Ù V 27 - files-cdn.cnblogs.com

8¹

1ÊÜ© �ÆØ 1

1�ÔÙ �5� 3

17.1 Ø, {Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

17.2 ÜþÈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

17.3 gd+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

17.4 È, {È . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

17.5 íÑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

17.6 .£ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1�lÙ �Æ�Ä�Vg 27

18.1 �Æ�½ÂÚ~f . . . . . . . . . . . . . . . . . . . . . . . . 27

18.2 ¼f�½ÂÚ~f . . . . . . . . . . . . . . . . . . . . . . . . 30

1

1ÊÜ©

�ÆØ

1

�ó

�ÆØ¿Ø´éÎ�êÆ, �åg�êÿÀ�ïÄ. �ÆØ�Ñy�

��êÆNX��w¯K��#�Ý, éõSNÏd {z, éõSN

��z. ù4�ÆØé¯¤�uÐ¤��\�Æ�. �´3�ã{¤

�Ï¥, �ÆØ��5¿vk�¿©@£, �ÆØ�Ý�@�´“Ä�¢

{”(abstract nonsense), Ï�,«§Ýþ, 3üX/ïÄ�Æ��L§¥,

¤k�·KÃ�´½Â�ÓÂ�E, ��n), �¿Ã“k¿Â”�?. �´,

Ù¢ù��ÓÂ�E¢Sþ�nØïÄ�Ø�Á=, ò�#�ÀÜz8

¤�Î�ÀÜ, �ØU��/�äd�.

�ÆØØ'58Ü��D�N��(�, '%8Üm��, é+

ó=Ó�, éÿÀ�m ó=ëYN�. ù��Ð?´ò(�äNXÛ�

lm5.

�ù½¦·�?Ød�N+|¤�L��“8Ü”, XdÈ��{g,

¬Úu�8ÜØþ�¯K, ë�??!, ·��Ø2[Ä, =æ^“a”ù�

c5�?n.

�;��ÆØ3ïÄ“¢{”��ú, �©ÄkÐ«��~f;�(Ü

��5�Ü©, ùÜ©¤��@#��ó, ø�uúnz8ÜØ�ó, dd

�±�½�ÆØMá�ÄÅ. ùÜ©~fk#kÎ, �&¬éù@�ók�

��ÐÚ�rº. ��·�2�Ñ�Æ�(�½Â.

2

1�ÔÙ �5�

�Ù�SN´�5�(universal property), ½¡�k5�. î��½Â

kJe, ��&ÖöÆS��ÙM¥Ò¬k�5��Vg. �ÆØ'5é

��m��, Ø'%äN�(�, ýZ�~fX�5�ê�Ó�, ��

ê�Ó, ü�5�m�Ó�, �´�5�ê¥��5C��nØ�¿Ø²�.

ÄÙ�Ï, ·�A��\'5��´Ä“g,”½ö“;�”, ~X+Ó�Ä�

½n`G/Kerϕ = Imϕ, ¢Sþ�I�Ñg + Kerϕ 7→ ϕ(g). �´äNXÛ

£ã“g,”, ·�kc�´Ny3i¡þ, ·�e¡¬^�5�î�£ã.

�Ü©SN½NI��þ��mn), g�$�×�, �ÖöÜnSü

�m, 5¿>E, Ø�r»\ .

17.1 Ø, {Ø

~ 17.1 (Ø) ®�Abel+A,B, Ó�ϕ : A→ B, K

(K, ι)|K = Kerϕ, ι´�¹N�

÷vXe��5�(universal property)

∀Abel+X, Ó�α : X → A,

s. t. ϕ ◦ α = 0

∃!Ó�α′ : X → K

s. t. α = α′ ◦ ι

Kι // A

ϕ // B

X

α

>>

α′

OO

3

4 1�ÔÙ �5�

Ù¥0´ò¤k��N�"��N�. �?Û÷vþã5��(K ′, ι)¥�K

ÑÓ�.

y² Ï�ϕ ◦ α = 0, �

∀x ∈ X,ϕ(α(x)) = 0⇒ α(x) ∈ Kerϕ

��I½Âα′ = α=�. α′��5´w,�. e¡y²~f�e�Ü©,

�(K ′, ι′)�÷vþã�5�

∃Ó�α : K ′ → K,β : K → K ′

s. t. ι = ι′ ◦ β, ι′ = ι ◦ α⇒ ι = ι ◦ α ◦ β, ι′ = ι′ ◦ α ◦ β

Kι //

β

Aϕ // B

K ′ι′

>>

α

HH

Ï�

¦mã��K → K ′

�Ó�´��.

⇒ α ◦ β = id

Ónβ ◦ α = id

Kι // A

ϕ // B

K

ι=ι◦α◦β

>>

id

VV

α◦β

HH

l α : K → K ′´Ó�. �

±þÒ´“�5�”��{ü��~f, =?Û÷vϕ ◦ α = 0�Ó

�Ñ7L²KLι �A, �Ò´`(Kerϕ, ι)´÷vϕ ◦ α = 0�“��”��

éAbel+ÚÓ�. �÷v^��KÚι3Ó�¿Âe´��.

·�lØyL§¥�±½�Xd?n¯K�A:—·�'5�´N�,

�é��.

e¡, X·�3ùa¯K¥~��, ·�òþã��Þã�¤k�ÞÑ

ò��N�, w÷véA�(J�é�´�o.

