category_._set

Category . set theory

context	$\mathcal{O},M$ … set
definiendum	$\langle \mathcal{O},M,id,* \rangle \in \mathrm{it}$
definition	$\mathrm{Mor}:\mathcal{O}\times\mathcal{O}\to M$
definition	$\circ:{\large\prod}_{A,B,C:\mathcal{O}}\,\mathrm{Mor}(B,C)\times\mathrm{Mor}(A,B)\to\mathrm{Mor}(A,C)$
definition	$id:{\large\prod}_{A:\mathcal{O}}\,\mathrm{Mor}_O(A,A)$
postulate	$\mathrm{Mor}(A,B)\cap\mathrm{Mor}(U,V)\ne\emptyset\implies U=A\land V=B$
postulate	$(g\circ f)\circ h=g\circ (f\circ h)$
postulate	$f\circ id_A=id_A\circ f=f$

Within set theory, we can define a category as quintuple given by two sets and two (polymorphic) maps into them.

The three axioms say the following: The hom-sets are pairwise disjoint, the composition is associative and $id$ denotes the identity.