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積範疇 .
數學 分支範疇論 中,兩個範疇
C
,
D
{\displaystyle {\mathcal {C,D))}
之積 ,是集合 的笛卡兒積 的延申。乘積以
C
×
D
{\displaystyle {\mathcal {C\times D))}
表示,其結果又稱積範疇 [ 1] (英語:product category )。定義雙函子及多函子 時,要用到積範疇。
積範疇
C
×
D
{\displaystyle {\mathcal {C\times D))}
的組成部分有:
物件 ,為
有序對
(
A
,
B
)
{\displaystyle (A,B)}
,其中
A
{\displaystyle A}
是
C
{\displaystyle {\mathcal {C))}
的物件,而
B
{\displaystyle B}
是
D
{\displaystyle {\mathcal {D))}
的物件;
態射 ,由物件
(
A
1
,
B
1
)
{\displaystyle (A_{1},B_{1})}
至物件
(
A
2
,
B
2
)
{\displaystyle (A_{2},B_{2})}
的態射為:
有序對
(
f
,
g
)
{\displaystyle (f,g)}
,其中
f
:
A
1
→
A
2
{\displaystyle f:A_{1}\to A_{2))
是
C
{\displaystyle {\mathcal {C))}
的態射,
g
:
B
1
→
B
2
{\displaystyle g:B_{1}\to B_{2))
是
D
{\displaystyle {\mathcal {D))}
的態射;
態射間的複合運算,是逐個分量的複合:
(
f
2
,
g
2
)
∘
(
f
1
,
g
1
)
=
(
f
2
∘
f
1
,
g
2
∘
g
1
)
;
{\displaystyle (f_{2},g_{2})\circ (f_{1},g_{1})=(f_{2}\circ f_{1},g_{2}\circ g_{1});}
物件上的恆等態射,由各分量上的恆等態射組成:
1
(
A
,
B
)
=
(
1
A
,
1
B
)
.
{\displaystyle 1_{(A,B)}=(1_{A},1_{B}).}
兩個小範疇 之積,是其作為小範疇範疇
C
a
t
{\displaystyle \mathbf {Cat} }
的物件的乘積 。定義域為積範疇的函子 ,也稱為雙函子 。重要例子有Hom函子 ,其定義域為某範疇
C
{\displaystyle {\mathcal {C))}
及其對偶範疇
C
o
p
{\displaystyle {\mathcal {C))^{\mathrm {op} ))
之積:
H
o
m
:
C
o
p
×
C
→
S
e
t
.
{\displaystyle \mathrm {Hom} :{\mathcal {C))^{\mathrm {op} }\times {\mathcal {C))\to \mathbf {Set} .}
正如二元笛卡兒積 可以推廣到n 元笛卡兒積 ,範疇的二元積亦同樣可以推廣到
n
{\displaystyle n}
元積。若不別同構之異 ,則二元範疇積可交換 及可結合 ,故此
n
{\displaystyle n}
元推廣在理論上並無定義額外的新事物。
高階範疇論
基本概念 n -範疇
弱 n-範疇
雙範疇
三範疇
四範疇
闞複形
∞-廣群
∞-拓撲斯
強 n -範疇
範疇化 概念
2-群
2-環
En -環
(對稱 ) 么半範疇
n -群
n -么半群
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