~ 17.2 ({Ø) ®�Abel+A,B, Ó�ϕ : A→ B, ¦

(C, π)|C´Abel+, π : B → C´Ó�N�

17.1 Ø, {Ø 5

÷vXe��5�

∀Abel+X, Ó�N�α : B → X,

s. t. α ◦ ϕ = 0

∃!Ó�N�α′ : C → X

s. t. α = α′ ◦ π

Aϕ // B

α

π // C

α′

��X

Ù¥0´ò¤k��N�"��N�.

) ·�y²

(C, π)|C = B/ Imϕ, π´g,N�

´÷vþã�5��kSé.

∀a ∈ A,α(ϕ(x)) = 0⇒ Imϕ ⊆ Kerα

�â+Ó�Ä�½nÚ1�Ó�½nk

X ⊇ Imα ∼=B

Kerα∼=

B/ Imϕ

Kerα/ Imϕ

�α′ : B/ Imϕ→ X=�. ��5: ÷vþã�5�, 7,k

α′ : B/ Imϕ → X

b+ Imϕ 7→ α(b)

ù´��û½Â�N�. �

½Â 17.3 ({Ø) ®�Abel+A,B, Ó�ϕ : A→ B, P

CoKerϕ = B/ Imϕ

¡�ϕ�{Ø(cokernel).

SK 17.1

6 1�ÔÙ �5�

SK 17.1.1 (�) ®�Abel+A,B, Ó�ϕ : A→ B, ¦y: ImϕkXe�5

�.

∀Abel+X, Ó�N�α : B → X,

s. t. α ◦ ι = 0

∃!Ó�N�α′ : Imϕ→ X

s. t. α = α′ ◦ π

Kerϕι // B

α!!

π // Imϕ

α′

��X

=�´Ø�{Ø.

SK 17.1.2 ({�) ®�Abel+A,B, Ó�ϕ : A → B, Uì�©ÚSK�

�{}Á(½“{�”. (J«: ={�´{Ø�Ø, (JÓ�u�. )

SK 17.1.3 ®�Abel+A,B,A′, B′, o�Ó�

ϕ : A→ B ϕ′ : A′ → B′

µ : A→ A′ ν : B → B′

s. t. ϕ ◦ µ = ν ◦ ϕ′

Aµ //

ϕ

��

A′

ϕ′

��B ν

// B′

Pι : Kerϕ → A, ι′ : Kerϕ′ → A′´�¹N�, π1 : B → CoKerϕ, π2 : B′ →CoKerϕ′´g,N�

∃µ : Kerϕ→ Kerϕ′,

s. t. ι′ ◦ µ = µ ◦ ι∃ν : CoKerϕ→ CoKerϕ′,

s. t. π′ ◦ ν = ν ◦ π

Kerϕµ //

ι

��

Kerϕ′

ι′

��A

µ //

ϕ

��

A′

ϕ′

��B ν

//

π

��

B′

π′

��CoKerϕ

ν// CoKerϕ′

(J«: y²ϕ′ ◦ [Kerϕ→ A′] = 0)

17.2 ÜþÈ 7

17.2 ÜþÈ

e¡0�ÜþÈ. Äk�ÑV�5�Vg.

½Â 17.4 (V�55) ®�F -�5�mV,W,U , eN�f : V ×W → U÷

vf(x1 + x2,y) = f(x1,y) + f(x2,y) x1,x2 ∈ V,y ∈Wf(x,y1 + y2) = f(x,y1) + f(x,y2) x ∈ V,y1,y2 ∈W

f(ax,y) = af(x,y) a ∈ F,x ∈ V,y ∈Wf(x, ay) = af(x,y) a ∈ F,x ∈ V,y ∈W

K¡f´V�5N�.

~ 17.5 (ÜþÈ) ®�F -�5�mV,W , (½

(T, τ) | T´F -�5�m, τ : V ×W → T´V�5N�

÷vXe�5�

∀F -�5�mU ,

V�5N�β : V ×W → U

∃!�5N�λ : T → U

s. t. β = λ ◦ τ

V ×W τ //

β##

T

λ

��U

) ��5�mF |V×W |, ��BP

(v,w) = (δ(x,y))(x,y)∈V×W δ(x,y) =

1, (x,y) = (v,w)

0 (x,y) 6= (v,w)

=¢5^(v,w)L«3(v,w)?�13Ù¦?�0��þ. PVW�V ×WÜ¤�f�m, K?ÛVW¥��ÑäkXe/ª

x =

n∑i=1

ai(vi,wi) (∗)

8 1�ÔÙ �5�

Ù¥n ∈ N, ai ∈ F,vi ∈ V,wi ∈W , P

{ (x1 + x2,y)− (x1,y)− (x2,y) | x1,x2 ∈ V,y ∈W }∪ { (x,y1 + y2)− (x,y1)− (x,y2) | x ∈ V,y1,y2 ∈W }∪ { (ax,y)− a(x,y) | a ∈ F,x ∈ V,y ∈W }∪ { (x, ay)− a(x,y) | a ∈ F,x ∈ V,y ∈W }

Ü¤�f�m�D, -T = VW/D, ��B, exX(∗)/ª, ·�PT¥��

x +D =

n∑i=1

ai(vi ⊗wi) (∗∗)

KN´�y

τ : V ×W → VW → T

(x,y) 7→ (x,y) 7→ x⊗ y

´V�5�. Ï�(x1 + x2,y)− (x1,y)− (x2,y) ∈ D(x,y1 + y2)− (x,y1)− (x,y2) ∈ D

(ax,y)− a(x,y) ∈ D(x, ay)− a(x,y) ∈ D

l (x1 + x2)⊗ y = x1 ⊗ y + x2 ⊗ y

x⊗ (y1 + y2) = x⊗ y1 + x⊗ y2

(ax)⊗ y = a(x⊗ y)

x⊗ (ay) = a(x⊗ y)

e¡y²(T, τ)÷v�5�. ½Â

λ : T → U

x⊗ y 7→ β(x,y)

÷vβ = λ ◦ τ . Ï�T¥��ÑX(∗∗)/ª, �þ½Â��N�, N´�y

Ùû½Â5. ��5�´w,�. �

17.2 ÜþÈ 9

½Â 17.6 (ÜþÈ) ®�F -�5�mV,W , Pþ~(½�T = V ⊗W , ¡

�VÚW�ÜþÈ, Ù¥��¡�Üþ(tensor).

