Subobject classifier: Difference between revisions

Revision as of 05:24, 5 May 2006

Revision as of 05:27, 5 May 2006

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== Further examples ==

Every [[topos]] has a subobject classifier. For the topos of [[sheaves]] of sets on a [[topological space]] ''X'', it can be described in these terms: take the [[disjoint union]] Ω of all the [[open set]]s ''U'' of ''X'', and its natural mapping π to ''X'' coming from all the [[inclusion map]]s. Then π is a [[local homeomorphism]], and the corresponding sheaf is the required subobject classifier (in other words the construction of Ω is by means of its [[espace étalé]]). One can also consider Ω to be, in a (tautological) sense, the graph of the membership relation obtaining between points ''x'' ofand open sets ''XU'' and open setsof ''UX''.

[[category:Topos theory]]