===== Category of open sets =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $\langle X,\mathcal T\rangle$ ... topological space |
| @#AAFFAA: inclusion   | @#AAFFAA: $\mathrm{Op}(X)$ ... category |
| @#FF9944: definition  | @#FF9944: $\mathrm{Ob}_{\mathrm{Op}(X)}\equiv \mathcal T$ |
| @#FFFDDD: for all     | @#FFFDDD: $V,U\in\mathrm{Ob}_{\mathrm{Op}(X)}$ |
| @#FF9944: definition  | @#FF9944: $\mathrm{Op}(X)[V,U]\equiv\{i:V\to U\ |\ i(x)=x\}$ |

==== Discussion ====
In the category of open sets, the arrows are the inclusion functions. In the case $V\subseteq U$, the hom-set $\mathrm{Op}(X)[U,V]$ is the singleton $\{i\}$ and otherwise it's empty.

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29|Sheaf]]

==== Parents ====
=== Element of ===
[[Cat]], [[Thin category]]