ÜþÈ��53u, §òV ×Wþ�V�5N�ÚV ⊗Wþ��5N��éAå5, l ¦¢S¥�«V�5¯K=z��N´?n��5¯

K.(k·�Ù��Ýóä��)

ÜþÈ´;.�(�E,�^�5�£ãå5{ü�êÆ(�.

SK 17.2

SK 17.2.1 y²~17.5¥÷v�5��(T, τ)¥�T´Ó��.

SK 17.2.2 y²3~17.5¥,

VW =

{(a(x,y))(x,y)∈V×W

∣∣∣∣∣ ∀(x,y) ∈ V ×W,a(x,y) ∈ F�kk��(x,y)¦�a(x,y) 6= 0

}

SK 17.2.3 ®�F -�5�mV,W,U , ¦y:

(1)V ⊗W ∼= W ⊗ V ;

(2)(V ⊗W )⊗ U ∼= W ⊗ (V ⊗ U);

(3)(V ⊕W )⊗ U ∼= (V ⊗ U)⊕ (W ⊗ U). (J«: ¦^�5�)

SK 17.2.4 Á�ÑÃ¡ÜþÈ�½Â.

SK 17.2.5 ®�F -�5�mV,W , dimV = n,dimW = m, ¦y: dimV ⊗W = mn. (J«: y²{ei ⊗ ej}(i,j)´�|Ä, |^�5�. )

SK 17.2.6 ®�Abel+A,B,C, ϕ : A×B → C�¡�V\5�, �

ϕ(a1 + a2, b) = ϕ(a1, b) + ϕ(a2, b) ϕ(a, b1 + b2) = ϕ(a, b1) + ϕ(a, b2)

10 1�ÔÙ �5�

¦ò~17.5¥�^cXeO�

F -�5�m Abel+

V�5N� V\5N�

�5N� Ó�N�

Á(½Abel+�ÜþÈ.

17.3 gd+

�0�gd+, ·�kÚ\��k'8ÜØ��E|.

½Â 17.7 (i1L) ®�8ÜX, 5½X0 = {∧}, ¡

∞⊔i=0

X |i|

�±X�i1L(alphabet))¤ücL, z��¡�i(word). Ù¥, ��

Bå�, P

(x1, . . . , xn) = x1x2 . . . xn

¡∧��i.

~ 17.8 (gd+) ®�8ÜX, (½

(F, ι)|F´+,Ó�N�ι : X → F

÷vXe��5�

∀+G, N�α : X → G

∃!Ó�N�α′ : F → G

s. t. α′ ◦ ι = α

Xι //

α

F

α′

��G

17.3 gd+ 11

) PX = X × {1,−1}, �Bå�, P

x := (x, 1) x−1 := (x,−1) (x−1)−1 := x

K�Ä±X�i1L)¤�ücLA, 3Aþ½Â'X

a . . . b cd e . . . f

∼ a . . . b e . . . f:⇐⇒ c−1 = d

¡��, PÙ�d4��'X≡(ë�SK??), PF = A/ ≡, 3Fþ, �½

Â¦{(��i�©�,

(x1 . . . xn) · (y1 . . . ym) = x1 . . . xny1 . . . ym

Kw,´µ4, (Ü�, kü �∧, x1 . . . xnk_x−1n . . . x−11 , Ï�

(x1 . . . xn) · (x−1n . . . x−11 ) ≡ ∧

�F�¤+, �

ι : X → F

x 7→ x

e¡y²�5�. ½Â

α′ : F → G

x 7→ x

÷vα′ ◦ ι = α. Ï�F¥?Û��Ñk/ªx1 . . . xn, �p�“≡”'X��

��Ó, ù½Â��û½Â�N�. ��5�´w,�. �

¤¢gd+, =r14?ÛX¥�ü��±$�, �Ø�?Û“'

X”, =gd. ¤¢'X, ~X+ZnÒk'Xn1 = 0, gd+vk?Û'

XÏd�¡�gd. nÜ5`, X�gd+Ò´¹X��Ø�?Û“'

X”�+. �â�5�, gd+ò��8Ü��+�N��+��

+�Ó�N��éAå5.

Xdw5, ~ó¤��“°JÚgØU�\”, Ù¢[�5��±3�½

�|Üe¤á.

12 1�ÔÙ �5�

½Â 17.9 (gd+) ®�8ÜX, Pþ~(½�F = F (X), ¡�X)¤�

gd(free)+.

X�uF (X)gXÄ�u�5�m�'X, Ï��ÑÄN Û�, �

��5�m��5C�Ò�(½.

·K 17.10 ?Û+3Ó�¿ÂeÑ´gd+�û+.

y² ®�+G, �ÄF (G), égd+¦^�5�Xe

∃!Ó�N�α : F (G)→ G

s. t. ϕ ◦ ι = id

Gι //

id""

F (G)

ϕ

��G

l ϕ´÷Ó�(ë�SK??), l G ∼= F (G)/Kerϕ. �

SK 17.3

SK 17.3.1 y²��|¤�8Ü)¤�gd+Ó�uZ.

