Elementary category theory

We need to develop a fine theory of (n-)category theory in this lecture. But that will take time. Meanwhile, we use an easy part of elementary category theory as a convenient language.

DEFINITION 5.1   A category $\mathcal{C}$ is a collection of the following data

1. A collection $\operatorname{Ob}(\mathcal{C})$ of objects of $\mathcal{C}$ .
2. For each pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$ , a set
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)$$
   of morphisms.
3. For each triple of objects $X,Y,Z \in \operatorname{Ob}(\mathcal{C})$ , a map(``composition (rule)'')
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)\times \operatorname{Hom}_{\mathcal{C}}(Y,Z)\to \operatorname{Hom}_{\mathcal{C}}(X,Z)$$

satisfying the following axioms

1. $\operatorname{Hom}(X,Y)\cap \operatorname{Hom}(Z,W) =\emptyset$ unless $(X,Y)=(Z,W)$ .
2. (Existence of an identity) For any $X\in \operatorname{Ob}(\mathcal{C})$ , there exists an element $\operatorname{id}_X\in \operatorname{Hom}(X,X)$ such that
   $$\operatorname{id}_X \circ f=f,\quad g \circ \operatorname{id}_X=g$$
   holds for any $f \in \operatorname{Hom}(S,X), g \in \operatorname{Hom}(X,T)$ ( $\forall S,T \in \operatorname{Ob}(\mathcal{C})$ ).
3. (Associativity) For any objects $X,Y,Z,W \in \operatorname{Ob}(\mathcal{C})$ , and for any morphisms $f\in \operatorname{Hom}(X,Y), g\in \operatorname{Hom}(Y,Z), h \in \operatorname{Hom}(Z,W)$ , we have
   $$(f\circ g)\circ h =f\circ (g\circ h).$$