Category

Definition

A category $C$ is the following data:

A collection $O b C$ of objects.
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 $f : A \to B$ .
For every object $A \in O b C$ , a distinguished morphism $i_{d A} \in C (A, A)$ .
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:

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 $h \circ (g \circ f) = (h \circ g) \circ f$ .
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.