SK 17.3.2 y²~17.8¥, ≡�²Lk�g��½V\xx−1½x−1x�z�pz.

SK 17.3.3 (Ø��i) ®�gd+F , e¡·�y²gd+�z�iÑk

��Q�i, ¡�éAQ�i. Q�i=Ø�3��i�i.

¦y: z�iÑk��éAQ�i. ±e´y²òV:

(1)z�iWÑ�3��éAQ�iU .

(2)�i�éAQ�i��⇐⇒¤ki�éAQ�i��.

(3)?Ûi��ÚV\xx−1½x−1x´��½�-��.

SK 17.3.4 (gdAbel+) Á�EgdAbel+. (J«: Ó�u∑

i∈I Z, ë

�SK??)

17.4 È, {È 13

SK 17.3.5 (lÑÿÀ) Áò~17.8¥�^cXeO�

+ ÿÀ�m

Ó�N� ëYN�

Á(½“gd”ÿÀ�m. (J«: IKÑñ�Y. )

SK 17.3.6 (¢#¼f) Áò~17.8¥�^cXeO�

+ 8Ü

8Ü +

Ó�N� N�

Á(½d�÷v�5��é�. (J«: r+þ�(�¢#K=�. )

SK 17.3.7 (ü��) ®�+G,H, Ó�N�ϕ : G→ H, ¦y: ϕ´ü��

¿©7�^�´

∀+K, Ó�N�µ, ν : K → G,ϕ ◦ µ = ϕ ◦ ν ⇐⇒ µ = ν

SK 17.3.8 (ÝK+) y²: gd+G÷vXe�ÝK5

∀+A,B, ÷��ϕ : A� B

Ó�N�β : G→ B

∃Ó�N�α : G→ A

s. t. ϕ ◦ α = β

P

α

��

β

��A ϕ

// // B

·��±^Xe�ó£ã:

þã·K´`, ?ÛG�A�?Ûû+�Ó�N�Ñ�±J,(ë�S

K??)�G�A�Ó�N�.

17.4 È, {È

ü�8Ü�CartesiusÈ��±^�5��x.

14 1�ÔÙ �5�

~ 17.11 (È) ®�8ÜA,B, y²

(C, π, ρ)|C = A×B, π : A×B → A, ρ : A×B → B´ÝKN�

÷vXe��5�

∀8ÜDN�α : D → A, β : D → B

∃!N�γ : D → C

s. t. π ◦ γ = α, ρ ◦ γ = β

A Cπoo ρ // B

D

α

``

β

>>

γ

OO

y² �

γ : D → C = A×Bx 7→ (α(x), β(x))

w,π ◦ γ = α, ρ ◦ γ = β, ��5�Î´w,�. �

e¡·�ò�Þ��, ïÄ“{È”. �d·�kÚ\��k'8ÜØ�

�E|.

½Â 17.12 (Ã�¿) ®�8Üx{Xi}i∈I , ¡⊔i∈I

(Xi × {i})

�8Üx{Xi}i∈IÃ�¿, ¿{üP�⊔i∈I Xi.

�Ò´`=B�x8Ükú��, ·��±À�ØÓ��, þ¡

¤��Ã�´�z��þI\, l ?Û8Üx, ·�Ñ�±À�´Ø

��, l kÃ�¿.

~ 17.13 ({È) ®�8ÜA,B, (½

(C, ι, κ)|C´8Ü, N�ι : A→ C, κ : B → C

17.4 È, {È 15

÷vXe��5�

∀8ÜDN�α : A→ D,β : B → D

∃!N�γ : C → D

s. t. γ ◦ ι = α, γ ◦ κ = β

Aι //

α

C

γ

��

Bκoo

β~~D

) dþ½Â, ·��±�½A,BÃ�, ·�y²

C = A tB, ι : A→ A tB, κ : B → A tB´�¹N�

éuα : A→ D,β : B → D, ��E

γ(x) =

α(x) x ∈ A

β(x) x ∈ B

w,γ ◦ ι = α, γ ◦ κ = β, ��5�Î´w,�. �

�Ú�PÒ,·�¬3�«�¸e,P÷vþã�5��é��AqB,

T�¦ÈÎÒΠ��, �Ã�¿ÎÒ�q. ~X3+��¸e, ÷vT�5�

�+KØ´Ã�¿ùo{ü.

~ 17.14 (gdÈ) ®�+A,B, (½

(C, ι, κ)|C´+, Ó�N�ι : A→ C, κ : B → C

÷vXe��5�

∀+DÓ�N�α : A→ D,β : B → D

∃!Ó�N�γ : C → D

s. t. γ ◦ ι = α, γ ◦ κ = β

Aι //

α

C

γ

��

Bκoo

β~~D

) P

A∗ = A− {1} B∗ = B − {1}

16 1�ÔÙ �5�

Ù¥ü�1©O�gA,B�ü �. �±A∗ t B∗�i1L�ücLΣ, ½Â

'X

a . . . b cd e . . . f

∼ a . . . b g e . . . f:⇐⇒ c, dÓ�áuA½B, cd = g

a . . . b cd e . . . f

∼ a . . . b e . . . f:⇐⇒ c, dÓ�áuA½B, cd = 1

�¡��, PÙ�d4��'X≡(ë�SK??), PC = Σ/ ≡, 3Cþ, ¦

{(�Ó��i�©�. e¡5y²�5�. éuÓ�N�α : A → D,β :

B → D, ½Â

γ(x) =

α(x) x ∈ A

β(x) x ∈ B

Ï�C¥?Û��Ñk/ªx1 . . . xn, �p�“≡”'X��Ó, ù½

Â��û½Â�N�. ��5�´w,�. �

¤¢{È, å��´8Ü¥�Ã�¿��^, Ï�A,B��3g��“'

X”, ��¡��gdÈ.

