subset A of X, there exists a infimum and supre-mum). So there exists a top (>) and a bottom (⊥) element. A homomor-phism of lattices is a map that preserves the infima, the suprema and thetop and bottom elements.


We call Sup the category with objects the complete lattices and withmorphisms the sup-maps. We call Inf the category with objects the completelattices and with morphisms the inf-maps.

A semilattice is a commutative monoid (see Definition 2.1.1) where everyelement is idempotent, so that x · x = x for all x ∈ X. One denotes suchmonoids by (X, ·, e). A homomorphism of semilattices is a map thatpreserves the operations ·, e. The category of semilattices and semilatticehomomorphisms is denoted by

SLat.

1.7 Filters, ultrafilters, cliques and ultracliques

Definition 1.7.1. We call F a filter on the set X, if it is a collection ofsubsets of X satisfying

(1) A,B ∈ F ⇒ A ∩B ∈ F ,

(2) X ∈ F ,

(3) A ∈ F , A ⊆ B ⇒ B ∈ F .

We call F a proper filter if F is a filter and satisfies

(4) ∅ /∈ F .

For every map f : X → Y and every filter F on X, one defines the imagefilter f [F ] on Y by

f [F ] :=↑PX {f(A) | A ∈ F} = {B ⊆ Y | f−1(B) ∈ F}.

Thus f [F ] is the filter generated by the filterbase {f(A) | A ∈ F}.To show that this definition is properly, we prove the last equality of the

definition. That is why we take a subset B ⊆ Y such that there exist a subsetA ∈ F and f(A) ⊆ B. Then we see immediately f−1(B) ∈ F , because a filteris an up-set. Conversely, for a subset B ⊆ Y such that f−1(B) ∈ F , we havef · f−1(B) ⊆ B. Thus B ∈ f [F ].

For every A ⊆ X, one has the following principal filter on X:

A =↑PX A = {B ⊆ X | A ⊆ B}.

Definition 1.7.2. An ultrafilter U on a set X is a maximal element withinthe set of proper filters on X, ordered by inclusion. That is, U is a properfilter on X such that if F is a proper filter on X with U ⊆ F , then U = F .


A more convenient characterization for ultrafilters is the following one:

Lemma 1.7.3. For a proper filter U on X, the following statements areequivalent:

(i) U is an ultrafilter on X;

(ii) for all A,B ⊆ X, if A ∪B ∈ U then A ∈ U or B ∈ U ;

(iii) for every subset A ⊆ X, one has A ∈ U or A{ ∈ U (where A{ = X \ Adenotes the complement of A in X).

Proof. This is proven in the course “Topology” [5]. �

For a map f : X → Y and an ultrafilter U on X, the image f [U ] is alsoan ultrafilter on Y . For every x ∈ X, the principal filter x =↑PX {x} is anultrafilter on X.

We now give some important propositions on ultrafilters, which we willuse later on.

Proposition 1.7.4. Every proper filter F on X is contained in an ultrafilterU on X.

Proof. Suppose F is a proper filter and define

X := {G | G a proper filter on X,F ⊆ G}.

Then (X ,⊆) is an partially ordered set with the property that every totallyordered subset has an upperbound in X . Thus Zorn’s Lemma gives us theexistence of the ultrafilter. �

In fact, this statement can be used to formulate a formally finer assertion:

Corollary 1.7.5. For a filter G and a proper filter F on X such that F ( G,there is an ultrafilter U on X with F ⊆ U but G * U .

Proof. Indeed, for some B ∈ G with B /∈ F , one considers the filter

F ′ =↑ {B{ ∩ A | A ∈ F},

which is proper since B{∩A = ∅ would imply A ⊆ B ∈ F . So there exists anultrafilter U containing F ′, and therefore also F ; as B{ ∈ U , we must haveB /∈ U . �

As an important consequence, we obtain:


Corollary 1.7.6. Every filter F on X is the intersection of all ultrafilterson X containing F .

Proof. We must show that every filter F ∈ FX may be obtained as

F =⋂{U ∈ βX | F ⊆ U}.

The inclusion “⊆” is clear. Moreover, when F is proper, so is the filter Gobtained on the right-hand side; therefore, if F ( G, there exists an ultrafilterU with F ⊆ U but G * U , contradicting the definition of G.

If ∅ ∈ F , then there do not exist ultrafilters finer than F , so

{U ∈ βX | F ⊆ U} = ∅.

In the complete lattice of all filters the infinum of the empty set is the biggestelement. For filters, it is the improper filter. Thus the corollary holds. �

We can also define the dual notion of a filter. We call it an ideal.

Definition 1.7.7. A collection of subsets J of X is an ideal if

(1) A,B ∈ J ⇒ A ∪B ∈ J ,

(2) ∅ ∈ J ,

(3) A ∈ J , B ⊆ A⇒ B ∈ J .

We call the ideal proper if

(4) X /∈ J .

Corollary 1.7.8. For a proper filter F and an ideal J on X such thatF ∩ J = ∅, there is an ultrafilter U with F ⊆ U and U ∩ J = ∅.

Proof. Since F is an up-set, the fact that F is disjoint from J translatesas A * J for all A ∈ F and J ∈ J , or equivalently as A ∩ J{ 6= ∅ for allA ∈ F , J ∈ J . Thus,

G := {A ∩ J{ | A ∈ F , J ∈ J }

is a proper filter containing F , and Proposition 1.7.4 yields the existence ofan ultrafilter U with G ⊆ U , and consequently F ⊆ U . If there was J ∈ J ∩U ,one would conclude J{ /∈ U , a contradiction. �

We now define new structures, called cliques and ultracliques, which haveas a structure a lot in common with filters and ultrafilters. The notions of


cliques and ultracliques are introduced and used independently by a rangeof authors and under different names. One of the main sources we used is“Neighborhood spaces” by D.C. Kent and Won Keun Min [8]. They call aclique a p-stack and an ultraclique an ultra-p-stack. Thampuran [15] in turncalls it extended filters with the pair-wise intersection property. De Groot,Jensen and Verbeek [6] named cliques and ultracliques respectively linkedsystems and maximal linked systems, because the elements of cliques andultracliques are pair-wise linked. We use the name Gavin J. Seal [4] haschosen.

Definition 1.7.9. A clique C on a set X is a subset of PX such that forall A,B ∈ PX :

(1) if ∅ /∈ C and A,B ∈ C, then A ∩B 6= ∅,

(2) A ∈ C, A ⊆ B ⇒ B ∈ C.

A clique is proper if ∅ /∈ C. The set of all cliques on X is denoted by CX.An ultraclique D on X is a maximal element in the set of proper cliques

on X, ordered by inclusion. That is, D is a proper clique on X and for allproper cliques C on X with D ⊆ C, we have D = C.

For ultracliques we have the following characterization, which can befound in “Neighborhood spaces” [8].

Lemma 1.7.10. For a proper clique D ∈ CX the following statements areequivalent:

(1) D is an ultraclique;

(2) if A ∩D 6= ∅, for all D ∈ D, then A ∈ D;

(3) B /∈ D ⇒ X \B ∈ D.

Proof. (1)⇒ (2). If A ∩D 6= ∅, for all D ∈ D and A /∈ D, then

D ∪ {B ⊆ X | A ⊆ B}

would be a clique strictly larger then D.(2) ⇒ (3). If B /∈ D, then by (2), there is D ∈ D such that B ∩D = ∅.

Thus D ⊆ X \B, which implies X \B ∈ D.(3) ⇒ (1). Suppose D ⊆ C, with C ∈ CX and C ∈ C. If C /∈ D, then by

(3) X \ C ∈ D and hence X \ C ∈ C, a contradiction. �


Similar to filters we define for a clique C ∈ CX and a map f : X → Ythe image clique

f [C] :=↑PX {f(C) | C ∈ C} = {B ⊆ Y | f−1(B) ∈ C},

where the last equality is proven entirely analogously as for the filter case.We now prove that f [C] is a clique, or an ultraclique, if C is one.

Lemma 1.7.11. For a map f : X → Y and C ∈ CX, f [C] is a clique. If Cis an ultraclique, so is f [C].

Proof. First, we remark that f(∅) = ∅ and therefore the image clique of aproper clique will be proper. Take A,B ∈ f [C] for a proper clique C, thenthere exists C,D ∈ C such that f(C) ⊆ A and f(D) ⊆ B. By the definitionof a clique is C ∩D 6= ∅, and so is f(C ∩D) 6= ∅. Therefore A ∩B 6= ∅.

Suppose that C is an ultraclique. If B ∩ f(C) 6= ∅ for all C ∈ C, thenf−1(B) ∩ C 6= ∅ for all C ∈ C and by Lemma 1.7.10, f−1(B) ∈ C, whichimplies B ∈ f [C] and thus f [D] is an ultraclique. �

The existence of “sufficiently many” ultraclique requires the axiom ofchoice (see 1.7.12), but these structures appear to be less diffcult to achievethen ultrafilters. For example, if the set X has three distinct elementsx, y, z ∈ X, then a non-principal ultraclique is given by

↑PX {{x, y}, {x, z}, {y, z}}.

Assuming the lemma of Zorn (or the axiom of choice), we can prove thereexist many ultracliques.

Lemma 1.7.12. Every proper clique is contained in an ultraclique.

Proof. Suppose C is a proper clique and define

X := {D | D a proper clique on X, C ⊆ D}.

Then (X ,⊆) is a partially ordered set with the property that every totallyordered subset has an upperbound in X. Zorn’s lemma gives us the existenceof the ultraclique. �

Every clique C can be written as the intersection of all ultracliques con-taining C, as with filters as the intersection of all ultrafilters finer than it.

Lemma 1.7.13. Every clique C on X can be written as

C =⋂{D ∈ CX | C ⊆ D,D an ultraclique}.


Proof. If C is proper, then C is a subset of the intersection of all ultracliquesthat contain C. Conversely, we suppose that A /∈ C, then we have for allC ∈ C that C * A. This means that for all C ∈ C there exists a point inC∩(X \A). Because of Lemma 1.7.12 there exists an ultraclique D such thatC ⊆ D and X \ A ∈ D. Thus A /∈ D.

If C is improper, then there are no ultracliques finer than C. In the com-plete lattice of all cliques the infinum of the empty set is equal to the biggestelement, the improper clique. �


Notes on Chapter 1

In the definition of a closure operation 1.5 one of the defining properties isthe idempotency of the map c. In most books the definition of idempotencyis c·c = c, but, like in our case, when we have reflexivity, both interpretationsof idempotency are equal, since

c · c ≤ c ≤ c · 1X ≤ c · c.

In many books and courses the notion of a filter is somewhat different fromthe one we use here. We have chosen to make a difference between properand improper filters. We need the improper filters, to construct the set ofall filters on the empty set. This is a pathological case we need to take intoaccount in the theory about Kleisli monoids (see 5).

Chapter 2

Monads

This chapter defines one of our most important structures troughout thisthesis, namely the concept of a monad. To define this structure, we base our-selfs on a well-known algebra structure, the monoid. To emphasize the simi-larities between monoids and monads more clearly, we first give the definitionof a monoid. For the monads we use a broader language, namely categorytheory, but these structures are constructed in the same way. Every monadis actually a monoid in a specific category. This chapter is based on ChapterII [4]. The proofs in this chapter are elaborated by the author.

2.1 Monads

As promised we now repeat the definition of a monoid and its morphisms,followed by the definition of a monad and its morphisms.

Definition 2.1.1. A monoid M is a set M that comes with a binary anda nullary operation

m : M ×M →M, e : {?} →M

that are associative and make e = e(?) a neutral element of M ; equivalently,the diagrams

M × (M ×M)∼= //

1M×m��

(M ×M)×M m×1M//M ×Mm

��M ×M m

//M

15

CHAPTER 2. MONADS 16

and

{?} ×M e×1M //

pr2∼= &&

M ×Mm

��

M × {?}1M×eoo

pr1∼=xx

M

commute.A homomorphism f : M → N of monoids preserves both operations:

f ·mM = mN · (f × f) and f · eM = eN .

Definition 2.1.2. A monad T = (T,m, e) on the category Set is given by afunctor T : Set→ Set and two natural transformations, the multiplicationand unit of the monad

m : TT → T, e : 1Set → T,

satisfying the multiplication law and the left and right unit laws:

m ·mT = m · Tm, m · eT = 1T = m · Te;

equivalently, these equalities mean that the diagrams

TTTTm //

mT��

TT

m

��TT m

// T

andT

eT //

1T !!

TT

m

��

TTeoo

1T}}T

commute.A morphism of monads α : S → T (where S = (S, n, d)) is a natural

transformation α : S → T that preserves the monad structure:

α · n = m · (α ∗ α), α · d = e

(with ∗ the horizontal composition or the Godement product).

When we look at the category of all endofunctors on Set. Then T is amonoid in this category [9].

We now give some examples of monads, using structures we know or havedefined in Chapter 1.


Examples 2.1.3. 1. The identity functor

1Set : Set→ Set : X 7→ X, f 7→ f

(with f : X → Y ) together with the map

1X : X → X : x 7→ x

forms a monad in a trivial way, namely I = (1Set, 1, 1) with 1 =(1X)X∈|Set|. We call it the identity monad.

2. The powerset functor

P : Set→ Set : X 7→ PX, f 7→ Pf,

with PX the powerset of X and Pf : PX → PY : A 7→ {f(a) | a ∈ A}for every f : X → Y, together with the union map

mX : PPX → PX : A 7→⋃A

and singleton map

eX : X → PX : x 7→ {x}

forms the powerset monad P = (P,m, e).

Proof. First we notice that the functions mX and eX are well-defined,because the union of a collection of subsets of X is a subset of X, alsothe singleton of an element of a set is a set.

Now we will prove that e and m are natural transformations. We takea function f : X → Y and we need to prove that

XeX //

f��

PX

Pf��

Y eY// PY

andPPX

mX //

PPf��

PX

Pf��

PPY mY// PY

are commutative.


So take an element x ∈ X and we can see that

(Pf · eX)(x) = Pf({x})= {f(x)}= eY (f(x))

= (eY · f)(x).

For the second diagram we take a collection A of subsets of X (A ∈PPX) and we compute

(Pf ·mX)(A) = Pf(⋃A)

=⋃{Pf(A) | A ∈ A}

= (mY · PPf)(A).

Now we can verify that this functor and these maps give a monad.First we prove the multiplication law. Therefore we take a set X andlook at the following diagram

PPPX

mPX��

PmX // PPX

mX��

PPX mX// PX

.

We have for A ∈ PPPX

(mX · PmX)(A ) = mX({mX(A) | A ∈ A })

= mX({⋃A | A ∈ A })

=⋃{⋃A | A ∈ A }

= {a ∈ X | ∃A ∈ A,∃A ∈ A : a ∈ A}

=⋃⋃

A

= mX(⋃

A )

= (mX ·mPX)(A ).

Finally we prove the left and right unit law,

PXePX //

1PX $$

PPX

mX��

PXPeXoo

1PXzzPX


Look at a subset A of X, then we get

(mX · ePX)(A) = mX({A}) =⋃{A} = A = 1PX(A),

and

(mX ·PeX)(A) = mX({{a} | a ∈ A}) =⋃{{a} | a ∈ A} = A = 1PX(A).

Thus P is a monad. �

3. Take for T the filter functor

F : Set→ Set : X 7→ FX, f 7→ Ff,

with FX the set of all filters on X and the image of f : X → Y thefunction Ff : FX → FY : F 7→ {B ⊆ Y | f−1(B) ∈ F}. Take for ethe principal filter map

eX : X → FX : x 7→ x = {A ⊆ X | x ∈ A}

and for m the map

mX : FFX → FX : F 7→ ΣF = {A ⊆ X | AF ∈ F}

with AF = {F ∈ FX | A ∈ F}. We call the map m the Kowalskysum. Then F = (F,m, e) is a monad. We call F the filter monad .

Proof. First we prove that F is a functor. In order to be well-defined(Ff)(F) should be a filter. This is the case, because

(Ff)(F) =↑PX {f(F ) | F ∈ F}.

Take B ⊆ Y such that f−1(B) ∈ F , then f(f−1(B)) ⊆ B and soB ∈↑PX {f(F ) | F ∈ F}. For the other inclusion we look at f(F ) withF ∈ F . Then F ⊆ f−1(f(F )) and thus f(F ) ∈ (Ff)(F).

The two conditions of a functor are easy to check. For a set X, we have

F (1X)(F) = {B ⊆ X | 1−1X (B) ∈ F} = {B ⊆ X | B ∈ F} = F = 1FX(F).

And for two functions f : X → Y and g : Y → Z and a filter F on X,we have

(Fg · Ff)(F) = Fg({A ⊆ Y | f−1(A) ∈ F})= {B ⊆ Z | g−1(B) ∈ {A ⊆ Y | f−1(A) ∈ F}}= {B ⊆ Z | f−1(g−1(B)) ∈ F}= F (g · f)(F).


It is obvious that the function eX is well-defined. In order to see thatthe function mX is well-defined, we need to prove that mX(F ) = ΣF ,with F ∈ F 2X, is a filter.

First we remark that every F ∈ F 2X contains FX. If we prove thatXF = FX, then we get that XF ∈ F and thus X ∈ mX(F ). Conse-quently ΣF is not empty. The statement XF = FX is very easy tosee, because XF is always a subset of FX by definition and every filteron X contains X and is therefore an element of XF.

Let A and B be elements of mX(F ), then we need to prove thatA ∩B ∈ mX(F ). We know that A and B ∈ mX(F ), so AF and BF areelements of the filter F . Therefore is AF ∩ BF ∈ F . If we prove that(A ∩ B)F = AF ∩ BF, then we know that A ∩ B ∈ mX(F ). Thus takea filter F ∈ (A ∩ B)F, then A ∩ B ∈ F . Because F is a filter, we haveA ∈ F and B ∈ F . Thus F ∈ AF and F ∈ BF and so F ∈ AF ∩ BF.Conversely, take a filter F ∈ AF ∩BF, then we know that F ∈ AF andF ∈ BF. By definition we have A ∈ F and B ∈ F . Because F is afilter, A ∩B is in F . Thus F ∈ (A ∩B)F.

Now let A and B be subsets of X such that A ⊆ B and A ∈ mX(F ).Then we have that AF ∈ F . Let F be a filter in AF, thus A ∈ F . Weknow that B ∈ F , because F is a filter. Therefore F ∈ BF. We seethat AF ⊆ BF if A ⊆ B. Due to the fact that F is a filter,we knowthat BF ∈ F . In other words B ∈ mX(F ).

This proves that mX(F ) is a filter and therefore mX is well-defined.

After checking that eX and mX are well-defined, we prove that e andm are natural transformations. So take a function f : X → Y and lookat the following diagrams.

X

f��

eX // FX

Ff

��

FFX

FFf

��

mX // FX

Ff

��Y eY

// FY FFYmY // FY

For x ∈ X we have

(Ff)(eX(x)) = (Ff)(x)

= {B ⊆ Y | f−1(B) ∈ x}= {B ⊆ Y | x ∈ f−1(B)}= {B ⊆ Y | f(x) ∈ B}= ˙f(x)

= eY (f(x)).


For the commutativity of the second diagram we need the followingequality:

(Ff)−1(BF) = {F ∈ FX | Ff(F) ∈ BF}= {F ∈ FX | B ∈ Ff(F)}= {F ∈ FX | f−1(B) ∈ F}= f−1(B)F.

For F ∈ FFX we get

(Ff)(mX(F )) = Ff({A ⊆ X | AF ∈ F})= {B ⊆ Y | f−1(B) ∈ {A ⊆ X | AF ∈ F}}= {B ⊆ Y | f−1(B)F ∈ F}= {B ⊆ Y | (Ff)−1(BF) ∈ F}= {B ⊆ Y | BF ∈ {C ⊆ FY | (Ff)−1(C ) ∈ F}}= mY ({C ⊆ FY | (Ff)−1(C ) ∈ F})= mY (FFf(F )).

To finalise the proof, we show the multiplication law and the left andright unit laws. For this we need the following equalities:

(BF)F = m−1X (BF), e−1X (BF) = B.

F ∈ (BF)F ⇐⇒ BF ∈ F

⇐⇒ B ∈ mX(F )

⇐⇒ mX(F ) ∈ BF

⇐⇒ F ∈ m−1X (BF).

x ∈ (eX)−1(BF) ⇐⇒ eX(x) ∈ BF

⇐⇒ B ∈ eX(x) = x

⇐⇒ x ∈ B.

For the multiplication law, we look at the following diagram, for a setX,

FFFX

mFX��

FmX // FFX

mX��

FFX mX// FX.


Take Φ ∈ FFFX, then we get

(mX ·mFX)(Φ) = mX({A ⊆ FX | A F ∈ Φ})= {B ⊆ X | BF ∈ {A ⊆ FX | A F ∈ Φ}}= {B ⊆ X | (BF)F ∈ Φ}= {B ⊆ X | m−1X (BF) ∈ Φ}= {B ⊆ X | BF ∈ {A ⊆ FX | m−1X (A ) ∈ Φ}}= mX({A ⊆ FX | m−1X (A ) ∈ Φ})= (mX · FmX)(Φ).

For the left and right unit laws, we look at the following diagram, fora set X,

FXeFX //

1FX $$

FFX

mX��

FXFeXoo

1FXzzFX

For every F ∈ FX we have

(mX · eFX)(F) = mX(F)

= {B ⊆ X | BF ∈ F}= {B ⊆ X | F ∈ BF}= {B ⊆ X | B ∈ F}= F

and

(mX · FeX)(F) = mX({F ⊆ FX | e−1X (F ) ∈ F})= {B ⊆ X | BF ∈ {F ⊆ FX | e−1X (F ) ∈ F}}= {B ⊆ X | e−1X (BF) ∈ F}= {B ⊆ X | B ∈ F}= F

This proves that F = (F,m, e) is a monad. �

4. The ultrafilter functor

β : Set→ Set : X 7→ βX, f 7→ βf,


with βX = {U ⊆ PX | U ultrafilter }, the set of all ultrafilters on Xand βf(U) = {B ⊆ Y | f−1(B) ∈ U}, together with the principal filtermap

eX : X → βX : x 7→ x = {A ⊆ X | x ∈ A}

and the map

mX : ββX → βX : U 7→ {A ⊆ X | A� ∈ U },

where A� = {U ∈ βX | A ∈ U}, forms a monad � = (β,m, e), calledthe ultrafilter monad . We call the map mX the Kowalsky sum

Proof. The proof for the ultrafilter monad is almost the same as theproof for the filter monad. There are two things in addition we needto check. First we need to prove that mX(U ) is proper for everyU ∈ ββX.Suppose that ∅ ∈ mX(U ), then we have that ∅� ∈ U , and

∅� = {U ∈ βX | ∅ ∈ U} = ∅.

Thus ∅ ∈ U , but this is impossible when U is an ultrafilter. Thereforewe have ∅ 6∈ mX(U ).

The second thing we need to prove is the ultrafilter condition ofmX(U ).So suppose we have A /∈ mX(U ) for A ⊆ X, then we have A� /∈ U .Because U is an ultrafilter, we know that βX \ A� ∈ U . The proof iscomplete, if βX \ A� = (X \ A)�. Indeed,

βX \ A� = {U ∈ βX | U /∈ A�}= {U ∈ βX | A /∈ U}= {U ∈ βX | X \ A ∈ U} ( U ultrafilter)

= (X \ A)�.

�

5. The up-set functor

U : Set→ Set : X 7→ UX, f 7→ Uf,

with UX = {S ⊆ PX |↑PX S = S}, all the up-closed sets for theinclusion order on PX and Uf(S) = {B ⊆ Y | f−1(B) ∈ S}, togetherwith the principal up-set map

eX : X → UX : x 7→ x = {A ⊆ X | x ∈ A},


and the map

mX : UUX → UX : S 7→ {A ⊆ X | AU ∈ S },

where AU = {S ∈ UX | A ∈ S}, forms a monad U = (U,m, e), calledthe up-set monad .

Proof. To check that the functor U is well-defined, we have to provethat (Uf)(S) is up-closed.

↑PX (Uf)(S) = {B ⊆ Y | ∃A ∈ (Uf)(S) : A ⊆ B}= {B ⊆ Y | ∃A : f−1(A) ∈ S and A ⊆ B}= {B ⊆ Y | f−1(B) ∈ S},

where the last equality holds due the fact that S is up-closed andf−1(A) ⊆ f−1(B) if A ⊆ B.

