Category

Definition

A category ${\mathcal {C}}$ is the following data:

• Objects: A collection $\operatorname {Ob} {\mathcal {C}}$ of objects.
• Morphisms: For any objects $A,B\in \operatorname {Ob} {\mathcal {C}}$, a collection ${\mathcal {C}}(A,B)$ of morphisms. Every element in ${\mathcal {C}}(A,B)$ 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 $f\in {\mathcal {C}}(A,B)$ is also written as $f:A\to B$. The collection ${\mathcal {C}}(A,B)$ is sometimes also denoted $\operatorname {Hom} _{\mathcal {C}}(A,B)$ or simply $\operatorname {Hom} (A,B)$.
• Identity morphism: For every object $A\in \operatorname {Ob} {\mathcal {C}}$, a distinguished morphism $\operatorname {id} _{A}\in {\mathcal {C}}(A,A)$. This is called the identity morphism of $A$.
• Composition rule: For $A,B,C\in \operatorname {Ob} {\mathcal {C}}$, a map, called composition of morphisms, from ${\mathcal {C}}(B,C)\times {\mathcal {C}}(A,B)$ to ${\mathcal {C}}(A,C)$. This map is denoted by $\circ$.

satisfying the following compatibility conditions:

• Associativity of composition: For $A,B,C,D\in \operatorname {Ob} {\mathcal {C}}$, with $f\in {\mathcal {C}}(A,B),g\in {\mathcal {C}}(B,C),h\in {\mathcal {C}}(C,D)$, we have $h\circ (g\circ f)=(h\circ g)\circ f$.
• Identity behaves as an identity: For $A,B\in \operatorname {Ob} {\mathcal {C}}$, with $f\in {\mathcal {C}}(A,B)$, we have $f\circ \operatorname {id} _{A}=\operatorname {id} _{B}\circ f=f$.

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.