½Â 17.15 (gdÈ, KÜÈ) ®�+A,B, Pþ~(½�C = A q B, ¡

�AÚB�gd(free)È, KÜ(amalgamated)È½{È(coproduct).

SK 17.4

SK 17.4.1 ò~17.11�^cXeO�:

(1)

8Ü +

N� Ó�N�

Áy²+��È÷v�5�.

17.4 È, {È 17

(2)

8Ü ÿÀ�m

N� ëY

Áy²¦È�m÷v�5�.

SK 17.4.2 ®�8ÜA,B, PF (X)L«±X�i1L�gd+, ¦y:

F (A)q F (B) ∼= F (A tB)

(J«: Ó�N�F (A)→ DÚA→ D��éA)

SK 17.4.3 ®�+A,B,C, ¦y:

(AqB)q C ∼= Aq (B q C)

(J«: �±k½ÂAqB q C2y²)

SK 17.4.4 Á�EÃ¡ÈÚ{È.

SK 17.4.5 �Ñ3ÿÀ�m(ëYN�)�¸e�Ã¡È´TychonoffÿÀ,

��ÿÀ. (ë�SK??)

SK 17.4.6 ò~17.13¥�^cXeO�:

8Ü Abel+

N� Ó�N�

Áy²A ⊗ B÷v�5�. ¿y²í2�Ã¡�¹, È��È, {È�S

K??½Â��Ú.

SK 17.4.7 ò~17.13¥�^cXeO�:

8Ü F -�5�m

N� Ó�N�

Áy²A ⊗ B(ÜþÈ)÷v�5�. (J«: �âþK��A × B → D �N

�, y²ù´V�5N�. )

18 1�ÔÙ �5�

SK 17.4.8 (©��m)

8Ü ÿÀ�m

N� ëYN�

e3AtBþ½ÂÿÀOp(AtB) = Op(A) tOp(B), ù�¡�©��m. ¦

y: d�A tB÷v�5�.

SK 17.4.9 e¡½Â+Ó��ÈÚ{È, ¦yXe�5�

∀+A,B,A′, B′

Ó�N�ϕ : A→ A′, ψ : B → B′

∃!Ó�N�σ : A⊗B → A′ ⊗B′

s. t.mã��

A

ϕ

��

A⊗Boo //

σ

��

B

ψ

��A′ A′ ⊗B′oo // B′

∀+A,B,A′, B′

Ó�N�ϕ : A→ A′, ψ : B → B′

∃!Ó�N�σ : AqB → A′ qB′

s. t.mã��

A //

ϕ

��

AqBσ

��

Boo

ψ

��A′ // A′ qB′ B′oo

ùü�σ©OP�ϕ⊗ ψÚϕq ψ.

17.5 íÑ

c¡·��ÑÃ�¿��5�, e¡·��Ñ“k�¿”�L�.

~ 17.16 ®�8ÜA,B, y²

(A ∪B,α, β)|A ∪B´8Ü, α : A→ A ∪B, β : B → A ∪B

17.5 íÑ 19

÷vXe��5�

∀8ÜC, N�µ : A→ C, ν : B → C,

s. t. µ ◦ ι = ν ◦ κ,∃!γ : A ∪B → C

s. t. γ ◦ α = µ, γ ◦ β = ν

C

A ∪B

γcc

Aαoo

µoo

B

β

OOν

QQ

A ∩B

ι

OO

κoo

Ù¥ι : A ∩B → A, κ : A ∩B → B´�¹N�.

y² �

γ(x) =

µ(x) x ∈ A

ν(x) x ∈ B

ù´û½Â�, Ï�

µ(x) = ν(x) x ∈ A ∩B

N´�yù÷v�5�. �

ùéu·��Äù��¯K, XJ^SO�KA ∩ B, S → A,S →B�N�Ø´ü�, @o(JXÛQ?

~ 17.17 (íÑ�) ®�8ÜS,A,B, N�ι : S → A, κ : S → B, (½

(T, α, β)|T´8Ü, N�α : A→ T, β : B → T , ÷vα ◦ ι = β ◦ κ

÷vXe��5�

∀8ÜC, N�µ : A→ C, ν : B → C,

s. t. µ ◦ ι = ν ◦ κ,∃!γ : T → C

s. t. γ ◦ α = µ, γ ◦ β = ν

C

T

γ``

Aαoo

µoo

B

β

OOν

OO

S

ι

OO

κoo

20 1�ÔÙ �5�

) k�A tB, ��½ÂA tBþ�'X

a ∼ b :⇐⇒ ∃s ∈ S, s. t. ι(s) = a, κ(s) = b

�ÄÙ�d4��'X≡(ë�SK??), PT = A tB/ ≡.

