===== Category . set theory =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $\mathcal{O},M$ ... set |
| @#FFBB00: definiendum | @#FFBB00: $ \langle \mathcal{O},M,id,* \rangle \in \mathrm{it}$ |
| @#FF9944: definition  | @#FF9944: $\mathrm{Mor}:\mathcal{O}\times\mathcal{O}\to M$ |
| @#FF9944: definition  | @#FF9944: $\circ:{\large\prod}_{A,B,C:\mathcal{O}}\,\mathrm{Mor}(B,C)\times\mathrm{Mor}(A,B)\to\mathrm{Mor}(A,C)$ |
| @#FF9944: definition  | @#FF9944: $id:{\large\prod}_{A:\mathcal{O}}\,\mathrm{Mor}_O(A,A)$ |
| @#55EE55: postulate   | @#55EE55: $\mathrm{Mor}(A,B)\cap\mathrm{Mor}(U,V)\ne\emptyset\implies U=A\land V=B$ |
| @#55EE55: postulate   | @#55EE55: $(g\circ f)\circ h=g\circ (f\circ h)$ |
| @#55EE55: postulate   | @#55EE55: $f\circ id_A=id_A\circ f=f$ |

==== Discussion ====
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.
==== Parents ====
=== Requirements ===
[[Function]]
=== Related ===
[[Cat]]