Category. Definition.

Category

A category $\mathrm{C}$ consists of:

• a class of objects, denoted as $\mathrm{Obj}(\mathrm{C})$
• a set, denoted as ${\mathrm{Hom}}_{\mathrm{C}}(A,B)$, for any objects $A$ and $B$ of $\mathrm{C}$. The elements are called morphisms.

The set of morphisms must have the following properties:

Identity

For each object $A\in \mathrm{Obj}(\mathrm{C})$, there exists (at least) one morphism [$?\in {\mathrm{Hom}}_{\mathrm{C}}(?,?)$], called the identity on $A$.

Composition

Morphisms can be composed: any two morphisms $f\in {\mathrm{Hom}}_{\mathrm{C}}(A,B)$ and $g\in {\mathrm{Hom}}_{\mathrm{C}}(B,C)$ [determine/imply what??].

Associativity of composition

For any $f\in {\mathrm{Hom}}_{\mathrm{C}}(A,B)$, $g\in {\mathrm{Hom}}_{\mathrm{C}}(B,C)$ and $h\in {\mathrm{Hom}}_{\mathrm{C}}(C,D)$, we have:

[$$?=?$$]

Identity law

The identity morphisms are identities with respect to composition. For any $f\in {\mathrm{Hom}}_{\mathrm{C}}(A,B)$, we have:

[$$f?=f,?f=f$$]

Morphism sets are disjoint

For any $A,B,C,D\in \mathrm{Obj}(\mathrm{C})$, then ${\mathrm{Hom}}_{\mathrm{C}}(A,B)$ and ${\mathrm{Hom}}_{\mathrm{C}}(C,D)$ are disjoint unless $A=C$ and $B=D$.