Category

Definition

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

1. a class $\mathrm{Ob}(\mathcal{C})$, whose members are called the objects of $\mathcal{C}$;
2. for each pair of objects $A, B$, a class $\Hom_{\mathcal{C}}(A, B)$ of morphisms from $A$ to $B$, written $f \colon A \to B$, with these classes pairwise disjoint;
3. for each triple of objects $A, B, C$, a composition map $\Hom_{\mathcal{C}}(B, C) \times \Hom_{\mathcal{C}}(A, B) \to \Hom_{\mathcal{C}}(A, C)$, written $(g, f) \mapsto g \circ f$;
4. for each object $A$, an identity morphism $1_A \in \Hom_{\mathcal{C}}(A, A)$.

These data must satisfy two axioms:

1. associativity: $h \circ (g \circ f) = (h \circ g) \circ f$ for all $f \colon A \to B$, $g \colon B \to C$ and $h \colon C \to D$;
2. identity: $f \circ 1_A = f = 1_B \circ f$ for all $f \colon A \to B$.