Definition 4.1.1. A category \(\mathscr C\) consists of:
- a collection \(\operatorname {ob}(\mathscr C)\) of objects;
- for each \(A,B\in \operatorname {ob}(\mathscr C),\) a hom-set \(\mathscr C(A,B)\) of morphisms from \(A\) to \(B\);
- for each \(A,B,C\in \operatorname {ob}(\mathscr C),\) a function \(\mathscr C(B,C) \times \mathscr C(A,B)\to \mathscr C(A,C)\) that maps \((g,f)\) to \(g\circ f,\) called composition;
- for each \(A\in \operatorname {ob}(\mathscr C),\) a morphism \(1_A\) of \(\mathscr C(A,A),\) called the identity on \(A,\) satisfying the following axioms:
- associativity: for each \(f\in \mathscr C(A,B), g\in \mathscr C(B,C),\) and \(h\in \mathscr C(C,D)\)