It is obvious that x is up-closed, so we can conclude that eX is well-defined.

Finally we verify that the image of S under the map mX is up-closed.Therefore we need to remark that if A ⊆ B, then also AU ⊆ BU. TakeS ∈ AU, then we have A ∈ S and ↑PX S = S. Because A ⊆ B, weknow that B ∈ S. Thus S ∈ BU.

Take an up-closed set S of UX (S ∈ UUX) and compute

↑PX mX(S ) = {B ⊆ X | ∃A ∈ mX(S ) : A ⊆ B}= {B ⊆ X | ∃A : AU ∈ S , A ⊆ B}= {B ⊆ X | ∃A :↑PPX AU = AU, A ⊆ B}= {B ⊆ X | ∃A : AU ⊆ BU and AU ∈ S }= {B ⊆ X | BU ∈ S }= mX(S ).

Because of the similarity with the filter monad, we don’t have to provethe conditions of a monad. The proof is completely analogous to theproof of the filter monad (changing AF by AU).

This proves that U = (U,m, e) is a monad. �

In the proofs above, we have seen that these monads are similar.This is easily explained by the fact that they are submonads of eachother. There exists a chain of monad morphisms all given objectwiseby inclusion maps:

I→ �→ F→ U.


6. For the concepts of cliques and ultracliques, it is clear that every ultra-clique is a clique and every clique is an up-set. We define the cliqueand ultraclique functor as follows:

C : Set→ Set : X 7→ CX, f 7→ Cf,

with CX the set of all cliques on X and

Cf(C) = {B ⊆ Y | f−1(B) ∈ C},

andκ : Set→ Set : X 7→ κX, f 7→ κf

is the functor, with κX the set of all ultracliques on X, namely therestriction of the clique functor. Like it was the case for filters, we getthe chain of natural transformations:

1Set → κ→ C → U.

This gives us the clique monad C = (C,m, e) and the ultraclique monad� = (κ,m, e). Here m and e are the restrictions of the m and e of theup-set monad to cliques and ultracliques respectively. So the chain offunctors becomes a chain of monad morphisms:

I→ �→ C→ U.

Every filter is a clique, thus we can construct another chain of monadmorphisms, given objectwise by inclusion maps:

I→ F→ C→ U.

Proof. C and κ are well-defined functors, since Lemma 1.7.11 gives usthat Cf(C) and κf(E) are cliques and ultracliques respectively.

For every x ∈ X, eX(x) is an ultraclique and therefore a clique, soe is well-defined. To prove that m is well-defined, we take a properclique C ∈ CCX and take two non-empty subsets A,B ⊆ X such thatA,B ∈ mX(C ). The assumption states that AC and BC are elementsof C . Thus there intersection is not empty, which means there existsa proper clique C ∈ CX such that C ∈ AC and C ∈ BC. Thus A ∈ Cand B ∈ C. By the definition of a clique, we get A ∩ B 6= ∅. If C isnot proper,then C is an up-set. We already know that mX preservesupsets, so mX is well-defined for proper and improper cliques.


If C is an ultraclique and A /∈ mX(K ), then we know that A� /∈ K .Thus, because of Lemma 1.7.10, we have that (A�){ ∈ K . It easilyfollows that (A�){ = (A{)�, thus we have A{ ∈ mX(K ).

The properties of a monad now follow from the inclusion maps into theup-set monad. �

Remark 2.1.4. For every monad T = (T,m, e) on Set there is a uniquemonad morphism from the identity monad I = (1Set, 1, 1) to T. It is given bythe unit e : 1Set → T of T.

The following proposition gives another way to write down the multipli-cation of a monad. This notation can, for instance, be found in [11].

Proposition 2.1.5. For the up-set monad and therefore for the filter monad,the ultrafilter monad, the clique monad and the ultraclique monad, we canwrite

mX(S ) =⋃A∈S

⋂S∈A

S

for S ∈ UUX,FFX, ββX,CCX or κκX.

Proof. We will prove the statement for the up-set monad:

A ∈ mX(S )⇔ AU ∈ S

⇔ ∃A ∈ S : A ⊆ AU

⇔ ∃A ∈ S ∀S ∈ A : A ∈ S

⇔ A ∈⋃A∈S

⋂S∈A

S.

�

2.2 Kleisli triples

There is an alternative presentation of monads, which we call a Kleisli triple.This representation can sometimes be useful. It is used for example in section5.4 The Kleisli extension.

Definition 2.2.1. A Kleisli triple (T, (−)T, e) on Set consists of

1. a function T : |Set| → |Set| sending X to TX,

2. an extension operator (−)T sending a map f : X → TY to a mapfT : TX → TY,


3. a map eX : X → TX for each set X,

subject to(gT · f)T = gT · fT, eTX = 1TX , fT · eX = f (2.1)

for all sets X, f : X → TY and g : Y → TZ.

One can setg ◦ f := gT · f,

so the previous conditions (2.1) are equivalent to requiring that this “Kleislicomposition” ◦ is associative and that eX acts as an identity (see section 4.3The Kleisli category).

Definition 2.2.2. A Kleisli triple morphism α : (S, (−)S, d)→ (T, (−)T, e)is given by a family of morphisms αX : SX → TX in Set, such that

αY · fS = (αY · f)T · αX , αX · dX = eX

for all f : X → SY. That is, a Kleisli triple morphism is a family morphismsthat preserve the Kleisli composition together with its unit.

Proposition 2.2.3. A Kleisli triple (T, (−)T, e) gives rise to a monad T =(T,m, e) and vice-versa.

Proof. Given a Kleisli triple (T, (−)T, e), set

Tf := (eY · f)T, mX := (1TX)T (2.2)

for all morphisms f : X → Y. By using (2.1) we see that (T,m, e) is a monad.First T is a functor, since, for f : X → Y and g : Y → Z,

Tg · Tf = (eZ · g)T · (eY · f)T

= ((eZ · g)T · eY · f)T

= (eZ · g · f)T

= T (g · f).

AndT1X = (eX · 1X)T = eTX = 1TX .

The morphisms eX form a natural transformation, i.e. for f : X → Y ,we have

Tf · eX = (eY · f)T · eX = eY · f.


The morphisms mX also form a natural transformation. For f : X → Y , wehave

mY · TTf = (1TY )T · (eTY · Tf)T

= ((1TY )T · eTY · Tf)T

= (1TY · Tf)T

= (Tf)T

= ((eY · f)T · 1TX)T

= (eY · f)T · (1TX)T

= Tf ·mX .

Finally we check the multiplication law and the left and right unit laws.The muliplication law follows from following computation

mX · TmX = (1TX)T · T ((1TX)T)

= (1TX)T · (eTX · (1TX)T)T

= ((1TX)T · eTX · (1TX)T)T

= (1TX · (1TX)T)T

= ((1TX)T)T

= ((1TX)T · 1TTX)T

= (1TX)T · (1TTX)T

= mX ·mTX .

Following computation gives us the left unit law

mX ·TeX = (1TX)T·(eTX ·eX)T = ((1TX)T·eTX ·eX)T = (1TX ·eX)T = eTX = 1TX ,

and the right unit law

mX · eTX = (1TX)T · eTX = 1TX .

Conversely, given a monad T = (T,m, e) on Set, one obtains a Kleislitriple (T, (−)T, e) via

fT := mY · Tf,

for all morphisms f : X → TY.


We will prove the three conditions of (2.1). Take f : X → TY andg : Y → TZ, then we have

(gT · f)T = mZ · T (mZ · Tg · f)

= mZ · TmZ · TTg · Tf= mZ ·mTZ · TTg · Tf (multiplication law)

= mZ · Tg ·mY · Tf (naturality of m)

= gT · fT.

We also haveeTX = mY · TeX = 1TX

by the left unit law.For f : X → TY we have

fT · eX = mY · Tf · eX= mY · eTY · f (naturality of e)

= 1TY · f (right unit law)

= f.

�

Chapter 3

Lax extensions

Chapter 3 provides the first notions, properties and examples of lax struc-tures. This type of structure is different from the ones we know, because wewill work with inequalities instead of equalities. This chapter is based onChapter III of “Monoidal Topology” [4], written by Dirk Hofmann, Gavin J.Seal and Walter Tholen. Further development of examples and additionalremarks in, for instance, proofs are elaborated by the author.

3.1 Lax functors and their transformations

We define now a special type of categories, which will be relevant for defininglax structures.

Definition 3.1.1. An ordered category C is a category C with each hom-set C(A,B) carrying an order (that is, reflexive and transitive relation ≤),such that the composition maps C(A,B)×C(B,C)→ C(A,C) : (f, g) 7→ g · fare monotone. We call a category C a separated ordered category if C isan ordered category and the orders on the hom-sets are separated.

Rel and Mod are ordered categories, with Rel(X, Y ) and Mod(X, Y ) orderedby inclusion for all sets X and Y .

Definition 3.1.2. A lax functor F : C → D of ordered categories is givenby functions

F : |C| → |D| and FA,B : C(A,B)→ D(FA, FB)

for all A,B ∈ |C|, such that

(1) FA,B is monotone,

30

CHAPTER 3. LAX EXTENSIONS 31

(2) Fg · Ff ≤ F (g · f),

(3) 1FA ≤ F1A,

for all A,B,C ∈ |C|, f : A→ B, g : B → C.The lax functor F is a 2-functor if the inequalities “≤” in (2) and (3)

may be replaced by “=”.An oplax functor F : C → D must satisfy condition (1) and

(2∗) F (g · f) ≤ Fg · Ff,

(3∗) F1A ≤ 1FA.

Definition 3.1.3. A lax transformation α : F → G, for two lax or oplaxfunctors F,G : C → D, is given by an |C|-indexed family of D-morphismsαA : FA→ GA such that, for all f : A→ B in C,

(4) Gf · αA ≤ αB · Ff,

while an oplax transformation satisfies

(4∗) αB · Ff ≤ Gf · αA.

3.2 Lax extensions of functors

We know that Rel can be seen as an extension of Set (see Section 1.1). Fora given monad T = (T,m, e) on Set, we now consider extensions of T to Rel.For this we first concentrate on the underlying Set-functor T , the naturaltransformations e and m will be considered afterwards, in 3.3.

Definition 3.2.1. Given a functor T : Set→ Set, a lax extension T : Rel→ Relof T to Rel is given by functions

TX,Y : Rel(X, Y )→ Rel(TX, TY )

for all sets X, Y (with TX,Y simply written as T ), such that

(1) r ≤ r′ =⇒ T r ≤ T r′,

(2) T s · T r ≤ T (s · r),

(3) Tf ≤ T f and (Tf)◦ ≤ T (f ◦),

for all sets X, Y, Z, relations r, r′ : X →7 Y, s : Y →7 Z and maps f : X → Y.


By setting TX = TX for all sets X, and observing that condition (3)yields 1TX ≤ T1X , one can define a lax extension of a Set-functor T equi-valently as a lax functor T : Rel → Rel that agrees with T on objects of Reland satisfies the extension condition (3).

Examples 3.2.2. 1. The identity functor on Set has a lax extension givenby the identity functor on Rel.

2. The powerset functor P : Set→ Set has lax extensions P , P : Rel→ Relgiven by

A (P r) B ⇐⇒ A ⊆ r◦(B) ⇐⇒ ∀x ∈ A ∃y ∈ B : x r y,

A (P r) B ⇐⇒ B ⊆ r(A) ⇐⇒ ∀y ∈ B ∃x ∈ A : x r y,

for every relation r : X →7 Y and all A ⊆ X,B ⊆ Y.

Proof. We will prove that P is a lax extension, the proof for P isanalogous.

Take two relations r, r′ : X →7 Y, such that r ≤ r′, this means that forevery B ⊆ Y, r◦(B) ⊆ r′◦(B). Then for A ⊆ X and B ⊆ Y :

A (P r) B ⇐⇒ A ⊆ r◦(B)

=⇒ A ⊆ r′◦(B)

⇐⇒ A (P r′) B.

To check the second condition we take two relations r : X →7 Y ands : Y →7 Z, and subsets A ⊆ X,C ⊆ Z, then we get

A (P s · P r) C ⇐⇒ ∃B ⊆ Y such that A (P r) B and B (P s) C

⇐⇒ ∃B ⊆ Y such that A ⊆ r◦(B) and B ⊆ s◦(C)

=⇒ A ⊆ r◦(s◦(C))

⇐⇒ A ⊆ (s · r)◦(C)

⇐⇒ A (P (s · r)) C.

For the third condition, we take a map f : X → Y and supposeA (Pf) B, for A ⊆ X and B ⊆ Y . From the assumption it follows thatf(A) = B. Thus we certainly have A ⊆ f ◦(B), so A (P f) B. Supposenow B (Pf)◦ A, then A (Pf) B, by definition of the opposite relation.Thus B = f(A), so B ⊆ (f ◦)◦(A) and therefore B (P (f ◦)) A.

Thus P is a lax extension. �


3. Every functor T on Set admits a largest lax extension to Rel given by

T>r : TX × TY → 2 : (S, T ) 7→ 1

for all relations r : X →7 Y.

Proof. It is easy to see that this defines a lax extension. �

Although a lax extension T preserves composition of relations only up toinequality, it operates more strictly on composites of relations with Set-maps,as the corollary to the following proposition shows.

Proposition 3.2.3. Given functions TX,Y : Rel(X, Y ) → Rel(TX, TY ) thatsatisfy the conditions (1) and (2) of the definition of a lax extension, thefollowing are equivalent:

(i) Tf ≤ T f and (Tf)◦ ≤ T (f ◦), for all f : X → Y (condition (3));

(ii) Tf ≤ T f and T (s · f) = T s · Tf, for all f : X → Y and s : Y →7 Z;

(iii) (Tf)◦ ≤ T (f ◦) and T (f ◦ · r) = (Tf)◦ · T r, for all f : X → Y andr : Z →7 Y.

The next condition is a consequence of any of the previous ones, and isequivalent to each of them if T also satisfies 1TX ≤ T1X :

(iv) T (s · f) = T s · Tf and T (f ◦ · r) = (Tf)◦ · T r, for all f : X → Y ,r : Z →7 Y and s : Y →7 Z.

Proof. For (i) =⇒ (ii) we take a map f : X → Y and a relation s : Y →7 Z,and we look at the following chain of inequalities:

T s · Tf ≤ T s · T f (Tf ≤ T f)

≤ T (s · f) (condition (2))

≤ T (s · f) · (Tf)◦ · Tf (1TX ≤ (Tf)◦ · Tf)

≤ T (s · f) · T (f ◦) · Tf ((Tf)◦ ≤ T (f ◦))

≤ T (s · f · f ◦) · Tf (condition (2))

≤ T s · Tf (f · f ◦ ≤ 1X).

Thus the inequalities are all equalities.


The implication from (i) to (iii) is proven analogously. For f : X → Yand r : Z →7 Y we have the following chain of inequalities:

(Tf)◦ · T r ≤ T (f ◦) · T r ((Tf)◦ ≤ T (f ◦))

≤ T (f ◦ · r) (condition (2))

≤ (Tf)◦ · Tf · T (f ◦ · r) (1TX ≤ (Tf)◦ · Tf)

≤ (Tf)◦ · T f · T (f ◦ · r) (Tf ≤ T f)

≤ (Tf)◦ · T (f · f ◦ · r) (condition (2))

≤ (Tf)◦ · T r (f · f ◦ ≤ 1X).

Thus the inequalities are all equalities.For (ii) =⇒ (iv) we have for f : X → Y and r : Z →7 Y that

Tf · T (f ◦ · r) ≤ T f · T (f ◦ · r) ≤ T (f · f ◦ · r) ≤ T r,

and thus T (f ◦ · r) ≤ (Tf)◦ · T r. Apply the lax extension T to the inequality1X ≤ f ◦ · f, then we get

1TX ≤ T1X ≤ T (f ◦) · Tf,

thus(Tf)◦ ≤ T (f ◦).

Now we have (Tf)◦ · T r ≤ T (f ◦) · T r ≤ T (f ◦ · r). Thus (iv) is proven.Similarly we prove (iii) =⇒ (iv). For f : X → Y and s : Y →7 Z we

haveT (s · f) · (Tf)◦ ≤ T (s · f) · T (f ◦) ≤ T (s · f · f ◦) ≤ T s,

and thus T (s · f) ≤ T s · Tf. Apply the lax extension T to the inequality1X ≤ f ◦ · f, then we get

1TX ≤ T1X ≤ (Tf)◦ · T f,

thusTf ≤ T f.

Now we have T s · Tf ≤ T s · T f ≤ T (s · f). Thus (iv) is proven.Finaly we prove (iv) =⇒ (i). We know that 1X ≤ f ◦·f and 1TX ≤ T (1X),

thus1TX ≤ T (1X) ≤ T (f ◦ · f) = T (f ◦) · Tf.

So(Tf)◦ ≤ T (f ◦).


We also have1TX ≤ T (1X) ≤ T (f ◦ · f) = (Tf)◦ · T f,

and thusTf ≤ T f.

This proves (i). �

Corollary 3.2.4. For a lax extension T : Rel→ Rel of a Set-functor T onehas

T (s · f) = T s · T f = T s · Tf, T (f ◦ · r) = T (f ◦) · T r = (Tf)◦ · T r

for all maps f : X → Y and relations r : Z →7 Y and s : Y →7 Z.

Proof. We have

T s · T f ≤ T (s · f) (condition (2))

= T s · Tf (equivalence (ii))

≤ T s · T f (equivalence (ii)),

and

T (f ◦) · T r ≤ T (f ◦ · r) (condition (2))

= (Tf)◦ · T r (equivalence (iii))

≤ T (f ◦) · T r (equivalence (iii)).

�

Definition 3.2.5. A lax extension T of T is flat if T1X = T1X = 1TX .

Lemma 3.2.6. If a lax extension T of T is flat, the following diagramscommute:

Rel T // Rel Rel T // Rel

Set

(−)◦

OO

T// Set

(−)◦

OO

Setop(−)◦

OO

T op// Setop

(−)◦OO

Proof. Indeed, if T is flat, by the previous Proposition one obtains

T f = T1Y · Tf = Tf and T (f ◦) = (Tf)◦ · T1X = (Tf)◦,

for all f : X → Y in Set. �

Remark 3.2.7. The lax extensions of the powerset functor and the largestlax extension of a functor T are not flat. The lax extension of the identityfunctor is flat.


3.3 Lax extensions of monads

Let us now turn our attention to the natural transformation e and m thatwe wish to extend from Set to Rel together with the functor T .

Definition 3.3.1. A triple T = (T ,m, e) is a lax extension of the monadT = (T,m, e) if T is a lax extension of T which makes both m : T T → T ande : 1Rel → T oplax transformations, that is:

(4) mY · T T r ≤ T r ·mX ,

(5) eY · r ≤ T r · eX .

for all relations r : X →7 Y.

By using both the adjunction rules (g · r ≤ t ⇐⇒ r ≤ g◦ · t andg · r◦ ≤ t ⇐⇒ g ≤ t · r for relations g, r, t) for the maps mX and eX , weobtain the following equivalent formulations of (4) and (5):

(4◦) T T r ·m◦X ≤ m◦Y · T r,

(5◦) r · e◦X ≤ e◦Y · T r.

Similarly, these conditions are equivalent to:

(4’) T T r ≤ m◦Y · T r ·mX ,

(5’) r ≤ e◦Y · T r · eX .

These inequalities then yield the following pointwise expressions:

(4∗) T T r(S ,T ) ≤ T r(mX(S ),mY (T )),

(5∗) r(x, y) ≤ T r(eX(x), eY (y)),

for all x ∈ X, y ∈ Y,S ∈ TTX,T ∈ TTY and relations r : X →7 Y.

Definition 3.3.2. One says that a lax extension T = (T ,m, e) of the monadT is flat if the lax extension T of the functor T is flat.

Examples 3.3.3. 1. The identity monad I on Set can be extended to theidentity monad I on Rel. It is a flat lax extension.


2. The lax extensions P , P provide non-flat lax extensions P, P of thepowerset monad P to Rel.

Proof. We will prove the statement for P and the statement for P

follows analogously. So take a relation r : X →7 Y and collections ofsubsets A ∈ PPX and B ∈ PPY . Then we have

A (P P r) B ⇐⇒ A ⊆ (P r)◦(B)

⇐⇒ ∀A ∈ A ∃B ∈ B ∀x ∈ A ∃y ∈ B : x r y

=⇒ ∀x ∈⋃A ∃y ∈

⋃B : x r y

⇐⇒⋃A (P r)

⋃B

⇐⇒ mX(A) (P r) mY (B).

This proves (4∗). To prove (5∗) we take x ∈ X and y ∈ Y and weknow that eX(x) = {x} and analogously for y. Thus the statementr(x, y) ≤ P r(eX(x), eY (y)) is trivial. �

3. Every monad T on Set admits a largest lax extension T> to Rel. It failsto be flat.

3.4 The Barr extension

Given a relation r : X ×Y → 2, we will denote its representation as a subsetof X×Y by R. With π1 : R→ X and π2 : R→ Y the respective projections,r is represented as a span, that is, as a diagram of the form

R

π2 ��π1~~X Y

and we have r = π2 · π◦1 in Rel.

Definition 3.4.1. The Barr extension of a functor T : Set → Set to Relis defined by

Tr := Tπ2 · (Tπ1)◦.

Elementwise, for T ∈ TX and U ∈ TY the Barr extension is given by

T Tr U ⇐⇒ ∃W ∈ TR : Tπ1(W) = T and Tπ2(W) = U .


Remarks 3.4.2. 1. The Barr extension T preserves the order on homsets.Indeed, if s ≤ r, then we may assume S ⊆ R with S the domain of thespan representing s; hence, with : S ↪→ R denoting the inclusion map,

Ts = Tπ2 · Ti · (Ti)◦ · (Tπ1)◦ ≤ Tπ2 · (Tπ1)◦ = Tr.

2. One easily verifies that T (r◦) = (Tr)◦ and Tf = Tf for all relationsr : X →7 Y and Set-maps f : X → Y. Moreover, given Set-mapsf : A→ X and g : Y → B one has

T (g · r) = Tg · Tr and T (r · f ◦) = Tr · (Tf)◦.

Proof. We have r◦ = π1 · π◦2 and so

T (r◦) = Tπ1 · (Tπ2)◦ = (Tπ2 · (Tπ1)◦)◦ = (Tr)◦.

For a map f : X → Y we have the following span,

R

π2 ��π1~~X

f// Y

and thus f · π1 = π2. So we have

Tf = Tπ2 · (Tπ1)◦ = Tf · Tπ1 · (Tπ1)◦ = Tf.

Let (ρ1, ρ2) be the projections representing g. Then we have the fol-lowing span

R

π2 ��π1~~

G

ρ2 ρ1��X Y g

// B.

ThusT (g · r) = Tρ2 · (Tρ1)◦ · Tπ2 · (Tπ1)◦ = Tg · Tr.

Analogously we prove the last statement (with (ρ1, ρ2) the projectionsrepresenting f):

T (r · f ◦) = Tπ2 · (Tπ1)◦ · Tρ1 · (Tρ2)◦ = Tr · Tf.

�


3. In the definition of Tr the pair (π1, π2) can be replaced by any othermono-source representing r, or even by any other source (p, q) withr = q · p◦ if T sends surjections to surjections.

Proof. Given any other factorization r = q·p◦ via maps p : P → X andq : P → Y, the equation r = q ·p◦ says precisely that the canonical mapP → X × Y has image R and therefore defines a surjection l : P → R.One has

Tq ·(Tp)◦ = T (π2 · l) ·(T (π1 · l))◦ = Tπ2 ·T l ·(T l)◦ ·(Tπ1)◦ ≤ Tπ2 ·(Tπ1)◦

with the equality holding if T l · (T l)◦ = 1TX , that is, if T l is surjective.�

Examples 3.4.3. 1. The Barr extension 1Set of the identity functor 1Set

on Set is simply the identity functor 1Rel on Rel.