α : A → T

a 7→ [a]

β : B → T

b 7→ [b]

Ù¥[a], [b]©O´a, b¤3��da. N´�y, α ◦ ι = β ◦ κ. e¡�y÷v

�5�,

γ([x]) =

µ(a), [x] = [a], a ∈ A

ν(b), [x] = [b], b ∈ B

ù´û½Â�, Ï�[x] = [a] 6= [b] ⇒ [x] = {a}, [x] = [b] 6= [a] ⇒ [x] = {b},[x] = [a] = [b], K

(x ∼ y ⇒ µ(x) = ν(y))⇒ (a ≡ b⇒ µ(a) = ν(b))

N´�yγ ◦ α = µ, γ ◦ β = ν. ��5�Î´N´�. �

þ¡L§¥¤��, Ã�´r(ι(s), κ(s))ù��:“Ê”3�å.

½Â 17.18 ·�¡þ~¥�÷v�5��é�¡�íÑ(pushout), ��ã

¡�íÑ�.

e¡·�r¯K��+þ.

~ 17.19 ®�+G,A,B, Ó�N�ι : G→ A, κ : G→ B, (½

(T, α, β)|T´+, Ó�N�α : A→ T, β : B → T , ÷vα ◦ ι = β ◦ κ

17.5 íÑ 21

÷vXe��5�

∀+C, Ó�N�µ : A→ C, ν : B → C,

s. t. µ ◦ ι = ν ◦ κ,∃!γ : T → C

s. t. γ ◦ α = µ, γ ◦ β = ν

C

T

γ``

Aαoo

µoo

B

β

OOν

OO

G

ι

OO

κoo

) �KÜÈAqB, �Ä

S = {ι(g)κ(g−1)|g ∈ G} ⊆ AqB

PΣ = {K|S ⊆ K � A q B}, PN = ∩Σ, =�N¹S��5f+��, ù�

Î´��5f+, �´¹S��5f+. �

AqB/N α(a) = aN β(b) = bN

w,÷vα ◦ ι = β ◦ κ. e¡y²�5�, éAqB¥�?Û��üix, �

γ′(x) =

µ(a), x ∈ A

ν(b), x ∈ B

ù(½��û½Â�Ó�.

∀s ∈ S, γ′(s) = e⇒ S ⊆ Ker γ′ ⇒ N ⊆ Ker γ′

d+Ó�1�½n

T ← Im γ′ ∼=AqBKer γ′

∼=AqB/NKer γ′/N

← AqB/N

ù(�

γ(x) =

µ(a), xN = aN, a ∈ A

ν(b), xN = bN, b ∈ B

22 1�ÔÙ �5�

�û½Â, ��5½´. �

þã¤��Ú8ÜÜ©aq, �=òéA:ÊÜ, ¿“)¤”��5f

+. 'u“)¤”�Vg, Öö�±ë�?Û��êÖ.

½Â 17.20 (KÜÈ) ®�+G,A,B, Ó�N�ι : G→ A, κ : G→ B, Pþ

~)¤�T = AqGB, ¡�AÚB�éuG�KÜÈ.

w,, ��KÜÈ�±w¤´�éu²�+{e}�.

SK 17.5

SK 17.5.1 ¦y:

F (A ∪B) = F (A)∐

F (A∩B)

F (B)

SK 17.5.2 (÷��) ®�+G,H, Ó�N�ϕ : G→ H, ¦y: ϕ´÷��

¿©7�^�´

∀+K, Ó�N�µ, ν : H → K,µ ◦ ϕ = ν ◦ ϕ ⇐⇒ µ = ν

(J«: �ÄH qImϕH��5�)

SK 17.5.3 ��Ó�

ϕ : Z → Qn 7→ n/1

¦y:

∀�R, Ó�N�µ, ν : Q→ R,µ ◦ ϕ = ν ◦ ϕ ⇐⇒ µ = ν

17.6 .£ 23

17.6 .£

e¡·�òíÑ��ÞÑ�L5.

~ 17.21 ®�8ÜS,A,B, N�π : A→ S, ρ : B → S, (½

(T, α, β)|T´8Ü, N�α : T → A, β : T → B, ÷vπ ◦ α = ρ ◦ β

÷vXe��5�

∀8ÜC, N�µ : C → A, ν : B → B,

s. t. π ◦ µ = ρ ◦ ν,∃!γ : C → T

s. t. α ◦ γ = µ, β ◦ γ = ν

Cγ

µ

��

ν

##

Tα //

β

��

A

π

��B ρ

// S

) �A×B�f8

T = {(a, b)|π(a) = ρ(b)}

N�α : (a, b) 7→ a, β : (a, b) 7→ b. w,÷vπ ◦ α = ρ ◦ β. e¡y²�5�,

�

γ : C → T

c 7→ (µ(c), ν(c))

ù´û½Â�. �½´. �

½Â 17.22 ·�¡þ~¥�÷v�5��é�¡�.£(pullback), ��ã

¡�.£�.

~ 17.23 ®�+G,A,B, Ó�N�π : A→ S, ρ : B → S, (½

(T, α, β)|T´+, +Ó�N�α : T → A, β : T → B, ÷vπ ◦ α = ρ ◦ β

24 1�ÔÙ �5�

÷vXe��5�

∀+C, +Ó�N�µ : C → A, ν : B → B,

s. t. π ◦ µ = ρ ◦ ν,∃!γ : C → T

s. t. α ◦ γ = µ, β ◦ γ = ν

Cγ

µ

��

ν

##

Tα //

β

��

A

π

��B ρ

// S

½Â 17.24 (n�È) ®�+G,A,B, Ó�N�π : A→ S, ρ : B → S, Pþ

~)¤�T = A⊗GB, ¡�AÚB�éuG�n�È(fibered product).

éõ(JÑ�±^.£�ÚíÑ�L«, e¡[ÞA~, �Öö�

¤�y.

