Category

A category $\mathscr{C}$ is a collection of objects, denoted $\text{obj}{\mathscr{C}}$ together with a collection of classes of morphisms between pairs of objects, each of which is denoted $\text{Mor}{(a,b)}$ for any pair $a,b \in \text{obj}{\mathscr{C}}$, such that the following data is given:

(1). For any objects $a,b,c \in \text{obj}{\mathscr{C}}$, there exist composition maps

associating to each pair of morphisms $(g,f)$ a morphism $g \circ f$ called the composition of the morphism $g$ with the morphism $f$.

(2). For any objects $a,b,c,d \in \text{obj}{\mathscr{C}}$, and for any morphisms $f \in \text{Mor{(a,b)}}$, $g \in \text{Mor{(b,c)}}$, and $h \in \text{Mor{(c,d)}}$, the composition maps between morphisms associate wherever they exist; that is:

whenever either $h \circ (g \circ f)$ or $(h \circ g) \circ f$ exists.

(3). For any object $a \in \text{obj}{\mathscr{C}}$, there is a morphism $id_a \in \text{Mor}{(a,a)}$, called the identity morphism on $a$, such that for any objects $b,c \in \text{obj}{\mathscr{C}}$, and for any morphisms $f \in \text{Mor}{(a,b)}$ and $g \in \text{Mor}{(c,a)}$, the following holds:

Given objects $a,b \in \text{obj}{\mathscr{C}}$, we denote a morphism $f \in \text{Mor}{(a,b)}$ by $f:a \rightarrow b$.