2. For the filter functor F : Set → Set, the Barr extension F can beconstructed as follows. We first notice that for filters F ∈ FX,G ∈ FYand a relation r : X →7 Y,

F (Fr) G ⇐⇒ ∃H ∈ FR : π1[H] = F and π2[H] = G,

where π1[H] is the image filter of H by the map π1. If such a filter Hexists, then for all A ∈ F , one has C = π−11 (A) ∈ H, and the set

r(A) := {y ∈ Y | ∃x ∈ A : x r y}

must be in G, as it contains π2(C) and π2(C) ∈ π2[H]. Similarly, oneobserves that r◦(B) ∈ F for all B ∈ G.Conversely, if r(A) ∈ G and r◦(B) ∈ F for all A ∈ F and B ∈ G,the sets CA,B = π−11 (A) ∩ π−12 (B) (with A running through F and Bthrough G) form a filter base for H ∈ FR such that π1[H] = F andπ2[H] = G.Therefore, the Barr extension of the filter functor is given by

F (Fr) G ⇐⇒ r[F ] ⊆ G and r◦[G] ⊆ F

for all F ∈ FX,G ∈ FY and relations r : X →7 Y, where r[F ] is thefilter generated by the filter base {r(A) | A ∈ F}.

3. In the previous example, if both F and G are ultrafilters, then,

r[F ] ⊆ G ⇐⇒ r◦[G] ⊆ F .


Indeed, for an ultrafilter G ∈ βY and A′ ⊆ Y, one has

A′ ∈ G ⇐⇒ ∀B ∈ G : A′ ∩B 6= ∅.

Hence, r[F ] ⊆ G means that for all A ∈ F and B ∈ G, one hasr(A) ∩ B 6= ∅, that is, A ∩ r◦(B) 6= ∅ and one obtains r◦[G] ⊆ F (theother implication also follows).

The Barr extension of the ultrafilter functor β is therefore described by

U (βr) V ⇐⇒ r[U ] ⊆ V ⇐⇒ r◦[V ] ⊆ U ,

for all U ∈ βX,V ∈ βY and relations r : X →7 Y .

4. The description of the Barr extension of the ultrafilter functor leadsto distinct extensions when we consider the filter functor instead. Weobtain two lax extensions of the filter functor, neither of which is theBarr extension of F . First, by setting

F (F r) G ⇐⇒ r◦[G] ⊆ F ⇐⇒ ∀B ∈ G ∃A ∈ F ∀x ∈ A∃y ∈ B : x r y,

for all relations r : X →7 Y and filters F ∈ FX,G ∈ FY, one obtainsa non-flat lax extension whose lax algebras (see 4.2) provide a conver-gence discription of the category of topological spaces (Corollary 5.5.4).

By contrast, the lax algebras with respect to the lax extension givenby

F (F r) G ⇐⇒ r[F ] ⊆ G ⇐⇒ ∀A ∈ F ∃B ∈ G ∀y ∈ B∃x ∈ A : x r y,

for all relations r : X →7 Y and filters F ∈ FX,G ∈ FY, and their laxhomomorphisms form a category isomorphic to the category of closurespaces (see Theorem 4.2.9).

Proof. The proof that F and F are lax extensions is completely ana-logous to those of P and P . �

We still have to adress the question of whether the Barr extension isactually a lax extension of the functor T or, even better, of the monad T. Tothis end the functor needs to satisfy an important additional condition thatwe now proceed to describe.


3.5 The Beck-Chevalley condition

A commutative diagram

Wh2 //

h1��

Y

g��

Xf// Z

is a Beck-Chevalley square or simply a BC-square, if the maps involvedsatisfy

h2 · h◦1 = g◦ · for equivalently, if

h1 · h◦2 = f ◦ · g,that is, if

Wh2 // Y

X

_h◦1

OO

f// Z

_g◦OO

or equivalently

W

h1��

Y�h◦2oo

g��

X Z�f◦oo

commutes in Rel.

Definition 3.5.1. 1. A Set-functor T satisfies the Beck-Chevalleycondition, or BC for short, if it sends BC-squares to BC-squares:

h2 · h◦1 = g◦ · f =⇒ Th2 · (Th1)◦ = (Tg)◦ · Tf

for all maps f, g, h1, h2 with h1 · f = g · h2.

2. A natural transformation α : S → T between Set-functors S andT satisfies the Beck-Chevalley condition, or BC for short, if itsnaturality diagrams

SX

Sf��

αX // TX

Tf��

SY αY// TY

are BC-squares for all maps f : X → Y , that is, if αX · (Sf)◦ =(Tf)◦ · αY , or equivalently, if Sf · α◦X = α◦Y · Tf.


Theorem 3.5.2. For a functor T : Set → Set, the following assertions areequivalent:

(i) the functor T satisfies BC,

(ii) the Barr extension T is a flat lax extension of T to Rel and a functorT : Rel→ Rel,

(iii) there is some functor T : Rel → Rel which is a lax extension of T toRel.

Moreover, any functor T : Rel→ Rel as in (iii) is uniquely determined, thatis, T = T .

Proof. Remark 3.4.2 says that the Barr extension preserves the order onthe hom-sets. From the same Remark we also know that (Tf)◦ = T (f ◦) andTf = Tf. If we verify that Ts · Tr = T (s · r), for relations r : X →7 Y ands : Y →7 Z, we have proven (ii).

So factorize r and s by r = π2 ·π◦1 and s = ρ2 ·ρ◦1. As the pullback (p1, p2) of

Rπ2 // Y S

ρ1oo yields a mono-source that moreover forms a factorizationp2 · p◦1 of the relation ρ◦1 · π2

Pp2

��

p1

��R

π2

��

π1

~~

Sρ2

��

ρ1

��X Y Z

one has T (ρ◦1 · π2) = Tp2 · (Tp1)◦ and Tp2 · (Tp1)◦ = (Tρ1)◦ · Tπ2, since T

satisfies BC.Consequently, one obtains (Remark 3.4.2)

Ts · Tr = Tρ2 · (Tρ1)◦ · Tπ2 · (Tπ1)◦ = Tρ2 · T (ρ◦1 · π2) · (Tπ1)◦

= T (ρ2 · ρ◦1 · π2 · π◦1) = T (s · r).

This proves (ii). The implication from (ii) to (iii) is trivial.Let us now prove (i), assuming (iii) holds. Let h2 ·h◦1 = g◦ · f be as in the

definition. Since functoriality make T also flat, one obtains with Corollary3.2.4

Th2 · (Th1)◦ = T (h2 · h◦1) = T (g◦ · f) = (Tg)◦ · Tf.For the same reasons one has

T r = Tπ2 · (Tπ1)◦ = Tr

for r = π2 · π◦1. �


3.6 The Barr extension of a monad

Theorem 3.5.2 proves that if T satisfies the Beck-Chevalley condition, theBarr extension T is a lax extension of T to Rel. It does not require muchmore effort to show that under the same assumption, the Barr extensionyields a lax extension of the monad T = (T,m, e).

Consider first a natural transformation α : S → T between functor S, T :Set → Set provided with their lax extensions S, T . Then, for a relationr : X →7 Y with r = π2 · π◦1, we have

αY ·Sr = αY ·Sπ2 · (Sπ1)◦ = Tπ2 ·αR · (Sπ1)◦ ≤ Tπ2 · (Tπ1)◦ ·αX = Tr ·αX ,

that is, α : S → T is oplax. From the same computation, we observe thatα : S → T is a natural transformation if all naturality diagrams

SX

Sf��

αX // TX

Tf��

SY αY// TY

form BC-squares.Therefore, if T belongs to a monad T = (T,m, e), then m and e become

oplax natural transformations in Rel: m : TT → T and e : 1Rel = 1Set → T .An issue remains with the domain of the multiplication, which should be T Trather then TT . Hence, in order to obtain a lax extension of the monad T toRel, we show that the identities 1TTX are the components of an oplax naturaltransformation T T → TT . It follows from Remark 3.4.2.(3) and the equalityTr = Tπ2 · (Tπ1)◦ that

TTr = TTπ2 · (TTπ1)◦ ≤ T (Tr)

for any relation r = π2 · π◦1, with equality holding if T preserves surjections.Thus the Barr extension T = (T ,m, e) is a lax extension of the Set-monad

T = (T,m, e) to Rel provided that T satisfies the Beck-Chevalley condition.

Theorem 3.6.1. For a monad T = (T,m, e) on Set, the following assertionsare equivalent

(i) the functor T satisfies BC,

(ii) the Barr extension yields a flat lax extension T = (T ,m, e) of T to Rel.


Proof. The implication (i) =⇒ (ii) follows from the previous discussionsince one knows from Theorem 3.5.2 that if T satisfies BC then T is a flatlax extension of T .

The converse implication is also an immediate consequence of the sameresult, since a monad is a flat lax extension exactly when its underlyingfunctor is one. �

Examples 3.6.2. 1. The identity functor 1Set on Set obviously satisfiesBC; it is also immediatly clear that the Barr extension 1Rel is a laxextension.

2. The filter functor F : Set→ Set satisfies BC. Indeed, suppose that

Wh2 //

h1��

Y

g��

Xf// Z

is a BC-square.

Since the square commutes, one immediately obtains the inequality

Fh2 · (Fh1)◦ ≤ (Fg)◦ · Ff.

For the other inequaltity, we must show that for all filters F ∈ FX andG ∈ FY

f [F ] = g[G] =⇒ ∃H ∈ FW : h1[H] = F and h2[H] = G.

But the sets h−11 (A) ∩ h−12 (B) (for A ∈ F and B ∈ G) form a base fora filter H satisfying h1[H] = F and h2[H] = G; indeed g◦ · f ≤ h2 · h◦1means that for any pair (x, y) ∈ A × B with f(x) = g(y), there is anelement x ∈ W satisfying h1(w) = x and h2(w) = y.

Thus, the Barr extension F is a lax extension of F to Rel.

3. The ultrafilter functor satisfies BC for similar reasons. In this case, tosee that (βg)◦ ·βf ≤ βh2 ·(βh1)◦, one obtains a filter H from ultrafiltersU ∈ βX, V ∈ βY as above. By Proposition 1.7.4, there is an ultrafilterW on W with H ⊆ W , such that U ⊆ h1[W ] and V ⊆ h2[W ]. Themaximality of ultrafilters yields the required equalities. Thus the Barrextension β of 3.4.3 is a flat lax extension of β.


Notes on Chapter 3

Throughout chapter III of “Monoidal Topology” [4], the authors use theconcept of a quantale V . A quantale V is a complete lattice which carriesa monoid structure with a neutral element k such that, when the binaryoperation is denoted as a tensor ⊗,

a⊗ (−) : V → V, (−)⊗ b : V → V

are sup-maps for all a, b ∈ V ; hence the tensor distributes over suprema:

a⊗∨i∈I

bi =∨i∈I

(a⊗ bi),∨i∈I

ai ⊗ b =∨i∈I

(ai ⊗ b).

The two-chain 2 = {0, 1} with ⊗ = ∧ and k = 1 is an example of a quantale.In most of the definitions in this chapter and the following chapters, one canchange 2 by any other quantale V and get some similar results, for exampleRel (∼= 2− Rel) can be extended to V − Rel. For our examples we only needthe quantale 2 and therefore we do not define this concept.

Chapter 4

Algebra structures and unitaryrelations

For a monad T = (T,m, e) we can give different algebra structures, namelyEilenberg-Moore algebras and lax algebras. After defining these algebrasand giving different examples, we investigate the defining structure of a laxalgebra, the relation a : TX →7 X. We ask the following question: “When dothese relations form an algebra structure?” The answer is formulated in thesections 4.4, 4.5 and 4.6.

In the second part of this chapter, we will address the problem formulatedin the introduction. We demonstrate a connection between the Eilenberg-Moore algebras and the lax algebras for the ultrafilter monad. We use theresult of Barr [1] to prove the result of Manes [10], although, historicallyseen, Barr based himself on the paper of Manes.

This chapter is based on chapter III of “Monoidal Topology” [4], writtenby Dirk Hofmann, Gavin J. Seal and Walter Tholen. Further developmentof examples and additional remarks in, for instance, proofs are elaborated bythe author.

4.1 Eilenberg-Moore algebra

Given a monad T = (T,m, e) on Set, a T-algebra (or Eilenberg-Moorealgebra) is a pair (X, a), where X is a set and a is the structure map:

a : TX → X

satisfiesa · Ta = a ·mX and 1X = a · eX ,

46

CHAPTER 4. ALGEBRA STRUCTURES ANDUNITARY RELATIONS47

diagramatically:

TTXTa //

mX��

TX

a��

XeX //

1X !!

TX

a��

TX a// X X

A T-homomorphism f : (X, a)→ (Y, b) is a set-morphism f : X → Y suchthat

f · a = b · Tf,

diagrammatically

TXTf //

a��

TY

b��

Xf// Y

The category of T-algebras and T-homomorphisms is denoted by SetT andis also called the Eilenberg-Moore category of T.

We give a first example of an Eilenberg-Moore algebra without a proof.

Example 4.1.1. Setting T = P, we can identify SetP with Sup. Namely, astructure morphism a : PX → X of a P-algebra (X, a) defines a completelattice on X via ∨

A := a(A),

and every P-homomorphism f : (X, a) → (Y, b) is then a sup-map. Onthe other hand, when X is a complete lattice, the map

∨: PX → X is

a P-algebra structure, and sup-maps between complete lattice become P-homomorphisms.

Another important example of Eilenberg-Moore categories are �-algebras.Later on in this chapter, we will prove Set� ∼= CompHaus in an elegant way,since the Barr extension of � to Rel is flat (see Section 4.8).

We now give some other examples, without proofs, of Eilenberg-Moore alge-bras of monads we have encountered. To do this, we need the following newconcepts.

We call a subset A ⊆ Z of a complete lattice Z a filter in Z if

• x, y ∈ A =⇒ x ∧ y ∈ A,

• > ∈ A, and


• x ∈ A, x ≤ y =⇒ y ∈ A

for all x, y ∈ Z, where the infinum is denoted by ∧. A filter on Z is a filterin PZ. This corresponds with the definition given in section 1.7.

We call a complete lattice X cocontinuous if the restriction of the infi-mum map

∧X : UpX → X

to the set FilX of filters in X has a right adjoint �X :

∧X a�X : X → FilX.

We call a complete lattice X completely distributive if the left adjoint∨of the down-closure, as in 1.6, has itself a left adjoint, i.e., if there is a

map⇓: X → DnX

with⇓ a ⊆ S ⇔ a ≤

∨S

for all a ∈ X,S ∈ DnX.

Examples 4.1.2. 1. The category of Eilenberg-Moore algebras of F, thefilter monad, is isomorphic to the category of cocontinuous lattices,

SetF ∼= Cntco.

2. The Eilenberg-Moore algebras of the up-set monad U on Set are thecompletely distibutive lattices and maps that preserve all infima andsuprema:

SetU ∼= Dst.

4.2 Lax algebra

Let T = (T ,m, e) be a lax extension to Rel of a monad T = (T,m, e) on Set.A (T, 2)-relation a : TX →7 X is transitive if it satisfies

a · T a ·m◦X ≤ a ⇐⇒ a · T a ≤ a ·mX .

A (T, 2)-relation a : TX →7 X is reflexive if it satisfies

e◦X ≤ a ⇐⇒ 1X ≤ a · eX .


Definition 4.2.1. A (T, 2)-category or lax algebra is a pair (X, a) con-sisting of a set X and a transitive and reflexive (T, 2)-relation a : TX →7 X;that is a set X with a relation a : TX →7 X satisfying the two laws for anEilenberg-Moore algebra laxly:

TTX

≥

�T a //

mX��

TX

_a��

and XeX //

1X

≤

!!

TX

_a��

TX �a

// X X.

Remark 4.2.2. Note that this notion depends in fact not just on T but alsoon T, hence, whenever needed we will refer to a (T, 2)-category more preciselyas a (T, 2, T)-category.

Definition 4.2.3. A map f : X → Y between lax algebras (X, a) and (Y, b)is a (T, 2)-functor if it satisfies

f · a ≤ b · Tf (⇐⇒ a ≤ f ◦ · b · Tf).

Diagrammatically this means that f is a lax homomorphism of lax algebras:

TX

≤

Tf //

_a��

TY_b��

Xf// Y.

Remark 4.2.4. The identity map 1X : (X, a) → (X, a) and the compositeof (T, 2)-functors are (T, 2)-functors. Hence, (T, 2)-categories and (T, 2)-functors form a category, denoted by (T, 2)−Cat.

Example 4.2.5. 1. Taking the identity monad with the lax extension toRel, the reflexivity and transitivity of a (I, 2)-category are equivalentwith the reflexivity and transitivity of an order relation on X. A (I, 2)-functor defines a monotone map between the two sets. Thus

(I, 2)−Cat ∼= Ord.

2. With T = P laxly extended by P , a transitive and reflexive relationa : PX →7 X must satisfy the conditions

(A ⊆ a◦(B) and B a x =⇒ (⋃A) a x) and {x} a x

for all x ∈ X,B ⊆ X,A ⊆ PX.


Since {x} a y may be re-written as {{x}} ⊆ a◦({y}), one defines anorder on X by

x ≤ y ⇐⇒ {x} a y.We claim that this order determines a completely, since

A a y ⇐⇒ ∀x ∈ A : {x} a y ⇐⇒ A ⊆↓ y. (4.1)

Indeed, when A a y and x ∈ A one has {{x}} ⊆ a◦(A), hence {x} a yby transitivity. When {x} a y for all x ∈ A one uses {{x} | x ∈ A} ⊆a◦({y}) to obtain A a y by transitivity.

Conversely, starting with an order ≤ on X, (4.1) defines a (P, 2, P)-category structure a on X which reproduces the original order.

To be a (P, 2)-functor a map must be monotone with respect to theinduced orders (4.1). As a consequence, one obtains an isomorphism

(P, 2, P)−Cat ∼= Ord

which leaves underlying sets invariant.

3. Trading P for P , for a transitive and reflexive relation a : PX →7 Xwe may define a closure operation c on PX by

x ∈ c(A)⇔ A a x.

Indeed, because a is reflexive, we have {x} a x for every x ∈ X and thusx ∈ c({x}), which gives us 1X ≤ c. We also know that c(A) ⊆ a(A)and thus by definition of P we have A Pa c(A). Take an elementx ∈ c(c(A)), then we know that c(A) a x. Together with the fact thatA Pa c(A) and a is transitive, we get A a x. Thus c is idempotent.

Conversely, given c, this definition yields a (P, 2, P)-category structurea on X.

One has an isomorphism

(P, 2, P)−Cat ∼= Cls.

4. For the largest lax extension T> of a monad T, the only (T, 2,T>)-category structure t on a set X is given by t(T , y) = 1 for all T ∈ TX,y ∈ X.Indeed, for any (T, 2,T>)-category structure a on X, one has

a = a · e◦TX ·m◦X ≤ e◦X · T>a ·m◦X ≤ a · T>a ·m◦X ≤ a,


so

a(T , y) = e◦X · T>a ·m◦X(T , y) =∨

T ∈m−1X (T )

T>a(T , eX(y)) = 1

since eTX(T ) ∈ m−1X (T ) 6= ∅.So one has an isomorphism

(T, 2,T>)−Cat ∼= Set.

5. As seen in Example 3.4.3.4, we can define the lax extension of the filtermonad F as follows

F (F r) G ⇐⇒ r◦[G] ⊆ F ,

for all relations r : X →7 Y and filters F ∈ FX,G ∈ FY. The category ofthe lax algebras associated with this monad and extension is isomorphicwith the category of topological spaces. To prove this statement, weneed some more advanced techniques, which are stated in Chapter 5.The proof of the isomorphism between Top and (F, 2, F)−Cat is givenin Corollary 5.5.4.

6. The category of the lax algebras of the ultrafilter monad � extendedwith the Barr extension is isomorphic to topological spaces. This willbe proven in Section 4.7.

7. The category of lax algebras of the up-set monad, the category of laxalgebras of the clique monad and the category of lax algebras of theultraclique monad with appropriate extensions are all isomorphic tothe category of closure spaces. The results are proven in Chapter 5and Chapter 6.

The description of the lax algebra of the powerset monad with the laxextension P appears for the first time in “Canonical and op-canonical laxalgebras” by Gavin J. Seal [12]. Other examples stated above, are also citedin this article.

A final example is given by the other lax extension of the filter functordefined in Example 3.4.3.4. Here we define F r for a relation r : X →7 Y as

F F r G ⇔ r[F ] ⊆ G,

for F ∈ FX and G ∈ FY.We first need a lemma about the Kowalsky sum of a specific filter.


Lemma 4.2.6. For all A ⊆ X

mX(AF) = A.

Proof. For a subset A of X the following equations hold

mX(AF) = {B ⊆ X | BF ∈ AF}= {B ⊆ X | AF ⊆ BF}= {B ⊆ X | A ⊆ B}= A,

where the third equality follows from the fact that if every filter containingA also contains B, we have that A ⊆ B. �

Theorem 4.2.7. Let (X, a) be an (F, 2, F)-category. Define a map ca :PX → PX as follows

x ∈ ca(A)⇔ A a x

for all A ⊆ X. Then (X, ca) is a closure space.

Proof. Take a subset A ⊆ X and set F = AF. To show idempotency ofca, take x ∈ ca(ca(A)). Then we have that ˙ca(A) a x. By transitivity of the

relation a and Lemma 4.2.6 it remains to show that a[F ] ⊆ ˙ca(A). Since

{AF} is a basis for F it suffices to show a(AF) ∈ ˙ca(A). For y ∈ ca(A) wehave A a y. This proves

ca(A) ⊆ a(AF) = {x ∈ X | ∃F ∈ AF : F a x}

or equivalentlya(AF) ∈ ˙ca(A).

In order to prove reflexivity of ca take x ∈ A. This immediately impliesA ∈ x and consequently a[AF] ⊆ x. By reflexivity and transitivity of therelation a it follows that mX(F ) a x. By Lemma 4.2.6 this is equivalentwith A a x and therefore x ∈ ca(A). This proves reflexivity and as a resultca defines a closure operation. �

Theorem 4.2.8. Let (X, c) be a closure space. Define a relation ac : FX →7X by

F ac x⇔ x ∈⋂A∈F

c(A)

for F ∈ FX and x ∈ X. Then (X, ac) is an (F, 2, F)-category.


Proof. Define a relation ac : FX →7 X as above. For every x ∈ X andA ∈ x, it follows from the reflexivity of the map c that x ∈ c(A). Thus

x ∈⋂A∈x

c(A).

So thanks to the definition of ac we have x ac x. This states the reflexivityof the relation a.

In order to prove transitivity of ac, take F ∈ FFX and G ∈ FX, andsuppose ac[F ] ⊆ G and G ac x hold. For every A ∈ mX(F ) there existsG ∈ G, by ac[F ] ⊆ G, such that

G ⊆ ac(AF) = {x ∈ X | ∃F ∈ AF : F ac x}

= {x ∈ X | ∃F ∈ AF : x ∈⋂B∈F

c(B)}.

Since A ∈ F for every F ∈ AF, it follows thatG ⊆ c(A). From the assumptionG ac x, we know that x ∈ c(H) for all H ∈ G, in particular x ∈ c(G).Combining these two results and the idempotency of the map c we have

x ∈ c(G) ⊆ c(c(A)) ⊆ c(A).

Thusx ∈

⋂A∈mX(F )

c(A).

It follows that mX(F ) ac x and the transitivity is proven. So (X, ac) is a(F, 2, F )-category. �

Combining the two theorems above, we can state the following result.

Theorem 4.2.9. There is an isomorphism

(F, 2, F)−Cat ∼= Cls.

Proof. Theorems 4.2.7 and 4.2.8 give the transition between (F, 2, F)-categoriesand closure spaces.

Let f : (X, a) → (Y, b) be a (F, 2, F)-functor. For all y ∈ f(ca(A)) thereexists x ∈ X such that f(x) = y and x ∈ ca(A). By the transition explained in

Theorem 4.2.7 we have A a x. Since f is a (F, 2, F)-functor and f [A] = ˙f(A)we have

˙f(A) b y.

Thus y ∈ cb(f(A)) and f is a continuous function.


Take a continuous function f : (X, c)→ (Y, d) between two closure spaces(X, c) and (Y, d) and suppose F (f · ac) y. This means there exists x ∈ Xsuch that F ac x and f(x) = y. By the transition stated in Theorem 4.2.8we know that x ∈ c(A) for all A ∈ F . Since f is a continuous function wehave for all A ∈ F

f(x) ∈ f(c(A)) ⊆ d(f(A)).

The collection {f(A) | A ∈ F} forms a basis for the image filter f [F ] andconsequently

f(x) ∈⋂

B∈f [F ]

d(B).