~ 17.25 ®�8ÜA,B, y²Xe�´.£�

A ∩B ⊆ //

⊆��

B

⊆��

A⊆// A ∪B

~ 17.26 ®�8ÜA,B, N�f : A → B, S ⊆ B, y²Xe�´.£�

f−1(S)f //

⊆��

S

⊆��

Af

// B

~ 17.27 ®�+G,H, Ó�ϕ : G→ H, y²Xe�´.£�

Kerϕ

⊆��

// {e}

��G ϕ

// H

17.6 .£ 25

~ 17.28 ®�+G,H, Ó�ϕ : G→ H, y²Xe�´íÑ�

CoKerϕ {e}oo

H

π

OO

Gϕoo

OO

SK 17.6

SK 17.6.1 (½~17.23¥�(T, α, β). (J«: ��D�8ÜÜ©�(J

�T±+(�. )

SK 17.6.2 y²~17.25, ~17.26, ~17.27, ~17.28´.£½íÑ�.

SK 17.6.3 ¦y:

A

φ

��

α // B

ψ

��X χ

// Y

B

ψ

��

β // C

ω

��Y η

// Z

´.£�⇒A

ψ

��

β◦α // C

ω

��X

η◦χ// Z

´.£�

SK 17.6.4 ®�+G, ¦y:

(1)H,S ≤ G. e��´íÑ�

H ∩ S ⊆ //

⊆��

H

⊆��

S⊆

// G

(2)N,K �G. e��´.£�

G/KN G/Noo

G/K

OO

G

OO

oo

26 1�ÔÙ �5�

SK 17.6.5 ®�8ÜX, ü��d'X≡,∼, éA�g,N�©O�π, ρ,

¦y: e��´.£�

X/ ≈ X/ ∼oo

X/ ≡

OO

X

π

OO

ρoo

Ù¥, ≈´≡ ∪ ∼��d4�(ë�SK??).

SK 17.6.6 ®�8ÜA,B,C,D|¤�.£�:

Aψ //

��

B

��C ϕ

// D

¦y: ϕ´ü�⇒ψ´ü�. (J«: |^�5�)

1�lÙ �Æ�Ä�Vg

e¡·�5�ã�Æ�½Â.

18.1 �Æ�½ÂÚ~f

½Â 18.1 (�Æ) ��Æ(category)C´�±e]�:

(1)a: é�(object) Ob C(2)a: ��(morphism) Mor C(3)N�: ½Â�(domain) dom : Mor C → Ob CN�: ��(codomain) cod : Mor C → Ob C

(4)N�:ð��(identity) id : Ob(C)→Mor(C)(5)N�:EÜ(composite) ◦ : C →Mor Ca: T�(composable)é C = {(g, f)|f, g ∈Mor C, cod f = dom g}

Ù¥EÜ$�÷v

(1)(�½Â�) dom(g ◦ f) = dom f

(��) cod(g ◦ f) = cod g

(2)(ü Æ) g idA = g �A = cod g

(3)((ÜÆ) h ◦ (g ◦ f) = (h ◦ g) ◦ f

µ5 18.2 �Bå�, ·��½±ePÒ

27

28 1�lÙ �Æ�Ä�Vg

A ∈ C :⇐⇒ A ∈ Ob C

Af // B :⇐⇒ dom f = A, cod f = B

f : A→ B :⇐⇒ dom f = A, cod f = B

HomC(A,B) := {f ∈Mor C|f : A→ B} = dom−1A ∩ cod−1B

gf := g ◦ fhgf := (hg)f = h(gf)

gf∃ :⇐⇒ cod f = dom g

µ5 18.3 þã½Â¥��N��A3��ã¥´

Bg // C

A

f

OO

gf

?? A

A

idA

OO

Bgoo

g__ A

gf

��f

��

hgf // D

B g//

hg

44

C

h

OO

dom(g ◦ f) = dom f

cod(g ◦ f) = cod gg idA = g h ◦ (g ◦ f) = (h ◦ g) ◦ f

~ 18.4 (8Ü�Æ) 8Ü�ÆSet´�

� ObSet := {A|A´8Ü}� MorSet := {f : A→ B|f´lA�B�N�}

idA�Aþ�ð�N�, EÜ�N��EÜ. �Bå�/ª/½Â∅→ A��

��¹N�.

~ 18.5 (�ê(�) +�ÆGrp´�

� ObGrp := {A|A´+}� MorGrp := {f : A→ B|f´lA�B�+Ó�N�}

idA�Aþ�ð�N�, EÜ�N��EÜ.

Ó�, aq/kAbel+�ÆAb, ��ÆRng, ��ÆCRng.

18.1 �Æ�½ÂÚ~f 29

~ 18.6 (ÿÀ(�) ÿÀ�m�ÆTop´�

� ObTop := {A|A´ÿÀ�m}� MorTop := {f : A→ B|f´lA�B�ëYN�}

idA�Aþ�ð�N�, EÜ�N��EÜ.

Ó�, aq/kHausdorffÿÀ�m�ÆHaus.

~ 18.7 ('X�Æ) 'X�ÆRel´�

� ObRel := {A|A´8Ü}� MorRel := {R : A→ B|R´AÚBþ�'X}

idA�Aþ�ü 'X, EÜ�'X�EÜ.

~ 18.8 (Ý�Æ) ®��N�R, MatR´�

�ObMatR = N�MorMatR = {(aij)m×n|m,n ∈ N, aij ∈ R}

idn�n�ü , EÜ�Ý�¦{.

