Category: Difference between revisions

Latest revision as of 23:25, 9 December 2008

Definition

A category $C$ is the following data:

Objects: A collection $O b C$ of objects.
Morphisms: For any objects $A, B \in O b C$ , a collection $C (A, B)$ of morphisms. Every element in $C (A, B)$ is termed a morphism from $A$ (i.e., with source or domain $A$ ) to $B$ (i.e., with target or co-domain $B$ ). The morphism sets for different pairs of objects are disjoint. Note that $f \in C (A, B)$ is also written as $f : A \to B$ . The collection $C (A, B)$ is sometimes also denoted $H_{o m C} (A, B)$ or simply $H o m (A, B)$ .
Identity morphism: For every object $A \in O b C$ , a distinguished morphism $i_{d A} \in C (A, A)$ . This is called the identity morphism of $A$ .
Composition rule: For $A, B, C \in O b C$ , a map, called composition of morphisms, from $C (B, C) \times C (A, B)$ to $C (A, C)$ . This map is denoted by $\circ$ .

satisfying the following compatibility conditions:

Associativity of composition: For $A, B, C, D \in O b C$ , with $f \in C (A, B), g \in C (B, C), h \in C (C, D)$ , we have $h \circ (g \circ f) = (h \circ g) \circ f$ .
Identity behaves as an identity: For $A, B \in O b C$ , with $f \in C (A, B)$ , we have $f \circ i_{d A} = i_{d B} \circ f = f$ .

The collections of objects and morphisms need not be sets. If the collection of objects is a set, and the collection of morphisms between any two objects is a set, the category is termed a small category. If the collection of morphisms between any two objects is a set, the category is termed a locally small category.

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 A '''category''' <math>\mathcal{C}</math> is the following data:
-* A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
+* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
-* For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>.
+* '''Morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
-* For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>.
+* '''Identity morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
-* For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
+* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
 satisfying the following compatibility conditions:
-* For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
+* '''Associativity of composition''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
-* For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
+* '''Identity behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
 The collections of objects and morphisms need not be sets. If the collection of objects is a set, ''and'' the collection of morphisms between any two objects is a set, the category is termed a [[small category]]. If the collection of morphisms between any two objects is a set, the category is termed a [[locally small category]].