This means f [F ] ad y and so the function f is a (F, 2, F)-functor. �

The description of the category of closure spaces as a (F, 2, F)-categorystated above, appears for the first time in “Canonical and op-canonical laxalgebras” by Gavin J. Seal [12], but is proven in a different way. The proofgiven above is elaborated by the author.

4.3 The Kleisli category

The objects of the Kleisli category SetT associated to the monad T =(T,m, e) on Set are the objects of Set. A morphism f : X ⇀ Y in SetT issimply a set-morphism f : X → TY. The Kleisli composition of f : X ⇀ Yand g : Y ⇀ Z in SetT is defined via the composition in Set as

g ◦ f := mZ · Tg · f.

The identity 1X : X ⇀ X in this category is just the componenteX : X → TX of the unit e.Proof. We will now prove the associativity of the Kleisli composition. Sotake f : X ⇀ Y , g : Y ⇀ Z and h : Z ⇀ V in SetT. Then

h ◦ (g ◦ f) = h ◦ (mZ · Tg · f) = mV · Th · (mZ · Tg · f).

And on the other hand

(h ◦ g) ◦ f = mV · T (h ◦ g) · f= mV · T (mV · Th · g) · f= mV · TmV · TTh · Tg · f= mV ·mTV · TTh · Tg · f (multiplication law)

= mV · Th ·mZ · Tg · f (naturality of m).


�

A Kleisli category has a forgetful functor GT : SetF → Set defined onobjects and morphisms by

X 7→ TX, (f : X ⇀ Y ) 7→ (mY · Tf : TX → TY ).

This functor has a left adjoint FT : Set→ SetF :

X 7→ X, (f : X → Y ) 7→ (eY · f : X ⇀ Y ).

The unit ηT : 1Set → GTFT = T of this adjunction is e and the componentsof the counit εT : FTGT → 1SetF are simply the morphisms 1TX : TX → TXin Set.Proof. Because a map f : X ⇀ Y in SetT is a map f : X → TY in Set,we get that mY · Tf is a map from TX to TY . So GT is well-defined. Theimage of the unit of SetT(X,X), eX : X → TX, under GT is

GT(eX) = mX · TeX = 1TX ,

due to the left unit law. Take now two maps in SetT f : X ⇀ Y andg : Y ⇀ Z and look at the following equalities:

GT(g ◦ f) = mZ · T (g ◦ f)

= mZ · T (mZ · Tg · f)

= mZ · TmZ · TTg · Tf= mZ ·mTZ · TTg · Tf (multiplication law)

= mZ · Tg ·mY · Tf (naturality of m)

= GT(g) ·GT(f).

Thus GT is a functor.Next we see that FT is a functor. The image of a map f : X → Y under

FT is a map from X to TY and thus a map in SetT(X,Y ). The image of1X : X → X is

eX · 1X = eX ,

the unit of SetT(X,X). Take two functions f : X → Y and g : Y → Z andlook at the following equalities:

FT(g) ◦ FT(f) = mZ · T (FT(g)) · FT(f)

= mZ · TeZ · Tg · eY · f= Tg · eY · f (left unit law)

= eZ · g · f (naturality of e)

= FT(g · f).


Thus FT is a functor.Finally we check that FT is a left adjoint of GT. We need to prove

that (FTX, eX) is a reflection. So for all f ∈ Set(X,TY ), we know thatf ∈ SetF(X, Y ). Since e is natural, we get

mY · Tf · eX = mY · eTY · f = 1TY · f = f.

Therefore we have the following commutative diagram

X

f

��

eX // GTFTX = TX

GTf=mY ·Tfvv

X

∃!fx

GTY = TY Y

�

Example 4.3.1. Setting T = P, we can identify SetP with Rel. Namely,a SetP-morphism from X to Y is just a relation r : X → PY. Here theKleisli composition is the relational composition, since for r : X → PY ands : Y → PZ we get

s ◦ r(x) := mZ · Ps · r(x)

=⋃

(Ps · r)(x)

=⋃

(P (s(r(x))))

=⋃{s(r(x))}

= s(r(x))

= s · r(x).

And thuss ◦ r = s · r.

4.4 Kleisli convolution

A (T, 2)-relation is a relation r : TX →7 Y also denoted by r : X ⇀7 Y.In order to compose such relations, we introduce the Kleisli convolution of(T, 2)-relations as a variation of the Kleisli composition. Let us emphasizethat associativity of this operation turns out to depend on the monad laxextension, so that sets with relations r : TX →7 Y only form a category inparticular cases.


Definition 4.4.1. Given a lax extension T = (T ,m, e) of a Set-monadT = (T,m, e), the Kleisli convolution s ◦ r : X ⇀7 Z of (T, 2)-relationsr : X ⇀7 Y and s : Y ⇀7 Z is the (T, 2)-relation defined by

s ◦ r := s · T r ·m◦X .

Remark 4.4.2. When T = I, then s ◦ r = s · r is just the relationalcomposition of relations.

The set of all (T, 2)-relations fromX to Y inherits the order of Rel(TX, Y ) :

r ≤ r′ ⇐⇒ ∀(T , y) ∈ TX × Y : r(T , y) ≤ r′(T , y),

and the Kleisli convolution preserves this order in each variable:

r ≤ r′, s ≤ s′ =⇒ s ◦ r ≤ s′ ◦ r′

for all r, r′ : X ⇀7 Y and s, s′ : Y ⇀7 Z. The (T, 2)-relation e◦X : X ⇀7 X is alax identity for this composition: one has

e◦Y ◦ r = e◦Y · T r ·m◦X ≥ r · e◦TX ·m◦X = r,

with equality holding if e◦ = (e◦X)X : T → 1 is a natural transformation, and

r ◦ e◦X = r · T e◦X ·m◦X ≥ r · (TeX)◦ ·m◦X = r,

with equality holding if T is flat. In particular, e◦X ◦ e◦X ≥ e◦X .

Lemma 4.4.3. For a lax extension T = (T ,m, e) to Rel of a monad T =(T,m, e) on Set one has:

T1X = T (e◦X) ·m◦X .

Proof. On one hand, we can exploit 1TX = 1◦TX = (mX ·TeX)◦ = (TeX)◦·m◦Xto obtain

T1X = T1X · 1◦TX (1TX = 1◦TX)

= T1X · (TeX)◦ ·m◦X≤ T (1X) · T (e◦X) ·m◦X ((TeX)◦ ≤ T (e◦X))

≤ T (e◦X) ·m◦X (T lax functor).

On the other hand

T (e◦X) ·m◦X ≤ T (e◦X · T1X) ·m◦X (1TX ≤ T1X)

= (TeX)◦ · T T1X ·m◦X (Corollary 3.2.4)

≤ (TeX)◦ ·m◦X · T1X (m oplax)

= T1X ,


which concludes the proof. �

We set1]X := e◦X ◦ e◦X ,

hence 1]X = e◦X · T (e◦X) ·m◦X = e◦X · T1X by the Lemma.We can prove

Proposition 4.4.4. If T = (T ,m, e) is a lax extension of the monad T =(T,m, e) to Rel, then

r ◦ e◦X = r · T1X = r ◦ 1]X and e◦Y ◦ r = 1]Y ◦ r

for all (T, 2)-relations r : X ⇀7 Y. In particular,

1]X ◦ 1]X = 1]X ,

so that (X, 1]X) is a (T, 2)-category.

Proof. We first observe that

r ◦ 1]X = r · T (e◦X · T1X) ·m◦X= r · (TeX)◦ · T T1X ·m◦X (Corollary 3.2.4)

≤ r · (TeX)◦ ·m◦X · T1X (m oplax)

= r · T1X (1TX = (TeX)◦ ·m◦X)

= r · T (e◦X) ·m◦X (Lemma 4.4.3)

= r ◦ e◦X .

This inequality suffices to prove the first set of equalities, since e◦X ≤ 1]Ximplies

r ◦ e◦X ≤ r ◦ 1]X .

The other equality follows directly from Corollary 3.2.4, as

1]Y ◦ r = e◦Y · T1Y · T r ·m◦X = e◦Y · T r ·m◦X = e◦Y · r.

Finally

1]X ◦ 1]X = 1]X · T1X = e◦X · T1X · T1X = e◦X · T1X = 1]X .

�


4.5 Unitary (T, 2)-relations

Our candidates for the identities of the Kleisli convolution are the (T, 2)-relations 1]X , but the array of (T, 2)-relations from X to Y that are leftinvariant by composition with these identities must still be determined.

Definition 4.5.1. A (T, 2)-relation r : X ⇀7 Y is right unitary if itsatisfies

r ◦ e◦X ≤ r,

while it is left unitary ife◦Y ◦ r ≤ r

holds. In terms of relational composition, these conditions amount to

r · T1X ≤ r

ande◦Y · T r ·m◦X ≤ r,

respectively. The (T, 2)-relation r is unitary if it is both left and right uni-tary.

The (T, 2)-relation e◦X itself is not unitary in general, but Proposition4.4.4 shows that we can replace it in the previous definitions by 1]X . It followsfrom the discussion preceding Lemma 4.4.3 that the inequalities appearingin the left and right unitary conditions are in fact equalities. Hence, a (T, 2)-relations r is right unitary, respectively left unitary, if r◦1]X = r, respectively1]Y ◦ r = r.

Let us examine (T, 2)-relations and (T, 2)-functors in the light of theKleisli convolution and unitary (T, 2)-relations. By definition, a (T, 2)-categorystructure is a relation a : TX →7 X such that a ◦ a ≤ a and e◦X ≤ a. Theseconditions imply that such a (T, 2)-relation a : X ⇀7 X is always unitary:

a ◦ 1]X = a = 1]X ◦ a,

since a◦e◦X ≤ a◦a ≤ a and e◦X ◦a ≤ a◦a ≤ a. As a consequenc, a : TX →7 Xis a (T, 2)-category structure if and only if

a ◦ a = a and 1]X ≤ a. (4.2)

Indeed, the first condition follows from transitivity: a ≤ a ◦ e◦X ≤ a ◦ a ≤ a,and the second condition follows from reflexivity: 1]X = e◦X ◦ e◦X ≤ a ◦ e◦X ≤ a(the converse resulting from e◦X ≤ 1]X). Hence, a (T, 2)-category structurea can also be seen as a monoid in the ordered set of unitary (T, 2)-relations


from X to X, considered as a category that is provided with the ◦-operation.But we recall that associativity of ◦ is guarenteed only under additionalhypotheses (see 4.6), a property that is needed to consider ◦ as a monoidalstructure.

By definition, a (T, 2)-functor f : (X, a) → (Y, b) is a map f : X → Ysatisfying f · a ≤ b · Tf. Since a (T, 2)-category structure b is right unitary,it satisfies b · T1Y = b ◦ e◦Y = b by Proposition 4.4.4. One then obtains byProposition 4.4.4 the equalities

b · T f = b · T1Y · T f = b · Tf.

Hence, the (T, 2)-functor condition can equivalently be expressed by usingthe lax extension of T :

f · a ≤ b · T f.

Setting f ] := f ◦ · 1]Y : Y ⇀7 X one can also express (T, 2)-functoriality ofa map f : X → Y via the Kleisli convolution as

a ◦ f ] ≤ f ] ◦ b.

4.6 Associativity of unitary (T, 2)-relations

With respect to the Kleisli convolution, the unitary (T, 2)-relation 1]X servesas an identity for all unitary (T, 2)-relations composable with 1]X . In generalhowever, unitary (T, 2)-relations do not compose associatively (see [4]).

Definition 4.6.1. A lax extension T to Rel of a monad T = (T,m, e) on Setis associative whenever the Kleisli convolution of unitary (T, 2)-relations isassociative. Explicitly, a lax extension T is associative whenever

t ◦ (s ◦ r) = (t ◦ s) ◦ r,

or equivalently,

t · T (s · T r ·m◦X) ·m◦X = t · T s ·m◦Y · T r ·m◦X (4.3)

for all unitary (T, 2)-relations r : X ⇀7 Y, s : Y ⇀7 Z and t : Z ⇀7 W.

For T associative, unitary (T, 2)-relations are closed under Kleisli con-volution:

(s ◦ r) ◦ 1]X = s ◦ (r ◦ 1]X) = s ◦ r and 1]Z ◦ (s ◦ r) = (1]Z ◦ s) ◦ r = s ◦ r.


Hence, in the presence of an associative lax extension T, we can form thecategory

(T, 2)−URel

whose objects are sets and whose morphism are unitary (T, 2)-relations thatcompose via the Kleisli composition.

We note that, like (T, 2)−Cat, also (T, 2)−URel depends on the lax extensionT; we write (T, 2, T)−URel whenever this dependency needs to be emphasized.

When the hom-sets (T, 2)−URel(X, Y ) are equipped with the pointwiseorder,

r ≤ r′ ⇐⇒ ∀(T , y) ∈ TX × Y : r(T , y) ≤ r′(T , y),

(T, 2)−URel becomes an ordered category.To verify which lax extensions are associative, we define the unitary (T, 2)-

relationr] := e◦Y · T r : TX →7 Y

associated to a relation r : X →7 Y. Notice that (1X)] = 1]X .

Lemma 4.6.2. Let T be a lax extension to Rel of a monad T = (T,m, e) onSet. Then

T (s] · T r) ·m◦X = T (s · r)

for all relations r : X →7 Y and s : Y →7 Z. In particular

T p] ·m◦X = T p

for all relations p : X →7 Y.

Proof. The first stated equality follows from

T (s · r) = T (s · r) · T (e◦X) ·m◦X (Lemma 4.4.3)

≤ T (s · r · e◦X) ·m◦X (T lax functor)

≤ T (e◦Z · T s · T r) ·m◦X (e◦ lax natural)

≤ T (e◦Z · T (s · r)) ·m◦X (T lax functor)

= (TeZ)◦ · T T (s · r) ·m◦X (Corollary 3.2.4)

≤ (TeZ)◦ ·m◦Z · T (s · r) (m◦ lax natural)

= T (s · r).

The particular case is obtained by setting s = p and r = 1X . �


Lemma 4.6.3. Let T be a lax extension to Rel of a monad T = (T,m, e) onSet. Then

m◦X · T1X = T1TX ·m◦X · T1X = T T1X ·m◦X · T1X .

Proof. Since 1TTX ≤ T1TX ≤ T T1X , we have

m◦X · T1X ≤ T1TX ·m◦X · T1X ≤ T T1X ·m◦X · T1X ≤ m◦X · T1X ,

by lax naturality of m◦. �

Proposition 4.6.4. Let T be a lax extension to Rel of a monad T = (T,m, e)on Set. The following assertions are equivalent:

(i) T is associative,

(ii) T : Rel→ Rel preserves composition and m◦ : T → T T is natural,

(iii) t ◦ (s ◦ r) = (t ◦ s) ◦ r for all relations t : TZ →7 W, s : TY →7 Z andright unitary relations r : TX →7 Y.

Proof. For (i) =⇒ (ii), consider relations r : X →7 Y and s :→7 Z. Wefirst prove that an associative lax extension preserves composition. Since allof r], s] and T1Z are unitary, one has

T1Z ◦ (s] ◦ r]) = (T1Z ◦ s]) ◦ r].

This identity is equivalent to T (s · r) = T s · T r. Indeed,

T1Z ◦ (s] ◦ r]) = T (s] · T r] ·m◦X) ·m◦X = T (s] · T r) ·m◦X = T (s · r)

by using Lemma 4.6.2 twice, and

(T1Z ◦ s]) ◦ r] = T s] ·m◦Y · T r] ·m◦X = T s · T r

by Lemma 4.6.2 again.To see that m◦ is natural, we compute

T1Y ◦ (T1Y ◦r]) = T1Y · T (T1Y · T r] ·m◦X) ·m◦X = T1Y · T T r ·m◦X = T T r ·m◦X

and(T1Y ◦ T1Y ) ◦ r] = T1Y · T T1Y ·m◦Y · T r] ·m◦X = m◦Y · T r

using Lemma 4.6.2 and 4.6.3. Since the Kleisli convolution is associative onunitary relations we obtain

T T r ·m◦X = m◦Y · T r.


For (ii) =⇒ (iii), we use right unitariness of r to write

t ◦ (s ◦ r) = t · T (s · T (r · T1X) ·m◦X) ·m◦X= t · T (s · T r · T T1X ·m◦X) ·m◦X= t · T (s · T r ·m◦X · T1X) ·m◦X (m◦ natural)

= t · T s · T T r · T (m◦X) · T T1X ·m◦X= t · T s · T T r · (TmX)◦ · T T1X ·m◦X= t · T s · T T r ·m◦TX ·m◦X · T1X

= t · T s ·m◦Y · T r ·m◦X · T1X

= t · T s ·m◦Y · T (r · T1X) ·m◦X= (t ◦ s) ◦ r.

The implication (iii) =⇒ (i) is immediately clear by definition of anassociative lax extension. �

Examples 4.6.5. The lax extension of the identity monad I is associative(with the usual composition). The lax extensions P, P and the largest laxextension T> are also associative (see [4]).

When we look at the Barr extension of a monad, then there exist a shorterway to prove whether the extension is associative, using the Beck-Chevalleycondition (see section 3.5).

Theorem 4.6.6. Suppose that T = (T,m, e) is a monad on Set such that Tand m satisfy BC. Then T is an associative lax extension of T to Rel.

Proof. In Theorem 3.6.1 we have proven that Ts ·Tr = T (s · r) if T satisfiesthe BC condition, since T is a flat lax extension. The discussion precedingTheorem 3.6.1 shows that if T satisfy BC and all naturality diagrams of mare BC-squares, then m : TT → T is a natural transformation. Since theBarr extension is flat, we can conclude that the Barr extension is associativeby Proposition 4.6.4. Taking into account that the naturality condition form◦, as stated in the proposition, can equivalently be described as naturalityof m : TT → T , if the lax extension satisfies T (r◦) = (Tr)◦. This is the casefor any Barr extension. �

We can now prove that the Barr extension of the ultrafilter monad � isan associative lax extension.Proof. We must show that the multiplication m of the ultrafilter monad �

satisfies BC. So take a function f : X → Y and take U ∈ βX and V ∈ ββY


with mY (V ) = βf(U). We must find W ∈ ββX with

ββf(W ) = V and mX(W ) = U .

By hypothesis is f(A)� ∩ B 6= ∅ for all A ∈ U and B ∈ V . We have for allA ⊆ X

βf(A�) = βf({W ∈ βX | A ∈ W})= {f [W ] | W ∈ βX,A ∈ W}= {V ∈ βY | f [W ] = V , A ∈ W}= {V ∈ βY | A ∈ f−1(V)}= {V ∈ βY | f(A) ∈ V}= f(A)�.

So βf(A�) ∩ B 6= ∅ and thus A� ∩ (βf)−1(B) 6= ∅. Therefore,

{A� | A ∈ U} ∪ {(βf)−1(B) | B ∈ V }

is a filter base. Any ultrafilter W containing this base has the desiredproperty. So by Theorem 4.6.6 � is associative. �

4.7 Fundamental example of a lax algebra

Our paradigmatic example of a (T, 2)-category comes from [1], in whichtopological spaces are represented as so-called relational algebras for theultrafilter monad; in our context this result reads as

(�, 2)−Cat ∼= Top,

with Top the category of topological spaces and continuous functions. Recallthat � stands for the ultrafilter monad and the required lax extension β ofβ : Set→ Set to Rel is described in Example 3.4.3.

There are many ways to prove the isomorphism mentioned above, dependingin particular on the choice of the standard presentation of topological spaces:open or closed sets, interior or closure operations, or neighborhood systems.Here we work with closure operations, while in Chapter 5 and Chapter 6, witha more developed theory at our disposal, we will present another approach.

The idea of the isomorphism between (�, 2)-categories and topologicalspaces is that a relation r : βX →7 X represents convergence and specifieswhich ultrafilters converge to which point of X. We can associate with r afinitely additive closure operation c : PX → PX, and conversely, show that


every finitely additive closure operation c determines a convergence relationr : βX →7 X.

A closure space is a pair (X, c) consisting of a set X and a closure opera-tion c : PX → PX on the powerset PX of X. We observe that such a closureoperation is a monotone map (or equivalently, an element of Ord(PX,PX))carrying a monoid structure with respect to the compositional structure, thatis,

c · c(A) ⊆ c(A), A ⊆ c(A) for all A ⊆ X.

Furthermore, c : PX → PX defines a topology on X if and only if c isfinitely additive:

c(A ∪B) = c(A) ∪ c(B), c(∅) = ∅ for all A,B ⊆ X.

To any relation r : βX →7 X, we can associate a finitely additive mapclos(r) : PX → PX given by

clos(r)(A) = {y ∈ X | ∃ U ∈ βX : A ∈ U and U r y}.

Conversely, given a map c : PX → PX, we define a relation conv(c) : βX →7 Xby setting

U conv(c) y ⇔ ∀A ∈ PX : (A ∈ U ⇒ y ∈ c(A)).

Note that we have the identities

conv(c)(A�) = c(A) and clos(r)(A) = r(A�) (4.4)

for all A ⊆ X, where A� = {U ∈ βX | A ∈ U} and

r(A�) = {x ∈ X | ∃ U ∈ A� : U r x}.

Lemma 4.7.1. If Set(PX,PX) is equipped with the pointwise order, thenthe maps

clos : Rel(βX,X)→ Set(PX,PX) and conv : Set(PX,PX)→ Rel(βX,X)

form an adjunction clos a conv. The fixpoints of clos · conv are the mapsc : PX → PX that are finitely additive.

Proof. Monotonicity of clos and conv follows immediately from the defini-tions. One also has

1 ≤ conv · clos and clos · conv ≤ 1.


Indeed, for U ∈ βX and y ∈ X,

U r y ⇒ ∀A ∈ PX : (A ∈ U ⇒ U r y)⇒ U conv(clos(r)) y

and U conv(c) y implies that y ∈ c(A) for all A ∈ U ; thus for any A ∈ PX,

clos(conv(c))(A) ={y ∈ X | ∃ U ∈ βX : A ∈ U& ∀B ∈ PX : B ∈ U ⇒ y ∈ c(B)} ⊆ c(A). (4.5)

This expression yields in particular that for c : PX → PX, one necessarilyhas clos · conv(c)(∅) = ∅ (since no ultrafilter U ∈ βX contains ∅), and

clos · conv(c)(A ∪B) = clos · conv(c)(A) ∪ clos · conv(c)(B),

as A ∪ B ∈ U implies that either A ∈ U or B ∈ U (Lemma 1.7.3), and theconverse holds because U is an up-set. Thus, the fixpoints of clos · conv mustbe finitely additive.

Consider now a map c : PX → PX that is finitely additive (and thereforemonotone) and A ∈ PX with y ∈ c(A). The set

J := {J ∈ PX | y /∈ c(J)}

is an ideal on X. By Corollary 1.7.8 (with the principal filter A = A), oneobtains the existence of an ultrafilter U ∈ βX with U ⊆ A and U ∩ J = ∅;in other words, A ∈ U and y ∈ c(B) for all B ∈ U , so one can conclude thatc(A) = clos · conv(c)(A) by (4.5) above. �

A crucial observation is that both maps clos and conv are homomorphismsof the monoids (�, 2)−URel(X,X) and SLat(PX,PX) whose operations aregiven by Kleisli convolution and map composition, respectively.

Proposition 4.7.2. The maps conv and clos satisfy

clos(s ◦ r) = clos(s) · clos(r), conv(d · c) = conv(d) ◦ conv(c)

clos(e◦X) = 1PX , conv(1PX) = e◦X

for all (�, 2)-relations r, s : X ⇀7 X and finitely additive maps c, d : PX →PX.

Proof. For U ∈ βX and x ∈ X, if U conv(1PX) x, then x ∈ A for allA ∈ U ; thus, conv(1PX) = e◦X . Since 1PX is finitely additive, it is a fixpoint ofclos · conv, so the previous equality yields 1PX = clos · conv(1PX) = clos(e◦X).

Consider now finitely additive maps c, d : PX → PX and let U ∈ βX, z ∈X. Set also

A :=↑PX {c(A) | A ∈ U},


which is a filter since c is monotone. Then (4.4), together with Corollary1.7.5, tells us

A ⊆ N ⇔ ∃U ∈ ββX : mX(U ) = U and U β(conv(c)) N

for all N ∈ βX. Assume first that U conv(d · c) z. Then A is disjoint fromthe ideal

J = {B ⊆ X | z /∈ d(B)}.