SK 18.1

SK 18.1.1 ½Â:

(1)f´ü��(monomorphism)�fg = fh ⇐⇒ g = h;

(2)f´÷��(epimorphism)�gf = hf ⇐⇒ g = h;

(3)fk�_�∃g, gf = id;

(4)fkm_�∃g, fg = id.

¦y:

(1)k�_⇒ü��; km_⇒÷��;

(2)��Ø�(. (J«: �ÄÿÀ�m�ëYN��¦. )


SK 18.1.2 ®�ÿÀ�mX, y²:

(1)A ⊆ X, f�mA÷vXe��5�, Pι : A→ X´�¹N�

éuD�Ù¦ÿÀ�A′ = A

eι : A′ → X´ëYN�,

KidA : A′ → A´ëYN�.

Aι // X

A′ι

>>

idA

OO

(2)X/ ∼´X�û8, û�mX/ ∼÷vXe��5�, Pπ : X →X/ ∼´g,N�

éuD�Ù¦ÿÀ�(X/ ∼)′ = X/ ∼eπ : X → (X/ ∼)′´ëYN�,

KidX/∼ : X/ ∼→ (X/ ∼)′´ëYN�.

X

##

π // X/ ∼

idX/∼

��(X/ ∼)′

SK 18.1.3 (:ÿÀ�Æ) ½Â:ÿÀ�ÆTop∗

� ObTop∗ := {(X,x)|x ∈ X,X´ÿÀ�m}� MorTop∗ := {f : (X,x)→ (Y, y)|fëY, �f(x) = y}

�yù´��Æ.

SK 18.1.4 éu��ýS8(ë�SK??)P , ½Â

�ObP = P

�HomP (x, y) =

{(x, y)} x ≤ y

∅ otherwise

�yù´��Æ.

18.2 ¼f�½ÂÚ~f

e¡·�0��Æm�“Ó�”.

18.2 ¼f�½ÂÚ~f 31

½Â 18.9 (¼f) ®��ÆA,B, N�

F :

ObA → ObB

MorA →MorB

�¡�¼f(functor)�

(1) ∀f : A→ B, F (α) : F (A)→ F (B)

(2) ∀A ∈ A, F (idA) = idF (A)

(3) ∀gf∃, F (gf) = F (g)F (g)

ÿI5¿�´, ¼fØ=�Ñé��m�éA, ��Ñ��m�

éA. ù4·�kÅ¬!Ø“p�(induce)Ñ”��N�. ù��á3 �

�ÆS¥õØ�ê, [{üÞA~.

~ 18.10 (�8¼f) �8¼f

P : Set // Set

A � // 2A

f : A→ B � // Pf

Pf : 2A → 2B

S 7→ f(S)

~ 18.11 (¢#¼f) ��Æ�Abel�Æ�¢#¼f

U : Rng // Ab

(R,+, ·) � // (R,+)

f � // f

Ó��kGrp→ Set, Top→ Set�¢#¼f.

~ 18.12 (gd+¼f) 8Ü�Æ�+�Æ�gd+¼f

F : Set // Grp

A � // F (A)

f : A→ B � // f : F (A)→ F (B)


Ù¥f´dXeL§p��.

ιB ◦ f : A→ F (B)

∃!f : F (A)→ F (B)

s. t. f ◦ ιA = ιB ◦ f

AιA //

!!f

��

F (A)

f

��B ιB

// F (B)

~ 18.13 +�Æ¥, éu�½�+B, A 7→ A× B/¤¼f, A 7→ A q B�/¤¼f.

·��±!Ø¼f�5�, ~X�âSK17.4.2, gd+¼f´�

±(preserve){È�. 2~X+�¢#¼f´�±È�, =+��È��8Ü

ó=ü�+��8Ü�CartesiusÈ. 2~Xgd+¼fòü�N�ü�,

ò÷�N�÷�(SK18.2.3), l 3Ó�¿Âe, A ⊆ B, KF (A) ⊆ F (B),

A´B�û8, KF (A)´F (B)�û8.

'u�ÆØ�k�õ�\�?Ø, �ë��õÖ7.

SK 18.2

SK 18.2.1 �y~18.12��÷vé¼f��¦.

SK 18.2.2 ò~18.13éé��*¿¤�¼f.

SK 18.2.3 y²gd+¼fòü�N�ü�, ò÷�N�÷�. (J«: ü

��âgd+�(�y², ÷��âSK17.5.2(Ü�5�)

SK 18.2.4 éuü� S8, ÀÙ��Æ(ë�SK18.1.4), K¼f´�

o?

¢Ú

Abel+�Æ, 28

HausdorffÿÀ�m�Æ, 29

��, 27

Ä�¢{, 2

ücL, 10

ü��, 13, 29

:ÿÀ�Æ, 30

;�, 3

½Â�, 27

é�, 27

�Æ, 27

�5�, 3

EÜ, 27

'X�Æ, 29

¼f, 31

ð��, 27

��Æ, 28

È, 14

8Ü�Æ, 28

��Æ, 28

Ý�Æ, 29

�i, 10

÷��, 22, 29

�8¼f, 31

©��m, 18

T�é, 27

+�Æ, 28

KÜÈ, 16, 22

V�55, 7

��, 27

ÝK+, 13

33

34 ¢Ú

ÿÀ�m�Æ, 29

�k5�, 3

Ã�¿, 14

n�È, 24

�, 6

¢#¼f, 31

{Ø, 5

{È, 16

{�, 6

Üþ, 9

ÜþÈ, 9

g,, 3

gdAbel+, 12

gdÈ, 16

gd+, 12

gd+¼f, 31

i, 10

i1L, 10

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