1. Categories

Definition. A category consists of

(i) a class ${\cal C}$ (whose members are called objects), together with

(ii) for each $A,B \in {\cal C}$, a set Morph$(A,B)$ (whose members are called morphisms), and

(iii) for each $A,B,C \in {\cal C}$, an operation $\circ$ between morphisms, so that for $\phi \in$   Morph$(A,B)$ and $\psi \in$   Morph$(B,C)$, $\phi \circ \psi \in$   Morph$(A,C)$ is defined,

with the properties

(a) $\phi \circ (\psi \circ \eta) = (\phi \circ \psi) \circ \eta$ for morphisms between successive objects, and

(b) for each $A \in {\cal C}$ there is an element 1$_ A \in$   Morph$(A,A)$ such that 1$_ A \circ \psi = \psi$ and $\phi \circ$   1$_ A = \phi$ for all $\psi \in$   Morph$(A,B)$ and $\phi \in$   Morph$(B,A)$, for any $B \in {\cal C}$.

In most familiar examples, the objects are sets, the morphisms are functions, and $\circ$ is composition (written composing to the right). The examples (1), (2), (3) are of this kind, but not all useful categories are.

We usually say ``the category ${\cal C}$'' with the associated morphisms and composition understood.

Exercise. Define what it should mean for two objects in a category to be isomorphic. Solution. $A \cong B$ means there exist morphisms $\phi \in$   Morph$(A,B)$ and $\psi \in$   Morph$(B,A)$ such that $\phi \circ \psi =$   1$_ A$ and $\psi \circ \phi =$   1$_ B$.