Applying Corollary 1.7.8, we see that there is an ultrafilterN ∈ βX containingA and disjoint from J . Hence N conv(d) z and there exists U ∈ ββX withU β(conv(c)) N and mX(U ) = U . We conclude that

U (conv(d) ◦ conv(c)) z.

Assume now that U (conv(d) ◦ conv(c)) z, so there is an N ∈ βX such thatA ⊆ N and N conv(d) z. From this we obtain z ∈ d · c(A) for every A ∈ U ,that is, U conv(d · c) z.

Finally, let r, s : X ⇀7 X be (�, 2)-relations and consider A ⊆ X, z ∈ X.If z ∈ clos(s ◦ r)(A), then we have U ∈ ββX and N ∈ βX with

A� ∈ U , U βr N , N s z;

and it follows that z ∈ clos(s)(r(A�)) = clos(s)(clos(r)(A)). Thus,

clos(s ◦ r) ≤ clos(s) · clos(r).

For the other inequality, we use that conv(d · c) = conv(d) ◦ conv(c) forfixpoints c, d of clos · conv, and in particular for c = clos(r), d = clos(s), andc · d :

clos(s) · clos(r) = clos · conv(clos(s) · clos(r))

= clos(conv(clos(s)) ◦ conv(clos(r))) ≤ clos(s ◦ r)

because conv · clos ≤ 1 (by Lemma 4.7.1). �

Lemma 4.7.3. Every (�, 2)-relation r : X ⇀7 X satisfies conv · clos(r) =e◦X ◦ r.

Proof. On one hand,

conv(clos(r)) = conv(1PX · clos(r)) = e◦X ◦ conv(clos(r)) ≥ e◦X ◦ r.

On the other hand, consider U ∈ βX, y ∈ X, and suppose that U conv(clos(r)) y.Then for all A ∈ U , we have y ∈ clos(r)(A) = r(A�), which implies that there


is an U ∈ ββX with mX(U ) = U and U (βr) eX(y). Hence, U (e◦X ◦ r) y,which shows that conv(clos(r)) ≤ e◦X ◦ r. �

As a consequence, we obtain, r = conv · clos(r) for every left unitary(�, 2)-relation r : X ⇀7 X, and in particular for every reflexive and transitive(�, 2)-relation.

Proposition 4.7.4. The fixpoints of the adjunction clos a conv are on onehand the finitely additive maps c : PX → PX, and on the other hand, theleft unitary relation r : βX →7 X.

Proof. This follows from Lemmas 4.7.1 and 4.7.3. �

Theorem 4.7.5. There is an isomorphism (�, 2)−Cat ∼= Top that commuteswith the underlying-set functors.

Proof. By Proposition 4.7.4, clos and conv define a one-to-one correspon-dence between finitely additive maps c : PX → PX and left unitary (�, 2)-relations a : X ⇀7 X. If c is a topological closure operation, so that 1PX ≤ cand c · c = c, then a := conv(c) satisfies

e◦X = conv(1PX) ≤ conv(c) = a

anda ◦ a = conv(c) ◦ conv(c) = conv(c · c) = conv(c) = a

by Proposition 4.7.2, that is, (X, a : βX →7 X) is a (�, 2)-category.Likewise, from e◦X ≤ a and a ◦ a = a one gets 1PX ≤ c and c · c = c for

c := clos(a), and therefore clos and conv actually define a bijective correspon-dence between (�, 2)-categorical structures and topological closure operationson X.

An easy verification shows that a (�, 2)-functor preserves the correspondingclosure operation - and is therefore continuous - while a continuous mappreserves the corresponding (�, 2)-categorical structure - and is therefore a(�, 2)-functor. �

Spelled out, the previous Theorem states that a topological space (X,OX)can be equivalently described as a pair (X, a), with a : βX →7 X a relationrepresenting convergence which, when we denote a and βa by −→, satisfies

(U −→ U and U −→ z ⇒ ΣU −→ z) and x −→ x,

for all x, z ∈ X,U ∈ βX and U ∈ ββX; here U −→ U ⇔ U ⊇ a◦[U ] and Σis the Kowalsky sum restricted to ultrafilters.


In this context, the continuous maps f : (X, a) → (Y, b) are exactly theconvergence-preserving maps, that is, the maps f : X → Y such that

U −→ x⇒ f [U ] −→ f(x)

for all x ∈ X and U ∈ βX.

4.8 �-algebras

For flat lax extensions of monads, there exists a nice result about the connectionbetween Eilenberg-Moore algebras and lax algebras, which is given in the nexttheorem.

Theorem 4.8.1. If T is a flat lax extension of T to Rel, then a T-algebra(X, a : TX → X) is also a (T, 2)-category. In this case, morphisms of T-algebras yield (T, 2)-functors between the corresponding (T, 2)-categories, andthere is a full embedding

SetT ↪→ (T, 2)−Cat.

Proof. In general a T-algebra (X, a), where a : TX → X satisfies a · eX = 1Xand a · Ta = a ·mX , is not a (T, 2)-category (X, a), which requires

a · T a ≤ a ·mX .

When T is a flat lax extension, then T a = Ta, since a is a function. Thereforethe T-algebra (X, a) is a (T, 2)-category. It follows easily that morphisms ofT-algebras yield (T, 2)-functors. To see that the embedding is full, we considerT-algebras (X, a) and (Y, b) with a (T, 2)-functor f : (X, a)→ (Y, b), that is,a function f : X → Y satisfying

f · a ≤ b · Tf

in Rel. As f · a and b · Tf are really Set-maps, the inequality means that thegraph of the first is contained in the second. But an inclusion of graphs ofSet-maps with the same domain is an equality. �

Because the Barr extension of � to Rel is flat (see Example 3.6.2.3), it ispossible to exploit the Theorem above and Theorem 4.7.5, which gives theequivalence between (�, 2)-categories and topological spaces, to obtain anelegant description of the category of �-algebras (that is, of Eilenberg-Moorealgebras associated to �, see 4.1).


Let us recall that a topological space X is compact if every open coverof X has a finite subcover, that is, for every open cover A there exists a finitesubset D ⊆ A with

⋃D = X. A topological space X is Hausdorff if, for all

x, y ∈ X with x 6= y, there exist open subsets A,B ⊆ X with x ∈ A, y ∈ Band A ∩B = ∅.

In the following results we freely exploit that a topological space X canequivalently be described via a set OX of open sets, a closure operationc : PX → PX, or a convergence relation a : βX →7 X. In fact, we can avoidthe closure operation by translating the definition of the convergence relationgiven in 4.7 as

U → x⇔ ∀A ∈ OX : (x ∈ A⇒ A ∈ U). (4.6)

This is true, because OX = {(c(B)){ | B ∈ PX} and

U → x⇔ ∀B ∈ PX : (B ∈ U ⇒ x ∈ c(B))

⇔ ∀A ∈ OX : ∃B ∈ PX : A = (c(B)){

and (B ∈ U ⇒ x ∈ c(B))

⇔ ∀A ∈ OX : (A /∈ U ⇒ x /∈ A) (U ultrafilter)

⇔ ∀A ∈ OX : (x ∈ A⇒ A ∈ U).

Proposition 4.8.2. The following statements are equivalent for a topologicalspace X:

(i) X is compact;

(ii) every ultrafilter on X converges, that is, 1βX ≤ a◦ · a.

Proof. We will first explain the last statement of (ii). Take two ultrafiltersU and V on X. They satisfy the relation a◦ · a if there exists an elementx ∈ X such that U a x and V a x. When we say 1βX ≤ a◦ · a, we say thata◦ · a is reflexive. So for every ultrafilter U on X there exists an elementx ∈ X such that U a x; every ultrafilter on X converges.

The proof of the equivalence of the statements (i) and (ii) is done, forinstance, in “Topology” [5]. �

Proposition 4.8.3. The following statements are equivalent for a topologicalspace X:

(i) X is Hausdorff;

(ii) every ultrafilter on X has at most one convergence point, that is, a · a◦ ≤ 1X .


Proof. We first explain the second statement of (ii). Take an ultrafilterU on X such that there exist x, y ∈ X such that U a x and U a y. Thenwe know that x a · a◦ y. And thus x 1X y, which means x = y. Thus everyultrafilter has at most one convergence point.

The proof of the equivalence of the statements (i) and (ii) can be found,for instance, in the course “Topology” [5]. �

The following theorem gives us one of the fundamental examples of anEilenberg-Moore algebra. We denote with CompHaus the category of alltopological spaces which are compact Hausdorff, with continuous functionsbetween such spaces.


Set� ∼= CompHaus

that commutes with the respective forgetful functors to Set.

Proof. Combining both propositions above, we see that a topological spaceX is compact Hausdorff if and only if its convergence relation a : βX →7 Xis actually a function a : βX → X. The rest of the proof is as in Theorem4.8.1. Since a is a function, we know from the flatness of � that βa = βa.The condition a · βa ≤ a ·mX becomes an inclusion of graphs of functions

a · βa ≤ a ·mX .

Since both functions have the same domain, this inequality is necessarily anequality.

Furthermore, a continuous map f : X → Y between compact Hausdorffspaces with convergence structures a : βX → X and b : βY → Y respectivelysatisfies f · a ≤ b · Tf, but since all relations involved are functions, thiscondition is actually an equality. �

4.9 Induced order on lax algebras

Given a lax extension T to Rel of a monad T on Set, the structure of a (T, 2)-category is a reflexive and transitive relation. This terminology suggest thatthe structure induces a natural order on the underlying set. Indeed, since a(T, 2)-category structure a : TX →7 X is left unitary, one has

e◦X · T a · eTX ≤ e◦X · T a ·m◦X = e◦X ◦ a = a.


The inequality in the other direction is just the expression of oplaxness ofthe unit e : 1Rel → T , so we have

a = e◦X · T a · eTX (4.7)

for any (T, 2)-category structure a : TX →7 X. This identity is used to provethe following result.

Proposition 4.9.1. Let T be a lax extension of a monad T on Set to Rel. Ifa : TX →7 X is a (T, 2)-category structure, then the relation

x ≤ y ⇔ 1 = a(eX(x), y),

(for all x, y ∈ X) defines an order on X, called the underlying orderinduced by a (or sometimes simply the induced order). The structure a isthen monotone in its second variable with respect to this order:

x ≤ y ⇒ a(T , x) ≤ a(T , y)

for all x, y ∈ X, T ∈ TX.

Proof. For the given relation ≤ on X, one immediately has x ≤ x since1 = a(eX(x), x) by reflexivity of a. By transitivity of a and the identity(4.7), if x ≤ y and y ≤ z, then

1 = a(eX(x), y) ∧ a(eX(y), z)

= T a(eTX(eX(x)), eX(y)) ∧ a(eX(y), z)

≤ a(eX(x), z) ≤ 1,

that is, x ≤ z, so the relation ≤ is also transitive. Finally, if x ≤ y thentransitivity of a also yields

a(T , x) = T a(eTX(T ), eX(x)) ≤ T a(eTX(T ), eX(x)) ∧ a(eX(x), y) ≤ a(T , y),

so that a is monotone with respect to this order. �

Examples 4.9.2. 1. For the identity lax extension of I to Rel, we have(I, 2)−Cat ∼= Ord (see Example 4.2.5.1). The underlying order on anordered set (X, a) induced by a returns the original order on X.

2. For the Barr extension of the ultrafilter monad � to Rel, Theorem 4.7.5states that (�, 2)−Cat ∼= Top. Here, the underlying order on (X, a) isgiven by (when we write a as −→)

x ≤ y ⇔ x −→ y.

Chapter 5

Kleisli Monoids

This chapter deals with the presentation of (T, 2)−Cat as the category T−Monof monoids in the hom-set of a Kleisli category that has the advantage ofavoiding explicit use of relations or lax extensions. We based ourselves onan important and well-known example, namely the filter monad F. The mostimportant results of this chapter are two isomorphisms, one for our mainexample

F−Mon ∼= Top

and one in generalT−Mon ∼= (T, 2)−Cat.

We will also present topological spaces by two filter-based counterpartsto the well-known defining structures like open and closed sets, finitely addi-tive closure operations and the convergence of ultrafilters (section 4.7). Thefirst one concentrates on neighborhood filters (Proposition 5.1.1) and willbe generalized in the concept of Kleisli monoids introduced in section 5.3.The second one focuses on filter convergence (Corollary 5.5.4) and uses anew general construction of a lax extension, namely the Kleisli extensionof a monad (section 5.4). Unlike the convergence of ultrafilters, these twopresentations avoid the Axiom of Choice.

The first time the isomorphism between the neighborhood systems and amonoid in the Kleisli category of the filter monad on Set (Proposition 5.1.1) isobserved, is by Gahler in [7]. He introduces the notion of a preordered monadthat is at the origin of the power-enriched monads we defined in section 5.2,and their associated category monadic topologies that, similarly to Kleislimonoids (see section 5.3), are defined as monoids in the Kleisli categoryof the filter monad. The similar result for closure spaces and the up-setmonad (Example 5.3.3.4) originally appeared in “A Kleisli-based aproach to

73

CHAPTER 5. KLEISLI MONOIDS 74

lax algebras” by Gavin J. Seal [13], in which the Kleisli extension (section5.4) is also defined.

All the theorems, propositions, proofs, etcetera in this chapter are statedin section 1 of Chapter IV of “Monoidal Topology” [4], written by DirkHofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal. Furtherdevelopment of examples and additional remarks in, for instance, proofs areelaborated by the author. The author elaborated the proofs of Proposition5.3.4 and Proposition 5.3.5 himself.

5.1 Topological spaces via neighborhood fil-

ters

It is convenient for the map τX : PX → FX : A 7→ A to be monotone. Ifwe have A ⊆ B, then B ⊆ A, so we define on the set FX of filters on X arefinement order:

F ⊆ G ⇐⇒ G ≤ F .

A filter G is finer than F or F is coarser than G, if G ⊇ F . With this orderthe map τX is monotone.

Given a topology OX and x ∈ X, the collection of all open sets thatcontain x spans the neighborhood filter ν(x) of x:

A ∈ ν(x) ⇐⇒ ∃U ∈ OX : (x ∈ U ⊆ A)

for all A ⊆ X. This defines a map ν : X → FX that sends a point of atopological space to its neighborhoodfilter and is such that

ν(x) ⊆ eX(x) = {A ⊆ X | x ∈ A}

for all x ∈ X, and thuseX ≤ ν (5.1)

in the pointwise refinement order.To relate ν with the filter monad multiplication, we define for A ⊆ X the

set AF of filters that contain A,

AF := {F ∈ FX | A ∈ F},

and recall that an open set is a neighborhood of each of its points; thus, in


particular, for all x ∈ X and A ⊆ X, one has

A ∈ ν(x) ⇐⇒ ∃B ∈ ν(x) : (y ∈ B =⇒ A ∈ ν(y))

⇐⇒ ∃B ∈ ν(x) : (y ∈ B =⇒ ν(y) ∈ AF)

⇐⇒ ∃B ∈ ν(x) : B ⊆ ν−1(AF)

⇐⇒ ν−1(AF) ∈ ν(x)

⇐⇒ AF ∈ Fν · ν(x)

⇐⇒ A ∈ mX · Fν · ν(x)

⇐⇒ A ∈ ν ◦ ν(x)

So we’ve got (ν(x) ⊆ ν ◦ ν(x))

ν ◦ ν ≤ ν (5.2)

By (5.1) and (5.2), a topology on X determines a monoid in the orderedhomset SetF(X,X) of the Kleisli category SetF.

Consider now a continuous map f : X → Y between topological spaces.If B ⊆ Y is a neigborhood of f(x), then there exists an open set U ⊆ Bcontaining f(x), so that f−1(U) is an open set with x ∈ f−1(U) ⊆ f−1(B)and f−1(B) is therefore an element of ν(x). Hence

f−1(B) ∈ ν(x) ⇐⇒ B ∈ Ff · ν(x), for all B ⊆ Y and x ∈ X.

For all x ∈ X this yields that ν · f(x) ⊆ Ff · ν(x) or

Ff · ν ≤ ν · f.

Instead of considering a map f : X → Y, one can look at its imagef\ = eY · f : X → FY by the left adjoint Set→ SetF to the forgetful functorSetF → Set. We then get

f\ ◦ ν ≤ ν ◦ f\ (5.3)

Proof. We will prove that Ff ·ν = f\◦ν and µ ·f = µ◦f\, so the equivalenceis clear.

First we look at the map f\ ◦ ν. We see

f\ ◦ ν = mY · F (f\) · ν = mY · F (eY · f) · ν = mY · FeY · Ff · ν.

Diagrammatically this is

Xν //

f\◦ν

��

Ff ·ν $$

FX

Ff��

F (eY ·f)

��

FY

FeY��

FY FFYmYoo


To prove the equality we need to show that

mY · FeY = 1FY

holds. This is the left unit law, so Ff · ν = f\ ◦ ν.Finally we look at the map µ ◦ f\. This is equal to

mY · Fµ · f\ = mY · Fµ · eY · f.

We can denote this diagrammatically as follows

Xf//

µ◦f\��

f\

))Y eY

//

µ}}

FY

Fµ��

FY FFYmYoo

If we prove that the diagram is commutative, then we get our equality. Todo this, we show mY · Fµ · eY = µ. For all y ∈ Y, we have

(mY · Fµ · eY )(y) = (mY · Fµ)(y)

= mY ({F ⊆ FY | µ−1(F ) ∈ y})= mY ({F ⊆ FY | y ∈ µ−1(F )})= {A ⊆ Y | AF ∈ {F ⊆ FY | y ∈ µ−1(F )}}= {A ⊆ Y | y ∈ µ−1(AF)}= {A ⊆ Y | µ(y) ∈ AF}= {A ⊆ Y | A ∈ µ(y)}= µ(y).

�

Not only do the neighborhoodfilters of topological spaces have propertiesnicely expressed in the language of the Kleisli category of F, but the con-ditions (5.1), (5.2) and (5.3) are sufficient to describe topological spaces andcontinuous maps.

Proposition 5.1.1. The category Top of topological spaces and continuousmaps is isomorphic to the category F−Mon whose objects are pairs (X, ν),with ν : X → FX a monoid in SetF(X,X) satisfying

ν ◦ ν ≤ ν, eX ≤ ν

and whose morphisms f : (X, ν)→ (Y, µ) are maps f : X → Y such that

f\ ◦ ν ≤ µ ◦ f\.


Proof. The previous discussion shows that the neighborhoodfilters of a topo-logical space define a map ν : X → FX satisfying the required conditions.

Conversely, given a monoid ν : X → FX in SetF(X,X), we define opensets as those U ⊆ X that are neighborhoods of each of their points:

U ∈ OX ⇐⇒ ∀x ∈ X : (x ∈ U =⇒ U ∈ ν(x)).

We check now that this defines a topology. Obviously ∅ is an open set,since there is no condition when U = ∅. Also X is an open set, due to thefact that ν(x) is a non-empty filter and therefore X ∈ ν(x) for every x ∈ X.

Take Ui ∈ OX for i ∈ I, then for every x ∈⋃i∈I Ui there exists a j ∈ I

such that x ∈ Uj. Since Uj ∈ OX we know that Uj ∈ ν(x). Because ν(x) isa filter, we have that ⋃

i∈I

Ui ∈ ν(x).

Thus⋃i∈I Ui ∈ OX.

Now we suppose that Ui ∈ OX with i ∈ I and I finite. Take x ∈⋂i∈I Ui,

hence Ui contains x for every i ∈ I, thus Ui ∈ ν(x). Since ν(x) is a filter, weknow that ⋂

i∈I

Ui ∈ ν(x).

So⋂i∈I Ui ∈ OX.

Finally we look at the morphisms. Let f : (X, ν) → (Y, µ) be a mapsuch that Ff · ν ≤ µ · f and U ∈ OY. Then for every x ∈ f−1(U) we havethat f(x) ∈ U and therefore U ∈ µ(f(x)). Due to the inequality we knowthat U ∈ (Ff · ν)(x). Thus f−1(U) ∈ ν(x), by definition. We now get thatf−1(U) ∈ OX.

One therefore has two functors Top→ F-Mon and F-Mon → Top whosecomposites are identical on Top and F-Mon. �

5.2 Power-enriched monads

The passage from neighborhood to convergence structure presented furtheron in 5.5 exploits the interaction of filters and relations via the principal filtermonad morphism τ : P → F whose components τX : PX → FX send a setA ∈ PX to the principal filter A ∈ FX. This monad morphism allows toplace the study of neighborhood systems, appearing in Proposition 5.1.1 asmorphisms of the Kleisli category SetF, in a more general context. In viewof this, the following proposition recalls that a monad morphism τ : P → T

relates SetT with both Rel ∼= SetP and Sup ∼= SetP via functors

Rel→ SetT → Sup.


Proposition 5.2.1. For a monad T = (T,m, e) on Set, one has a one-to-onecorrespondence between:

(i) monad morphisms τ : P→ T,

(ii) extensions E of the functor FT : Set→ SetT along the functor(−)◦ : Set→ Rel :

Rel E // SetT

Set

(−)◦

OO

FT

<<

(iii) liftings L of the functor GT : SetT → Set along the forgetful functorSup→ Set :

SetTL //

GT ""

Sup

��Set

(iv) complete lattice structures on TX such that Tf : TX → TY andmX : TTX → TX are supmaps for all maps f : X → Y and sets X.

Proof. To simplify the proof, we identify Rel with SetP (Example 4.3.1) andSup with SetP (Example 4.1.1).

(i) ⇐⇒ (ii). The functor E sends a SetP-morphism r : X → PY to themap

Er = τY · r : X → TY.

(i) ⇐⇒ (iii). The functor L sends a map f : X → TY to the P-homomorphism

mY · Tf : (TX,mX · τTX)→ (TY,mY · τTY ).

(iii) ⇐⇒ (iv). The functor GT of (iii) sends a map g : X → TY tomY · Tg : TX → TY, so that (with g = 1TY or g = eTY · f) condition (iv) isjust an elementwise restatement of (iii). �

For a morphism τ : P → T of monads on Set, condition (iii) equips theunderlying set TX of a free T-algebra with the separated order given by

R ≤ T ⇐⇒ mX · τTX({R, T }) = T , (5.4)

for all R, T ∈ TX. The hom-sets Set(X, TY ) become separated ordered setsvia the induced pointwise order

f ≤ g ⇐⇒ ∀x ∈ X : f(x) ≤ g(x),


for all f, g : X → TY. Composition on the right is always monotone, butcomposition on the left (−)T · f : SetT(Y, Z) → SetT(X,Z) may fail to beso (where (−)T = mY · T (−)). To remedy this and therefore make SetTinto a separated ordered category (Definition 3.1.1), it suffices that (−)T bemonotone; for all f, g : X → TY :

f ≤ g ⇐⇒ fT ≤ gT.

If this condition is satisfied, then the functors E : Rel→ SetT and L : SetT → Supof Proposition 5.2.1 becomes 2-functors between ordered categories.

Definition 5.2.2. A power-enriched monad is a pair (T, τ) composed ofa monad T on Set and a monad morphism τ : P→ T such that

f ≤ g ⇐⇒ fT ≤ gT (5.5)

for all f, g : X → TY . A morphism α : (S, σ) → (T, τ) of power-enriched monads is a monad morphism α : S→ T such that τ = α · σ :

Pσ

��

τ

��S α

// T.

When working with the power-enriched monads (T, τ), we will often as-sume a fixed choice of τ , and speak of “the power-enriched monad T”.

Examples 5.2.3. 1. The terminal monad I, whose functor sends all setsto a singleton {?}, is power-enriched. There is a unique map ! : P→ I,with components

!X : PX → {?}.

2. The powerset monad with the identity structure 1P : P → P is power-enriched. Hence, (P, 1P) is an initial object in the category of power-enriched monads and their morphisms. The order on the sets PXcoming from (5.4) is simply the subset inclusion because the supremumoperation is given by arbitrary union. For every A and B subsets of Xwe have (B ≤ A)

A = mX · τPX({A,B}) = mX{A,B} = A ∪B,

so B ⊆ A.


3. The filter monad F is power-enriched via the principal filter naturaltransformation τ : P → F that yields a monad morphism τ : P → F.The order on FX coming from (5.4) is the refinement order introducedin the beginning of this chapter.

