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Category of open sets

Set

context $\langle X,\mathcal T\rangle$ … topological space
inclusion $\mathrm{Op}(X)$ … category
definition $\mathrm{Ob}_{\mathrm{Op}(X)}\equiv \mathcal T$
for all $U,V\in\mathrm{Ob}_{\mathrm{Op}(X)}$
definition $\mathrm{Op}(X)[U,V]\equiv\{i:U\to V\ |\ i(x)=x\}$

Discussion

In the category of open sets, if one object $U$ is subset of another $V$, i.e. $U\subseteq V$, then the hom-set $\mathrm{Op}(X)[U,V]=\{i\}$ contains the only inclusion function and is empty otherwise.

Reference

Wikipedia: Sheaf

Parents

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