Category

Definition

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

• Objects: A collection $\mathrm{O}\mathrm{b}\mathcal{C}$ of objects.
• Morphisms: For any objects $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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 ${\mathrm{H}}_{\mathrm{o}}(A,B)$ or simply $\mathrm{H}\mathrm{o}\mathrm{m}(A,B)$.
• Identity morphism: For every object $A\in \mathrm{O}\mathrm{b}\mathcal{C}$, a distinguished morphism ${\mathrm{i}}_{\mathrm{d}}\in \mathcal{C}(A,A)$. This is called the identity morphism of $A$.
• Composition rule: For $A,B,C\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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 $A,B,C,D\in \mathrm{O}\mathrm{b}\mathcal{C}$, 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 $A,B\in \mathrm{O}\mathrm{b}\mathcal{C}$, with $f\in \mathcal{C}(A,B)$, we have $f\circ {\mathrm{i}}_{\mathrm{d}}={\mathrm{i}}_{\mathrm{d}}\circ f=f$.

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.