Proof. We first prove that the order on FX via (5.4) is equal to therefinement order. So let G ≤ F for G,F ∈ FX, then we have

F = mX · τFX({F ,G}) = mX˙{F ,G} = {A ⊆ X | AF ∈ ˙{F ,G}},

thus F ⊆ G.Finally we prove that F is power-enriched, thus (5.5) is true for allf, g : X → FY. We assume that f ≤ g and thus for every x ∈ X, wehave g(x) ⊆ f(x). Now take a filter F ∈ FX and look at the followingcomputation:

gF(F) = (mY · Fg)(F)

= mY ({G ⊆ FY | g−1(G ) ∈ F})= {A ⊆ Y | AF ∈ {G ⊆ FY | g−1(G ) ∈ F}}= {A ⊆ Y | g−1(AF) ∈ F}= {A ⊆ Y | {x ∈ X | ∃G ∈ AF : g(x) = G} ∈ F}= {A ⊆ Y | {x ∈ X | ∃G ∈ FY : g(x) = G and A ∈ G} ∈ F}= {A ⊆ Y | {x ∈ X | A ∈ g(x)} ∈ F}⊆ {A ⊆ Y | {x ∈ X | A ∈ f(x)} ∈ F}= fF(F),

where the inclusion holds due to the fact that g(x) ⊆ f(x) and F is afilter. �

4. The ultrafilter monad is not power-enriched: for the set X = ∅, oneobserves that βX = ∅ cannot be a complete lattice.

5. The up-set monad U has at least two different structure morphismsσ, τ : P→ U defined componentwise for A ∈ PX by

σX(A) = {B ⊆ X | A ∩B 6= ∅} and τX(A) = {B ⊆ X | A ⊆ B}.

It is easy to see that both σX(A) and τX(A) are up-closed sets (τ isjust the extension of the principal filter natural transformation). Theorder on UX induced by σ is given by subset inclusion, while the one


induced by τ is opposite, that is, τ induces the refinement order onup-sets:

∀S, T ∈ UX : S ≤ T ⇐⇒ S ⊇ T .

These morphisms demonstrate that the morphism P→ T given with apower-enriched monad is indeed a structure and not a property of themonad.

Proof. We first prove that the orders induced by σ and τ are respectivelythe subset inclusion and the refinement order. So for the morphism σand S, T ∈ UX with S ≤ T we have

T = mX · σUX({T ,S})= mX({S ⊆ UX | T ∈ S or S ∈ S })= {A ⊆ X | AU ∈ {S ⊆ UX | T ∈ S or S ∈ S }}= {A ⊆ X | T ∈ AU or S ∈ AU}= {A ⊆ X | A ∈ T or A ∈ S}= T ∪ S.=⇒ S ⊆ T .

And for the morphism τ and S, T ∈ UX with S ≤ T we have

T = mX · τUX({T ,S})= mX({S ⊆ UX | T ∈ S and S ∈ S })= {A ⊆ X | AU ∈ {S ⊆ UX | T ∈ S and S ∈ S }}= {A ⊆ X | T ∈ AU and S ∈ AU}= {A ⊆ X | A ∈ T and A ∈ S}= T ∩ S.=⇒ S ⊇ T .

The proof of condition (5.5), to be a power-enriched monad, is completelyanalogous to the proof for the filter monad. �

6. The clique monad C = (C,m, e) is power-enriched via the principalclique natural transformation τ : P→ C. The order on CX induced byτ is also the refinement order (opposite order on cliques).


Proof. Take two cliques C,D ∈ CX with C ≤ D, then we have

D = mX · τCX({C,D})= mX({C ⊆ CX | C ∈ C and D ∈ C })= {A ⊆ X | AC ∈ {C ⊆ CX | C ∈ C and D ∈ C }}= {A ⊆ X | C ∈ AC and D ∈ AC}= {A ⊆ X | A ∈ C and A ∈ D}= C ∩ D.=⇒ C ⊇ D.

The proof of condition (5.5), to be a power-enriched monad, is completelyanalogous to the proof for the filter monad. �

7. The ultraclique monad is not power-enriched: for the set X = {x, y, z},one observes that κX = {x, y, z, {{x, y} , {x, z} , {y, z} , {x, y, z}}} can’tbe a complete lattice, because κX does not have a biggest element.

5.3 T-monoids

Motivated by Proposition 5.1.1, we introduce the category of monoids in thehom-sets of a Kleisli category.

Definition 5.3.1. Let T = (T,m, e) be a monad on Set whose Kleisli categorySetT is a separated ordered category. The category T−Mon of T-monoids(or Kleisli monoids) has as object pairs (X, ν), where X is a set and itsstructure ν : X → TX is a transitive and reflexive Set-morphism:

ν ◦ ν ≤ ν, eX ≤ ν

(where ◦ is the composition of the Kleisli category SetT). A morphismf : (X, ν)→ (Y, µ) is a Set-morphism f : X → Y satisfying

Tf · ν ≤ µ · f or equivalently f\ ◦ ν ≤ µ ◦ f\,

where f\ = eY · f. In the case where T = (T, τ) is a power-enriched monad,the order on the hom-sets of SetT depends on τ ; however we will often assumethat τ is given implicitly and denote a category of Kleisli monoids by T−Monrather than by (T, τ)−Mon.


Remark 5.3.2. In presence of the extensivity condition, idempotency canbe expressed as an equality ν ◦ ν = ν, since

ν = ν ◦ eX ≤ ν ◦ ν ≤ ν.

Idempotent structures are also preserved by the functor GT = (−)T : SetT →Set :

νT · νT = (ν ◦ ν)T = νT.

Examples 5.3.3. 1. In the case where T = I is the terminal monad,Kleisli monoids are simply pairs (X, !X : X → {?}), and morphism aremaps f : X → Y . In other words, the category of Kleisli monoids isisomorphic to Set:

I−Mon ∼= Set.

2. In the case of the powerset monad (together with its identity structure1P), P−Mon is the category of the ordered sets.

A map ν : X → PX is precisely a relation on X, and the transitivityand reflexivity condition translate as transitivity and reflexivity of ν.Because the set PX is ordered by set-inclusion, ν is the down-set map

↓X : X → PX : x 7→↓X x = {y ∈ X | y ≤ x}.

A map f : X → Y is a morphism of P−Mon if and only if it preservesthe relations, that is, if and only if f is a monotone map. Hence,

P−Mon ∼= Ord.

3. Proposition 5.1.1 and Example 5.2.3.(3) show that when F is equippedwith the principal filter morphism τ : P → F, F−Mon is the categoryof topological spaces and continuous maps:

F−Mon ∼= Top.

4. With the principal filter morphism τ : P→ U of Examples 5.2.3.(5), thecategory of U-monoids is isomorphic to the category of interior space:

U−Mon ∼= Int

(see Proposition 5.3.4). In fact, the monad morphism σ : P → U ofExample 5.2.3.(5) yields

U−Mon ∼= Cls,

so that the structures τ and σ return isomorphic categories of U-monoids.


Proposition 5.3.4. The category Int of interior spaces and continuous func-tions is isomorphic to the category U−Mon, with power-enrichedment τ : P→ U,the principal filter morphism.

Proof. Define η : X → UX as

A ∈ η(x)⇔ ∃B ⊆ X : (d(B) = B and x ∈ B ⊆ A).

Then we have that η(x) is an up-set, because

↑PX η(x) = {C ⊆ X | ∃A ∈ η(x) : A ⊆ C}= {C ⊆ X | ∃B ⊆ X : d(B) = B and x ∈ B ⊆ A ⊆ C}= {C ⊆ X | C ∈ η(x)}= η(x).

For A ⊆ X and x ∈ X, we have the following chain of implications:

A ∈ η(x)⇒ x ∈ A⇒ A ∈ eX(x),

thus η(x) ⊆ eX(x), and soeX ≤ η.

To find the second condition for a U-monoid structure morphism, we lookat the following chain of equivalences:

A ∈ η(x)⇔ ∃B ∈ η(x) : (y ∈ B ⇒ A ∈ η(y)) (d(B) = B)

⇔ ∃B ∈ η(x) : (y ∈ B ⇒ η(y) ∈ AU)

⇔ ∃B ∈ η(x) : B ⊆ η−1(AU)

⇔ η−1(AU) ∈ η(x)

⇔ AU ∈ Uη · η(x)

⇔ A ∈ mX · Uη · η(x)

⇔ A ∈ η ◦ η(x).

Therefore we haveη ◦ η ≤ η.

For a continuous function f : (X, dX)→ (Y, dY ) we have

A ∈ ηY (f(x))⇔ ∃B ⊆ Y : (dY (B) = B and f(x) ∈ B ⊆ A).

We also know that

dX(f−1(B)) ⊆ f−1(B) = f−1(dY (B)) ⊆ dX(f−1(B)),


where the first inclusion follows from the fact that dX ≤ 1X , the equalityholds for subsets B such that B = dY (B) and the last inclusion follows fromcontinuity of f . By the previous statement there exists f−1(B) ⊆ X suchthat

dX(f−1(B)) = f−1(B) and x ∈ f−1(B) ⊆ f−1(A).

This means f−1(A) ∈ ηX(x), or A ∈ Uf · ηX(x).Conversely, we define an interior space (X, d) as follows

d : PX → PX : A 7→ d(A),

withx ∈ d(A)⇔ A ∈ η(x).

Then we have for A ⊆ X, x ∈ X

x ∈ d(A)⇔ A ∈ η(x)⇒ x ∈ A,

that is, d ≤ 1X .We also have

x ∈ d(A)⇔ A ∈ η(x)⇒ A ∈ η ◦ η(x)⇔ x ∈ d(d(A)),

so, d ≤ d · d.Finally we have for B ⊆ Y and f : X → Y, with ηY · f ≤ Uf · ηX ,

x ∈ f−1(dY (B))⇒ f(x) ∈ dY (B)

⇒ B ∈ ηY (f(x))

⇒ B ∈ Uf · ηX(x)

⇒ f−1(B) ∈ η(x)

⇒ x ∈ dX(f−1(B)).

Thus f is continuous.We have proven

Int ∼= U−Mon.

�

Proposition 5.3.5. There is an isomorphism

C−Mon ∼= U−Mon.

So C−Mon is isomorphic to Cls.


Proof. Because every clique is an up-set, each C-monoid (X, η) is an U-monoid.

Every U-monoid (X, ν) is also a C-monoid, since an up-set of the formν(x) is a clique. Take two non-empty subsets of X belonging to ν(x). Theyboth contain x (eX ≤ ν) and therefore their intersection is not empty. Soν : X → UX corestricts to ν : X → CX.

The definitions of the monads are similar, because C is a submonad of Uand with the observations above, we see that every C-monoid is isomorphicto an U-monoid and vice versa. �

Proposition 5.3.6. A morphism of power-enriched monads α : (S, σ) →(T, τ) induces a functor S−Mon → T−Mon that sends (X, ν) to (X,αX · ν)and commutes with the underlying-set functors.

Proof. The functor Setα: SetS → SetT sends ν to αX · ν. The claim nowfollows because of the fact that αX is monotone (Lemma 5.3.7). �

Lemma 5.3.7. Let α : (S, σ) → (T, τ) be a morphism of power-enrichedmonads, then αX is monotone for every set X.

Proof. Let S = (S, n, d) and T = (T,m, e) be two power-enriched monads,with structure σ, respectively τ and α a morphism between these monadswith the condition that τ = α · σ.

We first look at the following diagram

PSX

σSX��

PαX //

τSX

��

PTX

τTX��

SSX(α∗α)X//

αSX��

TTX

TSXTαX

::

We know that the big diagram commutes, because τ is a natural transfor-mation:

τTX · PαX = TαX · τSX .We also know that the lower diagram commutes, because of the definition ofthe Godement product ∗:

TαX · αSX = (α ∗ α)X .

Thus the upper square commutes:

τTX · PαX = TαX · τSX = TαX · αSX · σSX = (α ∗ α)X · σSX .


Then we look at the following diagram:

PSXPαX //

σSX��

PTX

τTX��

SSX(α∗α)X//

nX��

TTX

mX��

SX αX// TX

We know that the lower square commutes, because α is a monad morphismand preserves thus the monad structure (α · n = m · (α ∗ α)). So the wholediagram commutes.

We assume that J ≤ K, for J ,K ∈ SX. It means K = nX ·σSX({J ,K}).Then we get

mX · τTX({αX(J ), αX(K)}) = mX · τTX(PαX{J ,K})= αX · nX · σSX({J ,K})= αX(K).

So αX(J ) ≤ αX(K). Thus αX is monotone. �

Example 5.3.8. The monad morphism P → F between power-enrichedmonads induces a functor

Ord→ Top

that provides every ordered set (X,≤) with the Alexandroff topology, that is,the topology whose open sets are generated by the down-sets ↓ x for x ∈ X.

5.4 The Kleisli extension

Categories of lax algebras depend upon the lax extension of a monad T onSet to Rel. The Barr extension provides a construction of a lax extensionby viewing a relation r : X →7 Y as a composite r = q · p◦ (where p, q areprojection maps); the Kleisli extension introduced below exploits relationsas morphisms of the Kleisli category SetP = Rel. Hence, we will often beworking with maps r : X → PY representing relations r : X →7 Y andwill indifferently use the notation P = (P,∪, {−}) or (P, (−)P, {−}) for thepowerset monad and T = (T,m, e) or (T, (−)T, e) for an arbitrary monad onSet, together with the corresponding expressions for natural transformationsτ : P→ T.


Let us denote by (−)[ : Relop → SetP the functor that is identical on setsand sends a relation r : X →7 Y to the map r[ : Y → PX representing theopposite relation r◦ : Y →7 X :

x r y ⇐⇒ x ∈ r[(y).

By composition with the functors E : SetP → SetT and L : SetT → Sup ofProposition 5.2.1, one obtains a functor

(−)τ : Relop(−)[ // SetP

E // SetTL // Sup

that sends a setX to TX and a relation r : X →7 Y to the map rτ : TY → TXdefined by

rτ := mX · T (τX · r[) = (τX · r[)T.

Definition 5.4.1. Given a power-enriched monad (T, τ), the Kleisli extensionT of T to Rel (with respect to τ) is described by the function T = TX,Y :Rel(X, Y )→ Rel(TX, TY ) (indexed by sets X and Y ), with

T (T r) U ⇐⇒ T ≤ rτ (U) (5.6)

for all relations r : X →7 Y and T ∈ TX,U ∈ TY, or, equivalently,

(T r)[ =↓TX ·rτ : TY → PTX.

Examples 5.4.2. 1. In the case of the terminal power-enriched monad(I, !), the Kleisli extension of a relation r : X →7 Y is {?} →7 {?} withconstant value > (top element of an ordered set).

2. To obtain an explicit description of the Kleisli extension of the powersetmonad (P, 1P), observe that

A ⊆ r1P(B) ⇐⇒ A ⊆⋃

X · Pr[(B)

⇐⇒ ∀x ∈ A ∃y ∈ B : x ∈ r[(y) ⇐⇒ A ⊆ r◦(B),

for a relation r : X →7 Y and A ∈ PX,B ∈ PY, where

r◦(B) = {x ∈ X | ∃y ∈ B : x ∈ r[(y)}.

HenceA (P r)B ⇐⇒ A ⊆ r◦(B).

Here we obtain the lax extension P (see 3.2.2).


3. Let T = F be the filter monad and τ : P→ F the principal filter naturaltransformation. For a relation r : X →7 Y , A ⊆ X and G ∈ FY, wehave

A ∈ mX · F (τX · r[)(G) ⇐⇒ (τX · r[)−1(AF) ∈ G⇐⇒ {y ∈ Y | A ∈ τX · r[(y)} ∈ G⇐⇒ {y ∈ Y | r[(y) ⊆ A} ∈ G⇐⇒ ∃B ∈ G : r◦(B) ⊆ A.

This shows precisely that rτ (G) =↑PX {r◦(B) | B ∈ G}, so

F (F r) G ⇐⇒ F ⊇ rτ (G)

or, if we use the notation of an image filter r◦[G],

F (F r) G ⇐⇒ F ⊇ r◦[G]. (5.7)

The Kleisli extension of the filter monad returns the lax extension F(see 3.4.3).

4. The Kleisli extension of the up-set monad U, equipped with the principalfilter natural transformation τ : P → U, is obtained as for the filtermonad in the previous example so that

R (Ur) S ⇐⇒ R ⊇ r◦[S]

for all maps r : X → PY and up-sets R ∈ UX,S ∈ UY.

5. The Kleisli extension of the up-set monad U, equipped with the struc-ture morphism σ, is something similar to the lax extensions P and F .Indeed, for a relation r : X →7 Y , A ⊆ X and S ∈ UY we have

A ∈ mX · U(σX · r[)(S) ⇐⇒ (σX · r[)−1(AU) ∈ S⇐⇒ {y ∈ Y | A ∈ σX · r[(y)} ∈ S⇐⇒ {y ∈ Y | r[(y) ∩ A 6= ∅} ∈ S⇐⇒ {y ∈ Y | ∃x ∈ A : x r y} ∈ S⇐⇒ r(A) ∈ S.

Thus rσ(S) = {A ⊆ X | r(A) ∈ S}. This gives us

R (Ur) S ⇐⇒ R ⊇ rσ(S)

⇐⇒ r[R] ⊆ S.


6. The Kleisli extension of the clique monad C, equipped with the principalfilter natural transformation τ : P → C, is obtained as for the up-setmonad in Example 5.4.2.4 so that

C (Cr) D ⇐⇒ C ⊇ r◦[D]

for all maps r : X → PY and cliques C ∈ CX,D ∈ CY.

To prove that T : Rel → Rel is indeed a lax extension of the Set-functorT , it is convenient to express the former as a composite of lax functors. Inview of this, we remark that T r (for a relation r : X →7 Y ) can be written as

T r = (r[)∗ : TX →7 TY,

where (−)∗ : Ord → Modop is the functor that sends a monotone map f :X → Y to the module f ∗ = f ◦· ≤Y : Y →◦ X.

The Kleisli extension is therefore a functor

T op : Relop(−)τ // Sup // Ord

(−)∗ //Modop

(where Sup→ Ord is the forgetful functor). There is moreover a lax functorMod → Rel that assigns to a module its underlying relation: compositionof modules is composition of relations, identity modules are order relationsand 1X ≤ (≤X) for any ordered set X. Hence, with E : SetP → SetT andL : SetT → Sup denoting the functors from (ii) and (iii) of Proposition 5.2.1,the Kleisli extension T op can be composed as the top line of the commutativediagram

Relop(−)[ // SetP

E // SetTGT

""

L // Sup

��

// Ord

||

(−)∗ //Modop // Relop

Set(−)◦

cc

FP

OO

FT

;;

T// Set

(5.8)

in which all arrows except Modop → Relop are functors and Modop → Relop isa lax functor that fails only to preserve identities (the unnamed arrows areall forgetful).

Proposition 5.4.3. Given a power-enriched monad (T, τ), the Kleisli exten-sion T of T to Rel yields a lax extension T = (T ,m, e) of T = (T,m, e) toRel.


Proof. The fact that T : Rel→ Rel is a lax functor follows from its decom-position as lax functor preserving composition in the first line of (5.8). Thelax extension condition (Tf)◦ ≤ T (f ◦) can be deduced from the diagram

Relop(−)[ // Ord

��

(−)∗ //Modop // Relop

Set

(−)◦OO

T// Set

(−)◦

44

≥

in which the first line is T op.The second lax extension condition T (h◦ · r) = (Th)◦ · T r for all relations

r : X →7 Y and maps h : Z → Y (see Proposition 3.2.3.(3)) comes from theequivalences (for all T ∈ TX,U ∈ TZ):

T (T (h◦ · r)) U ⇐⇒ T ≤ rτ · (h◦)τ (U)

⇐⇒ T ≤ rτ · Th(U) ⇐⇒ T ((Th)◦ · T r) U .

To verify oplaxness of e : 1Rel → T , we use that τX = mX · TeX · τX =mX · τTX · PeX =

∨X · PeX (with

∨X : DnX → X : A 7→

∨XA and

∀x ∈ X :∨

XA ≤ x ⇐⇒ A ⊆↓ x). The order on TX is given by themorphism τ : P→ T (see (5.4)), therefore mX · τTX =

∨X . Given a relation

r : X →7 Y and x ∈ X, y ∈ Y with x r y, one has

eX(x) ≤∨

x′∈r[(y)

eX(x′)

= τX · r[(y)

= (τX · r[)T · eY (y) (see (2.1))

= rτ · eY (y)

as required.For proving oplaxness of m : T T → T , recall that mX · τTX · ↓TX=∨

X · ↓TX= 1TX , and note that

(rτ )T = (rτ · 1TY )T

= ((τX · r[)T · 1TY )T

= (τX · r[)T · 1TTY (see (2.1))

= rτ ·mY (Proposition 2.2.3).


Thus, if T ∈ TTX and U ∈ TTY are such that T (T T r) U , orequivalently T ≤ (T r)τ (U ) then

mX(T ) ≤ mX · (T r)τ (U )

= 1TTX · (τTX · ↓TX ·rτ )T(U )

= (1TTX · τTX · ↓TX ·rτ )T(U ) (see (2.1))

= (rτ )T(U )

= rτ ·mY (U ),

which concludes the proof. �

Remark 5.4.4. Since the Kleisli extension provides the monad T with a laxextension, there is a natural order on TX associated with T (see 4.9); onTX there is also the order (5.4) induced by the monad morphism τ : P→ T.Since the first order T1X is defined via the second:

T (T1X) U ⇐⇒ T ≤ U

(definition (5.6)) the orders are equivalent.

5.5 Topological spaces via filter convergence

In this subsection, we show that (F, 2)−Cat is isomorphic to F−Mon ∼= Top(Proposition 5.1.1), that is we present topological spaces as sets equippedwith a transitive and reflexive convergence relation a : FX →7 X. Thecorrespondence between convergence and neighborhoods can be formalizedvia maps

conv : Set(X,FX)→ Rel(FX,X) and nbhd : Rel(FX,X)→ Set(X,FX).

In fact, one can without further thought replace the filter monad F with apower-enriched monad (T, τ). By identifying Rel(TX,X) with Set(X,PTX),isomorphic as ordered sets (via (−)[), we define

conv(ν) =↓TX ·ν and nbhd(r) =∨

TX · r[

for all maps ν : X → TX and relations r : TX →7 X. In pointwise notation,these maps may be written as

T conv(ν) y ⇐⇒ T ≤ ν(y) and nbhd(r)(y) =∨{T ∈ TX | T ∈ r[(y)}

for all y ∈ X and T ∈ TX, as a direct generalization of the fact that, in atopological space X, a filter F converges to a point y precisely when F isfiner than the neighborhood filter of y.


Proposition 5.5.1. With Set(X,TX) and Rel(TX,X) ordered pointwise,the monotone maps defined above form an adjunction

nbhd a conv : Set(X,TX)→ Rel(TX,X)

for all sets X. Moreover, the fixpoints of (conv · nbhd) are precisely the uni-tary relations, and conv is fully faithful, so that the fixpoints of (nbhd · conv)are the maps ν : X → TX.

Proof. The equivalence

nbhd(r) ≤ ν ⇐⇒ r[ ≤ conv(ν)

(for all maps ν : X → TX and relations r : TX →7 X) follows directly fromthe adjunction

∨X a ↓TX (

∨A ≤ x ⇐⇒ A ⊆ ↓ x).

Similarly, from∨

X · ↓TX= 1TX follows that nbhd · conv = 1, that is, convis fully faithful (see 1.4.2).

Remark 5.4.4 shows that (T1X)[ = ↓TX . Hence, if r : TX →7 X isunitary, then

r[ = (↓TX)P · r[ (r right unitary in 4.4.4, (s · r)[ = (r[)P · s[))= (↓TX)P · (e◦X · T r ·m◦X)[ (r left unitary, e◦X · T r ·m◦X = r)

= (↓TX)P · ((m◦X)[)P · (T r)[ · eX ((e◦X)[ = {−}TX · eX , fP · {−}TX = f)

= (↓TX ·mX)P · (T r)[ · eX (gP · fP = (gP · f)P, (m◦X)[ = {−}TX ·mX ,

fP · {−}TX = f)

= (↓TX ·mX)P· ↓TTX ·(τTX · r[)T · eX (definition of T )

= (↓TX ·mX)P· ↓TTX ·τTX · r[ (fT · eX = f)

≥ (↓TX ·mX)P · {−}TTX · τTX · r[ (↓TTX≥ {−}TTX)

=↓TX ·mX · τTX · r[ (fP · {−}X = f)

= conv · nbhd(r) (mX · τTX =∨

TX)

≥ r[ (nbhd a conv)

and we may conclude that conv · nbhd(r) = r (via the understood identi-fication of Rel(TX,X) with Set(X,PTX)).

Conversely, if r is a fixpoint of conv · nbhd, then r is of the form conv(ν)for some ν : X → TX, and one sees that r · T1X ≤ r, so r is right unitary.To prove that r is left unitary, we must verify that e◦X · T r ≤ r ·mX .

Thus, suppose that T (T r) eX(y) holds. By definition of T , we have

T ≤ rτ · eX(y) or equivalently T ≤ τTX · r[(y).


Applying mX to each side of this inequality, we obtain

mX(T ) ≤ mX · τTX · r[(y).

This means precisely that mX(T ) ≤∨

TX · r[(y) = nbhd(r)(y), or

mX(T )(conv · nbhd(r)) y

which is mX(T ) r y by the fixpoint condition. �

Proposition 5.5.2. The adjoint maps nbhd and conv defined above aremonoid homomorphisms between SetT(X,X) and (T, 2)−URelop(X,X), thatis, they satisfy

nbhd(s ◦ r) = nbhd(r) ◦ nbhd(s), conv(µ) ◦ conv(ν) = conv(ν ◦ µ)

nbhd(1]X) = eX , conv(eX) = 1]X

for all unitary relations r, s : TX →7 X and maps µ, ν : X → TX.

Proof. The equality nbhd(1]X) = eX follows immediately from the definitionof 1]X , as

T 1]X y ⇐⇒ T (e◦X · T1X) y ⇐⇒ T ≤ eX(y)

for all T ∈ TX and y ∈ X. The multiplication mX = 1TTX of the monad T isa sup-map and 1TTX · τTX =

∨TX (Proposition 5.2.1), so

1TTX · τTX · PmX · ↓TTX=∨

TX · PmX · ↓TTX= mX ·∨

TX · ↓TTX= 1TTX .

By definition of T , we obtain

1TTX ·(τTX ·r[)T = 1TTX ·τTX ·PmX · ↓TTX ·(τTX ·r[)T = 1TTX ·τTX ·PmX ·(T r)[.(5.9)


Therefore

nbhd(s) ◦ nbhd(r) = (1TTX · τTX · r[)T · 1TTX · τTX · s[ (∨

TX = 1TTX · τTX)

= ((1TTX · τTX · r[)T)T · τTX · s[ (gT · fT = (gT · f)T)

= (1TTX · (τTX · r[)T)T · τTX · s[ ((gT · f)T = gT · fT)

= (1TTX · τTX · PmX · (T r)[)T · τTX · s[ (see (5.9))

= 1TTX · (τTX · PmX · (T r)[)T · τTX · s[ ((gT · f)T = gT · fT)

= 1TTX · τTX · (PmX · (T r)[)P · s[ (naturality of τ)

=∨

TX ·((

(m◦X)[)P· (T r)[

)P· s[ (PmX = ((m◦X)[)P,

1TTX · τTX =∨

TX)

=∨

TX · (s · T r ·m◦X)[ (Rel = SetP)

= nbhd(s ◦ r).

The equalities for conv follow directly from the fact that conv and nbhd areinverse of each other on fixpoints (Proposition 5.5.1). �

Theorem 5.5.3. Given a power-enriched monad (T, τ) equipped with itsKleisli extension T , there is an isomorphism

(T, 2)−Cat ∼= T−Mon

that commutes with the underlying-set functor.

Proof. For a (T, 2)-category (X, r), Proposition 5.5.2 implies that (X, nbhd(r))is a T-monoid, and conversely, if (X, ν) is a T-monoid, then (X, conv(ν))is a (T, 2)-category (one also use the fact that nbhd and conv are mono-tone). Moreover, Proposition 5.5.1 states that the objects are in bijectivecorrespondence.

We are therefore left to show that this correspondence is functorial.Consider first a (T, 2)-functor f : (X, r) → (Y, s), so that r · (Tf)◦ ≤ f ◦ · s.


We have

Tf · nbhd(r) = Tf ·∨

TX · r[

=∨

TX · PTf · r[ (f is a sup-map)

=∨

TX · (r · (Tf)◦)[ (PTf = ((Tf ◦)[)P)

≤∨

TX · (f ◦ · s)[ (f is a (T, 2)-functor)

=∨

TX · (s[)P · {−}X · f ((f ◦)[ = {−}X · f)

=∨

TX · s[ · f (gP · {−}X = g)

= nbhd(s) · f ;

hence, f is a morphism of T-monoids.Consider now f : (X, ν)→ (Y, µ) satisfying

Tf · ν ≤ µ · f.

Then T conv(ν) y means T ≤ ν(y), so we have

Tf(T ) ≤ Tf · ν(y) ≤ µ · f(y),

and can therefore conclude that Tf(T ) conv(µ) y; that is, f is a (T, 2)-functor between the corresponding (T, 2)-categories. �

Corollary 5.5.4. The category Top of topological spaces is isomorphic tothe category (F, 2)−Cat whose object are pairs (X, a), with a : FX →7 X arelation representing convergence and, when a and F a are denoted by −→,satisfying

F −→ F and F −→ z =⇒ ΣF −→ z

andx −→ x,

for all x, z ∈ X, F ∈ FX and F ∈ FFX; here F −→ F ⇐⇒ F ⊇ a◦[F ].The morphisms are the convergence-preserving maps f : X → Y :

F −→ y =⇒ f [F ] −→ f(y)

for all y ∈ X and F ∈ FX.

Proof. Proposition 5.1.1 together with the previous Theorem yield an iso-morphism between Top and (F, 2)−Cat, and the statement is just an explicit


desription of the latter category using the Kleisli extension of the filter monad((5.7)). �

Naturally the same statement holds for the category Cls of closure spaces,for which we give the more synthetic form below.

Corollary 5.5.5. For the up-set monad U equipped with the Kleisli exten-sion associated with the principal filter natural transformation, there is anisomorphism

Cls ∼= (U, 2)−Cat

that commutes with the underlying-set functors. An object (X, a) of (U, 2)−Catrepresents a convergence structure for closure spaces. When a and Ua aredenoted by −→, then we have

S −→ S and S −→ z =⇒ mX(S ) −→ z

andx −→ x,

for all x, z ∈ X, S ∈ UX and S ∈ UUX; here S −→ S ⇐⇒ S ⊇ a◦[S].

Proof. This is another application of Theorem 5.5.3 and the fact thatU−Mon ∼= Cls. �

Remark 5.5.6. For a power-enriched monad (T, τ) the monoid homomorphismsnbhd and conv can be extended to yield monotone maps

nbhd = nbhdX,Y :Rel(TX, Y )→ Set(Y, TX)

conv = convX,Y :Set(Y, TX)→ Rel(TX, Y )

that form an adjunction nbhd a conv for all sets X, Y. If we equip T with itsKleisli extension T, one has

nbhd(s ◦ r) = nbhd(r) ◦ nbhd(s), conv(µ) ◦ conv(ν) = conv(ν ◦ µ)


for all unitary relations r, s : TX →7 Y and maps µ, ν : Y → TX. Thereforethe Kleisli extension T is associative. Indeed, the associativity from the mapsµ, ν and η : Y → TX will correspond by this adjunction to the associativityof the unitary relations r, s, t : TX →7 Y.

Chapter 6

Initial extensions

We effectively established an isomorphism (F, 2)−Cat → (�, 2)−Cat, bothcategories being isomorphic to Top. It turns out that this isomorphism maybe thought of as induced by the monad morphism �→ F. More generally, inthis chapter we seek sufficient conditions for a monad morphism α : S → T

into a power-enriched monad T to induce an isomorphism

Aα : (T, 2)−Cat→ (S, 2)−Cat

when S and T are equipped with adequate lax extensions; here, Aα is thealgebraic functor of α. This step is facilitated by the construction of laxextensions induced by a lax extension T of T to Rel. Specifically, in 6.2, we“transfer” the lax extension from T to S along α. But first we will introducethe notion of algebraic functors.

The results stated in this chapter come out of section 2 of Chapter IV of“Monoidal topology” [4]. I developed myself the result about the clique andultraclique monad and elaborated the proofs.

6.1 Algebraic functors

Consider lax extensions S, T to Rel of monads S = (S, n, d) and T = (T,m, e)on Set. A morphism of lax extensions α : (S, S) → (T, T ) is a naturaltransformation α : S → T that extends to an oplax transformation S → T ,so that

αY · Sr ≤ T r · αXfor all relations r : X →7 Y.

A monad morphism α : S→ T which is also a morphism of lax extensionsα : S → T is denoted by α : (S, S) → (T, T ). Any such natural transfor-

98

CHAPTER 6. INITIAL EXTENSIONS 99

mation α induces a functor

Aα : (T, 2)−Cat→ (S, 2)−Cat,

sending (X, a) to (X, a · αX) and mapping morphisms identically. Indeed,one has

1X ≤ a · eX = a · αX · dXand

a · αX · S(a · αX) = a · αX · Sa · SαX≤ a · T a · αTX · SαX≤ a ·mX · αTX · SαX (a · T a ≤ a ·mX , 4.2.1)

= a · αX · nX . (α · n = m · (α ∗ α))

Moreover, a (T, 2)-functor f : (X, a)→ (Y, b) is an (S, 2)-functorf : (X, a · αX)→ (Y, b · αY ) :

f · a · αX ≤ b · Tf · αX = b · αY · Sf.

The functor Aα is called the algebraic functor associated with α.

Example 6.1.1. From section 4.8 we know that (�, 2)−Cat ∼= Top. We alsohave (I, 2)−Cat ∼= Ord. With the monad morphism e : I→ � we can associatethe algebraic functor

Ae : Top→ Ord

that sends a topological space (X, a) to the ordered set (X, a · eX) whoseorder is the underlying order of the topological space (X, a). Thus Ae is theforgetful functor to Ord.

6.2 Initial extensions

In 3.4 and 5.4 two constructions of a lax extension of a monad to Rel aregiven: the Barr extension and the Kleisli extension. In practice, the Barrextension can be extracted from the Kleisli extension of a larger monad. Forexample, the Barr extension of the ultrafilter functor β : Set→ Set :

U (βr) N ⇔ U ⊇ r◦[N ]

(for all relations r : X →7 Y,U ∈ βX,N ∈ βY ) is the restriction to ultrafiltersof the Kleisli extension of the filter functor

F (F r) G ⇔ F ⊇ r◦[G]


(for all F ∈ FX,G ∈ FY ).More generally, if α : S → T is a natural transformation of Set-functors,

and T is a lax extension of T to Rel, the initial extension of S along α isthe lax extension S given by

Sr := α◦Y · T r · αX ,

for any relations r : X →7 Y. In pointwise notation, the definition becomes

Sr(S,R) = T r(αX(S), αY (R)),

for all S ∈ SX,R ∈ SY.Before showing that S is indeed a lax extension of S if T is one of T

(Proposition 6.2.1), we briefly discuss the “initial” terminology. Recall from6.1 that a morphism of lax extensions α : (S, S) → (T, T ) is a naturaltransformation α : S → T that extends to an oplax transformation S → T :

αY · Sr ≤ T r · αX ,

for all relations r : X →7 Y.If U : LXT → SetSet denotes the forgetful functor from the category of

lax extensions to Rel that sends T to T , then the initial extension is an U -initial morphism. Indeed, consider a natural transformation λ : R→ S withlax extension R of R; then λ is a morphism of lax extensions if and only ifα · λ : R→ T is one:

αY ·λY · Rr ≤ T r ·αX ·λX ⇔ λY · Rr ≤ α◦Y · T r ·αX ·λX ⇔ λY · Rr ≤ Sr ·λXfor all relations r : X →7 Y.

Proposition 6.2.1. For a lax extension T to Rel of a Set-functor T , and anatural transformation α : S → T , the initial extension S of S along α isa lax extension of S. Furthermore, if T belongs to a lax extension to Rel ofa monad T = (T,m, e) and α : S → T is a monad morphism, then S alsobelongs to a lax extension of S = (S, n, d).

Proof. Since T preserves the order on the hom-sets Rel(X, Y ), it is imme-diatly clear that S does too. Because for r, r′ : X →7 Y, we have

r ≤ r′ ⇒ T r ≤ T r′ ⇒ α◦Y · T r · αX ≤ α◦Y · T r′ · αX ⇒ Sr ≤ Sr′.

If r : X →7 Y and s : Y →7 Z are relations, then

Ss · Sr = α◦Z · T s · αY · α◦Y · T r · αX≤ α◦Z · T s · T r · αX≤ α◦Z · T (s · r) · αX= S(s · r),


because T is a lax functor. As α is a natural transformation, we have

αY · Sf = Tf · αX

which can be written Sf ≤ α◦Y ·Tf ·αX , or equivalently (Sf)◦ ≤ α◦X ·(Tf)◦·αY ,in Rel; the extension conditions for S follow because they are satisfied for T .Indeed,

Sf ≤ α◦Y · Tf · αX ≤ α◦Y · T f · αX = Sr

and(Sf)◦ ≤ α◦X · (Tf)◦ · αY ≤ α◦X · T (f ◦) · αY = S(f ◦),

for f : X → Y.Finally, suppose that T yields a lax extension of the monad T. Since

e : 1Rel → T is oplax, then so is d : 1Rel → S; indeed, α is a monad morphism,so we have α · d = e and

r ≤ e◦Y · T r · eX = d◦Y · α◦Y · T r · αX · dX = d◦Y · Sr · dX ,

as expected. To verify oplaxness of n, we use that m · Tα · αS = α · n(m · (α ∗ α) = α · n):

SSr = α◦SY · T (α◦Y · T r · αX) · αSX= α◦SY · (TαY )◦ · T T r · TαX · αSX≤ α◦SY · (TαY )◦ ·m◦Y · T r ·mX · TαX · αSX (oplaxness of m)

= n◦Y · Sr · nX ,

as required. �

From this last result one infers that in presence of the initial extension Sof S, the maps αX : SX → TX become order-embeddings with respect tothe orders 4.9 induced by the lax extensions:

S ≤ R ⇔ αX(S) ≤ αX(R)

for all S,R ∈ SX. In fact, this condition witnesses the smooth interactionof the initial and Kleisli extensions, as we will see next.

Proposition 6.2.2. A morphism α : (S, σ) → (T, τ) of power-enrichedmonads becomes a morphism α : S → T of the Kleisli extensions to Rel.

When the sets SX and TX are equipped with the orders (5.4) induced byσ and τ respectively, the components αX are order-embeddings if and only ifthe initial extension of S along α is the Kleisli extension of S.


Proof. Observe for any relation r : X →7 Y that

αX · rσ = αX · nX · S(σX · r[)= mX · αTX · SαX · SσX · S(r[) (α · n = m · (α ∗ α))

= mX · αTX · S(τX · r[) (α power-enriched monad morphism)

= mX · T (τX · r[) · αY (α natural transformation)

= rτ · αY .

Therefore,

S ≤ rσ(R ⇒ αX(S) ≤ αX · rσ(R) = rτ · αY (R)

for all S ∈ SX,R ∈ SY (αX is monotone by Lemma 5.3.7). This im-plies that α is a morphism between the respective Kleisli extensions. If αXis order-embedding, the implication above is an equivalence, so that α isinitial. Conversely, if the initial extension of S is the Kleisli extension, theequivalence also holds, and we can conclude that αX is an order-embeddingby choosing r = 1X . �

Examples 6.2.3. 1. For every monad S on Set, there is a unique monadmorphism ! : S → I into the terminal monad. When the latter isequipped with its largest lax extension ?> : Rel → Rel (such that?>r(?, ?) = 1 for all relations r : X →7 Y as in 3.2.2(3)), the initialextension S of ?> along ! is given by

Sr(S,R) = 1,

for all S ∈ SX,R ∈ SY ; hence, S = S>.

2. It is obvious from Example 5.4.2(4) that the Kleisli extension F of Fcan be obtained as the restriction to filters of the Kleisli extension Uof the up-set functor U ; in this case, the components αX : FX → UXof α are the embeddings, and F is the initial extension of U along α.

Similarly, the Barr extension of the ultrafilter functor can be obtainedby restriction of the Kleisli extension F of the filter functor F , and thelax extension of the identity monad can also be seen as a restrictionof the ultrafilter functor (via the principal ultrafilter natural transfor-mation). Thus, the chain of natural transformations, whose respectivecomponents are all embeddings:

1Set → β → F → U

yields the following chain of initial extensions:

1Rel → β → F → U .


3. Similar to the Kleisli extension F of F the Kleisli extension C of theclique functor C can be obtained as the restriction of the Kleisli exten-sion U of the up-set functor U . The lax extension of κ is the initialextension of κ along κ→ C. This gives explicitly

D (κr) E ⇔ D ⊇ r◦[E ],

for all relations r : X →7 Y and ultracliques D ∈ κX and E ∈ κY.Again we get a chain of natural transformations

1Set → κ→ C → U

that yields a chain of initial extensions

1Rel → κ→ C → U .

6.3 Sup-dense and interpolating monad mor-

phisms

The isomorphism (S, 2)−Cat ∼= (T, 2)−Cat that we are aiming for requiresthat the monads S and T be sufficiently compatible. The conditions wepresent continue to be guided by the case where S = �,T = F and α : � ↪→ F

is the inclusion of the set of ultrafilters into the set of filters.Hence, consider a monad morphism α : S → T, where T = (T,m, e) is a

power-enriched monad with structure τ : P→ T and equipped with its Kleisliextension T , and S = (S, n, d) a monad equipped with its initial extension Salong α. To be able to exploit the adjunction

∨X a ↓TX as in Proposition

5.5.2, we introduce the transformation α∨ : PS → T via

α∨X :=∨

PαX = mX · τTX · PαX = αTX · τSX ,

or equivalently, α∨X(A ) =∨αX(A ) for all A ⊆ SX. Each α∨X preserves

suprema, and therefore has a right adjoint, denoted by α↓X : TX → PSX, sothat

α↓X(T ) = {S ∈ SX | αX(S) ≤ T } = α−1X · ↓TX (T )

for all T ∈ TX. The maps α↓X allows for a convenient description of the

initial extension S of S. Indeed, for a relation r : X →7 Y, we have

(Sr)[ = α↓X · rτ · αY ,

so the order relation S1X on SX is given by (S1X)[ = α↓X · αX .


The monad morphism α : S→ T is sup-dense if one has

α∨ · α↓ = 1TX ; (6.1)

in pointwise notation this says that every element of TX can be expressedas a supremum of αX-images of elements of SX :

∀T ∈ TX ∃A ⊆ SX : T =∨

αX(A ).

When S is a submonad of T and the embedding is sup-dense, we simply saythat S is sup-dense in T.

The morphism α : S→ T is interpolating for a relation r : SX →7 X if

α↓X · α∨X · r[ ≤ (↓SX ·nX)P · (Sr)[ · dX (6.2)

holds. This condition expands to

α↓X · α∨X · r[ ≤ (↓SX ·nX)P · (Sr)[ · dX

≤ (α↓X · αX · nX)P · α↓SX · rτ · αX · dX

= (α↓X · αX · nX)P · α↓SX · (τSX · r[)T · eX

= (α↓X · αX · nX)P · α↓SX · τSX · r[

and can be written pointwise as

αX(S) ≤∨{αX(R) | R r y} ⇒ ∃S ∈ SSX : S ≤ nX(S ) and αSX(S ) ≤ τSX ·r[(y)

for all S ∈ SX, y ∈ X.If S is a submonad of T, the previous condition naturally has a simpler

expression, and may be represented graphically by

S ≤ τSX · r[(y)_

��

S ≤ mX · τSX · r[(y)⇒ ∃S :

S ≤ mX(S )

A monad morphism α : S→ T is interpolating if it is interpolating forall relations r : SX →7 X. If S is a submonad of T and the embedding isinterpolating, we may simple say that S is interpolating in T.


Note that α is interpolating whenever it is a morphism of power-enrichedmonads α : (S, σ)→ (T, τ). Indeed, since {−}SSX ≤ α↓SX · αSX , we have

α↓X · α∨X · r[ = α↓X · α

TX · αSX · σSX · r[

= (α↓X · αTX · αSX)P · {−}SSX · σSX · r[

≤ (α↓X · αTX · αSX)P · α↓SX · αSX · σSX · r

[

= (α↓X ·mX · TαX · αSX)P · α↓SX · τSX · r[

= (α↓X · αX · nX)P · α↓SX · τSX · r[

Examples 6.3.1. 1. Any power-enriched monad T = (T,m, e) comeswith the interpolating monad morphism α = e : I → T. Indeed, usingthat αTX = 1TX and {−}X ≤ α↓X · αX , we have

α↓X · α∨X · r[ = α↓X · τX · r

[

= ({−}X)P · α↓X · τX · r[

≤ (α↓X · αX)P · α↓X · τX · r[.

2. If S = P is the powerset monad embedded in T = F via the principalfilter morphism τ : P → F, then the interpolation condition followsimmediately since τ is a morphism of power-enriched monads.

3. Consider the filter monad F with the principal monad morphism τ :P → F. Every filter is the supremum (that is, the intersection) of allultrafilters finer than it (see Corollary 1.7.6), so the ultrafilter monad� is sup-dense in F.

Let us verify that � is interpolating in F. For ultrafilters U ,V on X anda relation r : βX →7 X, suppose U ≤ Σrτ (V) (with Σ denoting themonad multiplication of �), that is, for all A ⊆ X and B ∈ V :

r[(B) ⊆ A� ⇒ A ∈ U .

If there existed A ∈ U and B ∈ V with A� ∩ r[(B) = ∅, we would haver[(B) ⊆ (A�){ = (A{)� so that A{ ∈ U , a contradiction. Therefore,A� ∩ r[(B) 6= ∅ for all A ∈ U and B ∈ V , and there exists an ultrafilterU on βX that refines both {A� | A ∈ U} and rτ (V). In particular,ΣX(U ) = U . By setting V = y, we observe that rτ (V) = τβX · r[(y), sothe interpolation condition is verified.

4. The ultraclique monad � is sup-dense and interpolating in C. Lemma1.7.13 gives us that every clique is the supremum (that is, the inter-section, since we work with reversed inclusion) of all ultracliques finerthan it. This proves that � is sup-dense in C.


To prove that � is interpolating in C, we take two ultracliques D, E onX and a relation r : κX →7 X. Suppose D ≤ mX · rτ (E). This meansthat

mX · rτ (E) = {A ⊆ X | A� ∈ rτ (E)} ⊆ D.

Thus for all A ⊆ X and for all B ∈ E we have that r[(B) ⊆ A� impliesthat A ∈ D.We have that A� ∩ r[(B) 6= ∅ for all D ∈ D and B ∈ E , because if itisn’t true, then there exist A ∈ D and B ∈ E with A� ∩ r[(B) = ∅.Then we have r[(B) ⊆ (A�){ = (A{)�, which means A{ ∈ D, which is acontradiction.

Lemma 1.7.12 and the fact that a set with the finite intersection propertygives rise to a clique, gives us the existence of an ultraclique D on κXthat refines both {A� | A ∈ D} and rτ (E). So

D ≤ {A� | A ∈ D} and D ≤ rτ (E).

Thus we have D ⊆ mX(D), but because D is an ultraclique, we havemX(D) = D. For E = y we have rτ (E) = τκX · r[(y) and thus D ≤τκX · r[(y). The interpolation condition is satisfied.

6.4 (S, 2)-categories as Kleisli monoids

Given a lax extension S to Rel of a monad S on Set, we call

(S, 2)−UGph

the category whose objects are pairs (X, r) with r : SX →7 X a unitaryrelation, so

d◦X · Sr · n◦X ≤ r and r · S1X ≤ r,

and whose morphisms f : (X, r)→ (Y, s) are maps f : X → Y satisfying

f · r ≤ s · Sf.

In addition, given a functor T : Set→ Set that lifts tacitly along the forgetfulfunctor Ord→ Set, we can consider the lax comma category

(1Set ↓ T )≤


whose objects are pairs (X, νX) with a map νX : X → TX, and whosemorphisms are maps f : X → Y with Tf · νX ≤ νY · f :

Xf //

νX��

≤

Y

νY��

TXTf// TY.

We now proceed to presenting conditions which will give us an isomorphismbetween the two categories (S, 2)−UGph and (1Set ↓ T )≤ that restricts to anisomorphism

(S, 2)−Cat ∼= T−Mon.

Consider now a monad morphism α : S → T, where T = (T,m, e) is amonad power-enriched by τ : P→ T and equipped with its Kleisli extensionT , and S = (S, n, d) is a monad equipped with its initial extension S along α.The sets SX and TX are equipped with the orders induced by the respectivelax extensions, and hom-sets of Set are ordered pointwise.

Following 5.5, we define for all sets X an adjunction

Set(X,TX)conv

>//Rel(SX,X)

nbhdoo

by exploiting the Ord-isomorphism Rel(SX,X) ∼= Set(X,PSX). Moreconcretely, we set

conv : Set(X,TX)→ Set(X,PSX) : ν 7→ α↓X · ν,

nbhd : Set(X,PSX)→ Set(X,TX) : r[ 7→ α∨X · r[.The adjunction nbhd a conv follows from α∨X a α↓X . Indeed, in pointwisenotation conv and nbhd can be expressed as

S conv(ν) y ⇔ αX(S) ≤ ν(y) and nbhd(r)(y) =∨

XαX(r[(y)),

for all relations r : SX →7 X, and maps ν : X → TX. Naturally, conv andnbhd restrict to mutually inverse isomorphisms between the sets of fixpointsof conv · nbhd and of nbhd · conv .

The fixpoints of nbhd · conv are exactly the maps ν : X → TX such that:

∀y ∈ X ∃A ⊆ SX : ν(y) =∨

αX(A ).

Hence, nbhd · conv = 1Set(X,TX) precisely when α is sup-dense. In turn, arelation r : SX →7 X is a fixpoint of conv · nbhd precisely when

∀y ∈ X : αX(S) ≤∨

αX(r[(y))⇒ S r y. (6.3)


One obtains the following generalizations of Proposition 5.5.1, Proposition5.5.2 and Theorem 5.5.3.

Lemma 6.4.1. For the adjunction nbhd a conv : Set(X,TX)→ Rel(SX,X)defined above, the following hold:

(1) Fix(nbhd · conv) = Set(X,TX) if and only if α is sup-dense;

(2) a relation r : SX →7 X is a fixpoint of conv · nbhd if and only if it isunitary and α is interpolating for r.

Proof. The first point is immediately clear from the previous discussion.For a unitary relation r : SX →7 X, we obtain as in the proof of Propo-

sition 5.5.1r[ = (↓SX ·nX)P · (Sr)[ · dX . (6.4)

If moreover α is iterpolating for r, then

α↓X · α∨X · r[ ≤ (↓SX ·nX)P · (Sr)[ · dX = r[,

and r is indeed a fixpoint of conv · nbhd .Conversely, if r : SX →7 X is a fixpoint of conv · nbhd, it is of the form

conv(ν) for a map ν : X → TX, so r · S1X ≤ r, and r is right unitary.Moreover, if S Sr dX(y) holds, we can apply mX · TαX to each side of theinequality αSX(S ) ≤ rτ · eX(y) to conclude that nX(S ) r y by the fixpointcondition (6.3), that is, r is left unitary. Indeed,

mX · TαX · αSX(S ) ≤ mX · TαX · rτ · eX(y)

⇔ αX(nX(S )) ≤ mX · TαX · (τSX · r[)T · eX(y) (α monad morphism

and definition rτ )

≤ mX · TαX · τSX · r[(y) (see (2.1))

≤ mX · τTX · PαX · r[(y)

≤∨

X · PαX · r[(y).

As r is unitary, (6.4) holds and implies that α is interpolating for r. �

Proposition 6.4.2. The adjoint maps nbhd and conv defined above satisfy

nbhd(s ◦ r) ≤ nbhd(r) ◦ nbhd(s), conv(µ) ◦ conv(ν) ≤ conv(ν ◦ µ)


for all relations r, s : SX →7 X, and maps µ, ν : X → TX (where 1]X =

d◦X · S1X). Moreover, if α is sup-dense, then nbhd(s◦ r) = nbhd(r)◦nbhd(s)for all relations r, s : SX →7 X. In this case, the equality conv(µ)◦ conv(ν) =conv(ν ◦ µ) also holds.


Proof. The displayed equalities follow from the fact that

S (1]X) y ⇔ S (d◦X · α◦X · T1X · αX) y ⇔ αX(S) ≤ eX(y)

for all S ∈ SX and y ∈ X, where d◦X · α◦X = e◦X . Indeed,

nbhd(1]X)(y) =∨

XαX((1]X)[(y)) =∨

X{T ∈ TX | T ∈ αX((1]X)[(y))}

=∨

X{T ∈ TX | ∃S ∈ SX : αX(S) = T and S 1]X y} = eX(y),

and

S conv(eX) y ⇔ αX(S) ≤ eX(y)⇔ S (1]X) y.

To show that nbhd(s ◦ r) ≤ nbhd(r) ◦ nbhd(s), we first note that

α∨X · PnX = αTX · τSX · PnX= αTX · TnX · τSSX= mX · TαX · TnX · τSSX= mX · T (αX · nX) · τSSX= (αX · nX)T · τSSX= (αX · 1SSX)T · τSSX (nX = 1SSX , see (2.2))

= ((αX · 1SX)T · αSX)T · τSSX (αY · fS = (αY · f)T · αX , see (2.2))

= (αTX · αSX)T · τSSX= αTX · αTSX · τSSX ((gT · f)T = gT · fT, see (2.1))

= αTX · α∨SX .

By composing these equalities with α↓X · rτ · αX on the right, we obtain

α∨X · PnX · α↓X · r

τ · αX = αTX · α∨SX · α↓X · r

τ · αX ≤ αTX · rτ · αX . (6.5)


We can now proceed as in Proposition 5.5.2:

nbhd(r) ◦ nbhd(s) = (α∨X · r[)T · α∨X · s[

= αTX · (τSX · r[)T · αTX · τSX · s[ ((gT · f)T = gT · fT)

= αTX · (rτ · αX)T · τSX · s[ (gT · fT = (gT · f)T)

= (αTX · rτ · αX)T · τSX · s[ (gT · fT = (gT · f)T)

≥ (α∨X · PnX · α↓SX · r

τ · αX)T · τSX · s[ (6.5)

= (α∨X · PnX · (Sr)[)T · τSX · s[ (definition initial extension)

= αTX · (τSX · PnX · (Sr)[)T · τSX · s[ ((gT · f)T = gT · fT)

= αTX · τSX · (PnX · (Sr)[)P · s[ (τ natural transformation)

= α∨X · (((n◦X)[)P · (Sr)[)P · s[ (PnX = ((n◦X)[)P)

= α∨X · (s · Sr · n◦X)[ (Rel = SetP)

= nbhd(s ◦ r).

The inequality for conv follows from the adjunction nbhd a conv .If α is sup-dense, then the inequality in (6.5) becomes an equality, so that

nbhd(r) ◦ nbhd(s) = nbhd(s ◦ r).

The claim for conv then follows from the equality nbhd · conv = 1Set(TX,X)

(see Lemma 6.4.1). �

Theorem 6.4.3. Let (T, τ) be a power-enriched monad together with a monadmorphism α : S→ T, and suppose that T is equipped with its Kleisli extensionT , and S with the initial extension of T along α. If α is sup-dense, then thereis a full reflective embedding (1Set ↓ T )≤ ↪→ (S, 2)−UGph that commutes withthe underlying-set functor and restricts to a full reflective embedding

T−Mon ↪→ (S, 2)−Cat.

If α is also interpolating, then this functor is an isomorphism.

Proof. For every map ν : X → TX, the relation conv(ν) is a fixpoint ofconv · nbhd, and is therefore unitary by Lemma 6.4.1. Similarly, a unitaryrelation r : SX →7 X yields a map nbhd(r) : X → TX. Thus, we can considerthe functors

C : (1Set ↓ T )≤ → (S, 2)−UGph

N : (S, 2)−UGph→ (1Set ↓ T )≤


defined on objects by C(X, ν) = (X, conv(ν)) and N(X, r) = (X, nbhd(r)),and leaving maps untouched (the fact that C and N send morphisms tomorphisms follows easily from the definition). The adjunction nbhd a convyields an adjunction N a C. Lemma 6.4.1 shows that if α is sup-dense, thenC is a full reflective embedding, and Proposition 6.4.2 yields that C restrictsto a functor T−Mon ↪→ (S, 2)−Cat. Finally, if α is interpolating, then C isan isomorphism by Lemma 6.4.1. �

Theorem 5.5.3 now appears as a direct consequence of this more generalresult, since α = 1T is both sup-dense and interpolating. Moreover, thecategory of Kleisli monoids provides a link between presentations of lax al-gebras.

Proposition 6.4.4. If α : S→ T is sup-dense as in Theorem 6.4.3, then thealgebraic functor

Aα : (T, 2)−Cat→ (S, 2)−Catis a full reflective embedding. If α is also interpolating, then Aα is an iso-morphism.

Proof. The isomorphism (T, 2)−Cat ∼= T−Mon of Theorem 5.5.3 composedwith the full reflective embedding T−Mon ↪→ (S, 2)−Cat of Theorem 6.4.3sends (X, a : TX →7 X) to (X, a · αX : SX →7 X). Hence, this compositionis precisely the algebraic functor Aα. When α is interpolating, Aα is anisomorphism by Theorem 6.4.3. �

Examples 6.4.5. 1. Depending on whether a relation r on a set X isseen as a map

r : X ×X → 2, r : X → PX or r : PX ×X → 2,

the category Ord of ordered sets is described respectively as any of thefollowing categories

(I, 2)−Cat ∼= P−Mon ∼= (P, 2)−Cat.

2. Whether ultrafilter convergence, neighborhood systems, or filter conver-gence is chosen as defining structure, the category Top of topologicalspaces appears as

(�, 2)−Cat ∼= F−Mon ∼= (F, 2)−Cat.

3. Whether ultraclique convergence, clique convergence, neighborhood cliques(see Section 6.5), up-set convergence or neighborhood up-sets is chosenas defining structure, the category Cls of closure spaces appears as

(�, 2)−Cat ∼= (C, 2)−Cat ∼= C−Mon ∼= (U, 2)−Cat ∼= U−Mon.


6.5 Compact Hausdorff closure spaces

In Section 4.8 we have proven the isomorphism between the category ofEilenberg-Moore algebras of the ultrafilter monad � and the category ofcompact Hausdorff topological spaces, using the flatness of the lax exten-sion of the ultrafilter monad �. We have shown in the previous section thatthe category of lax algebras of the ultraclique monad � is isomorphic to thecategory of closure spaces. We first prove that the ultraclique monad is flat.Then one might ask if we can prove a similar isomorphism for the ultracliquemonad. The answer to this question is positive. We can define Hausdorffnessand compactness for closure spaces and show that the category of Eilenberg-Moore algebras of the ultraclique monad is isomorphic to the category ofcompact Hausdorff closure spaces. To properly define Hausdorff for closurespaces, we first must define neighborhood cliques, similar to neighborhoodfilters.

More information about closure spaces, compactness and Hausdorffnesscan be found in the paper “Basic properties of closure spaces” [14] and thePhD thesis “A study of function spaces and compactness for the category ofaffine sets and its subconstructs” [3]. This section is developed by the authorusing the references stated above.

In section 1.5 we define a closure space (X, c) as a set X and a closureoperation c : PX → PX, i.e. c is a monotone map and satisfies

c · c ≤ c and 1PX ≤ c.

If the closure operation, in addition, satisfies c(∅) = ∅ and c(A ∪ B) =c(A) ∪ c(B), then c defines a topology on X. The condition c(∅) = ∅ is inour definition not required, however in [3] and [14], the references we used,it is a basic axiom of a closure space. In this chapter we need to take theabsence of this condition into account.

With a closure space (X, c) we define a collection of open sets O(X) as

O(X) = {A ⊆ X | c(X \ A) = X \ A}.

The collection of open sets contains the empty set and is closed for arbitraryunions.

If (X,O(X)) is a pair where X is a set and O(X) ⊆ PX contains theempty set and is closed for arbitrary unions, then c : PX → PX defined by

c(M) = {x ∈ X | ∀U ∈ O(X), x ∈ U : U ∩M 6= ∅}

is a closure operation.We now will define neighborhood cliques for closure spaces.


Proposition 6.5.1. For a closure space (X, c), for every x ∈ X, define theneighborhood clique V(x) of x by

V(x) := {U ∈ PX | x /∈ c(X \ U)}.

In terms of open sets, the neighborhood clique of x ∈ X is equal to

V(x) =↑PX {U ∈ O(X) | x ∈ U}.

For every x ∈ x, V(x) satisfies the following conditions:

(N1) ∀V ∈ V(x) : x ∈ V ;

(N2) V ∈ V(x), V ⊆ W ⇒ W ∈ V(x);

(N3) ∀V ∈ V(x) ∃W ∈ V(x) ∀y ∈ W : V ∈ V (y).

Condition (N1) is equivalent with the reflexivity of the closure operation,condition (N2) is equivalent with the monotonicity of the closure operationand condition (N3) is equivalent with the idempotency of the closure oper-ation.

Since the closure operation doesn’t necessarily satisfy c(∅) = ∅, we can’tbe sure that every neighborhood clique contains a subset of X. If every neigh-borhood clique has an element, and by condition (N2) contains the set X,then the closure operation will satisfy c(∅) = ∅. Conversely, if the closureoperation satisfies c(∅) = ∅, then X ∈ V(x) for every x ∈ X.

For morphisms in the category of closure spaces, we have the followingequivalent descriptions, implicitly seen in Example 6.4.5.3.

Proposition 6.5.2. For f : (X, cX)→ (Y, cY ) the following are equivalent

(i) f is continuous, i.e. f(cX(A)) ⊆ cY (f(A)) for all A ⊆ X;

(ii) K →X x⇒ κ(f)(K)→Y f(x) for K ∈ κX and x ∈ X.

With some basic descriptions for closure spaces developed, we now turnour attention to the ultraclique monad.

Proposition 6.5.3. The lax extension � of the ultraclique monad � = (κ,m, e)is flat.

Proof. Take two ultracliques K and L on X. Suppose

K κ1X L


holds. Then by definition of the lax extension of κ, we have

K ⊇ 1◦X [L]⇔ K ⊇ L.

Since L is an ultraclique, K and L are equal. Thus, by arbitrariness of Kand L, we have

κ1X = 1κX = κ1X .

�

Since the ultraclique monad is flat, we can apply Theorem 4.8.1 for theultraclique monad. The Eilenberg-Moore algebras of the ultraclique monad �

are closure spaces, with extra conditions. We now define ultraclique conver-gence, Hausdorffness and compactness for closure spaces properly and provethe isomorphism between the category of Eilenberg-Moore algebras for theultraclique monad and the category of compact Hausdorff closure spaces.

Since C−Mon ∼= (C, 2)−Cat, we can connect the neighborhood cliqueswith a convergence relation. This connection gives rise to following definition.

Definition 6.5.4. A clique C converges to x if

V(x) ⊆ C.

We denote this byC → x.

If an element x ∈ X has an empty neighborhood clique V(x), then forevery clique C ∈ CX

C → x.

Definition 6.5.5. We call a closure space X Hausdorff if for x 6= y, thereis V ∈ V(x) and W ∈ V(y) such that V ∩W = ∅.

Since the definition of a Hausdorff closure space X implies the existenceof an element in V(x) for every x ∈ X, Hausdorffness gives us an extracondition on the closure operation c, i.e. c(∅) = ∅.

If we denote a for→, where a : κX → X, we have the follwing proposition.

Proposition 6.5.6. The following are equivalent for a closure space X:

(i) X is Hausdorff;

(ii) every ultraclique on X converges to at most one point, i.e. a · a◦ ≤ 1X .


Proof. Suppose K is a ultraclique converging to both x and y, i.e.

K → x and K → y.

By definition of the convergence structure, this means V(x)∪V(y) ⊆ K. SinceK is an ultraclique we have

V ∩W 6= ∅

for every V ∈ V(x) and W ∈ V(y). Thus x = y, since X is Hausdorff.Conversely, if every ultraclique converges to at most one point, it means

that if x 6= y there doesn’t exist an ultraclique finer than V(x)∪V(y). It alsoimplies that for every z ∈ X the neighborhood V(z) 6= ∅. Therefore we canchoose V,W ∈ V(x)∪V(y) such that V ∩W = ∅. Since every element of V(x)contains x and every element of V(y) contains y, we can choose V ∈ V(x)and W ∈ V(y). Thus X is Hausdorff. �

Definition 6.5.7. A closure space X is compact if every open cover has afinite subcover.

When an element x ∈ X has an empty neighborhood clique V(x), thenthere don’t exist open sets, that contains x. Thus there doesn’t exist an opencover of X. So every closure space (X, c) such that c(∅) 6= ∅ is compact.

Proposition 6.5.8. For a closure space X 6= ∅ the following are equivalent:

(i) X is compact;

(ii) every ultraclique on X converges, i.e. 1κX ≤ a◦ · a.

Proof. First we assume that c(∅) 6= ∅ holds for the closure operation on X.Then is X compact and there exists an element x ∈ X such that V(x) = ∅.For every ultraclique K on X holds K → x. Conversely, every ultraclique onX converges, since V(x) = ∅ for some x ∈ X. Thus X is compact, since theredon’t exist open covers on X.

Secondly, we assume that c satisfies c(∅) = ∅. Then the equivalence alsoholds. Indeed, assume that X is compact. Let K ∈ κX and set

A = {A ⊆ X | A open, A /∈ K}.

If the set A covers X, then there would exist A1, A2, . . . , An ∈ A with

n⋃i=1

Ai = X ∈ K.


Thus,by Lemma 1.7.10, there exist Ak ∈ K for some k = 1, . . . , n . Thiscontradicts the definition of A and therefore A doesn’t cover X. We now canfind x ∈ X such that every open set A with x ∈ A belongs to K. This impliesthat K → x.

Conversely, now assume that every ultraclique K ∈ κX converges. Let Abe a set of open subsets of X such that no finite subset of A covers X, then

C =↑PX {(⋃F){ | F ⊆ A,F finite}

is a proper clique on X. Indeed, for F ,G ⊆ A finite, we have F ∪G ⊆ A andF ∪ G is finite. Thus

(⋃F) ∪ (

⋃G) =

⋃(F ∪ G) 6= X.

So we have

(⋃F){ ∩ (

⋃G){ = ((

⋃F) ∪ (

⋃G)){ 6= ∅.

By Lemma 1.7.12 there exists an ultraclique L such that C ⊆ L. The ultra-clique L converges to some x ∈ X by hypothesis. Hence, x /∈ A for anyA ∈ A, that is, A does not cover X. �

We denote the category of compact Hausdorff closure spaces with continuousfunctions as

CompHausCls.


Set� ∼= CompHausCls

that commutes with the respective forgetful functor to Set.

Proof. Applying both propositions above, we see that a closure space iscompact Hausdorff if and only if its convergence relation a : κX →7 X isactually a map a : κX → X. Theorem 4.8.1 now gives us the result. �

Bibliography

[1] M. Barr. Relational algebras. In Reports of the Midwest CategorySeminar, IV, volume 137 of Lect. Notes Math., pages 3955, Springer,Berlin, 1970.

[2] G. Birkhoff. Lattice Theory. American Mathematical Society,Providence, Rhode Island, 1940.

[3] V. Claes. A study of function spaces and compactness for the category ofaffine sets and its subconstructs. PhD thesis, Vrije Universiteit Brussel,2004

[4] M. M. Clementino, E. Colebunders, D. Hofmann, R. Lowen, R.Lucyshyn-Wright, G. J. Seal, W. Tholen. Monoidal Topology, Acategorical approach to order, metric and topology. Cambridge Univer-sity Press, 2014.

[5] E. Colebunders. Topologie. Dienst Uitgaven VUB, 2010.

[6] J. de Groot, G.A. Jensen, A. Verbeek. Superextensions. Technical reportZW-017, Math. Centrum Amsterdam Afd. Zuivere Wisk., 1968.

[7] W. Gahler. Monadic topology - a new concept of generalized topology. InRecent Developments of General Topology and its Applications, number67 in Math. Res., pages 136-149. Akedemie-Verlag, Berlin, 1992.

[8] D.C. Kent and W.K. Min. Neighborhood spaces. Int. J. Math. Math.Sci., 32(7):387-399, 2002.

[9] S. Mac Lane. Categories for the Working Mathematician, volume 5 ofGraduate Texts in Mathematics. Springer, New York, 1971.

[10] E.G. Manes. A triple theoretic construction of compact algebras. In Sem.on Triples and Categorical Homology Theory (ETH, Zurich, 1966/67),number 80 in Lect. Notes Math., pages 91-118, Berlin, 1969. Springer.

117

BIBLIOGRAPHY 118

[11] C. Pisani. Convergence in exponentiable spaces. Theory and Appli-cations of Categories, Vol. 5, No. 6, pp. 148-1622, 1999.

[12] G.J. Seal. Canonical and op-canonical lax algebras. Theory Appl. Categ.,14(10):221-243, 2005.

[13] G.J. Seal. A Kleisli-based approach to lax algebras. Appl. Categ. Struc-tures, 17(1):75-89,2009.

[14] B.M.R. Stadler, P.F. Stadler. Basic properties of closure spaces. J.Chem. Inf. Comput. Sci., 42: 577585, 2002.

[15] D.V. Thampuran. Extended topology: Filters and convergence. I. Math.Ann., 158: 57-68, 1965.

Index

(1Set ↓ T )≤, 107(S, 2)−UGph, 106(T, 2)-category, 49(T, 2)-functor, 49(T, 2)−Cat, 49(T, 2)−URel, 612-functor, 31C, 25Cls, 6CompHaus, 71F, 19I, 17Inf, 8Int, 6Mod, 4Ord, 3P, 17Rel, 2SLat, 8Set, 2SetT, 47SetT, 54Sup, 8T-algebra, 46T-homomorphism, 47T−Mon, 82Top, 64U, 24�, 23�, 25

algebraic functor, 99associative, 60

Barr extension, 37BC, 41BC-square, 41Beck-Chevalley condition, 41Beck-Chevalley square, 41

clique, 11clique monad, 25closure operation, 6closure space, 6cocontinuous, 48compact, 70, 115complement, 6complete, 7completely distributive, 48continuous for closure spaces, 6continuous for interior spaces, 6

down-set, 7dual relation, 2

Eilenberg-Moore algebra, 46Eilenberg-Moore category, 47

fibre, 2filter, 8filter in, 47filter monad, 19flat, 35, 36

graph, 2

Hausdorff, 70, 114homomorphism of lattices, 7homomorphism of monoids, 16

119

INDEX 120

homomorphism of semilattices, 8

ideal, 10idempotency, 6identity monad, 17image filter, 8induced order, 72inf-map, 7infimum, 7initial extension, 100interior operation, 6interior space, 6interpolating, 104isomorphism, 5

Kleisli category, 54Kleisli composition, 54Kleisli convolution, 57Kleisli extension, 88Kleisli monoids, 82Kleisli triple, 26Kleisli triple morphism, 27Kowalsky sum, 19, 23

lattice, 7lax algebra, 49lax comma category, 106lax extension, 31lax extension of a monad, 36lax functor, 30lax identity, 57lax transformation, 31left adjoint, 5left unitary, 59

module, 3monad, 16monoid, 15monotone, 3morphism of lax extensions, 98morphism of monads, 16

morphism of power-enriched monads,79

multiplication, 16

oplax functor, 31oplax transformation, 31opposite relation, 2order, 3order-preserving, 3ordered category, 30

power-enriched, 79powerset monad, 17principal filter, 8proper clique, 11proper filter, 8proper ideal, 10

refinement order, 74reflexive, 3, 48reflexivity, 6relation, 1relational algebras, 64right adjoint, 5right unitary, 59

semilattice, 8separated, 3separated ordered category, 30span, 37sup-dense, 104sup-map, 7supremum, 7

transitive, 3, 48

ultraclique, 11ultraclique monad, 25ultrafilter, 8ultrafilter monad, 23underlying order, 72unit, 16unitary, 59

INDEX 121

up-closed, 7up-closure, 7up-set, 7up-set monad, 24

Kleisli Monoids - Vrije Universiteit Brussel...monoid, commute. In Chapter2we give di erent examples...

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