Editing Subobject classifier (section)

== Further examples ==

=== Sheaves of sets ===
The category of [[sheaf (mathematics)|sheaves]] of sets on a [[topological space]] ''X'' has a subobject classifier Ω which can be described as follows:  For any [[open set]] ''U'' of ''X'', Ω(''U'') is the set of all open subsets of ''U''.  The terminal object is the sheaf 1 which assigns the [[Singleton (mathematics)|singleton]] {*} to every open set ''U'' of ''X.'' The morphism η:1 → Ω is given by the family of maps η<sub>''U''</sub> : 1(''U'') → Ω(''U'') defined by η<sub>''U''</sub>(*)=''U'' for every open set ''U'' of ''X''.  Given a sheaf ''F'' on ''X'' and a sub-sheaf ''j'': ''G'' → ''F'', the classifying morphism ''χ<sub> j</sub>'' : ''F'' → Ω is given by the family of maps ''χ<sub> j,U</sub>'' : ''F''(''U'') → Ω(''U''), where ''χ<sub> j,U</sub>''(''x'') is the union of all open sets ''V'' of ''U'' such that the restriction of ''x'' to ''V'' (in the sense of sheaves) is contained in ''j<sub>V</sub>''(''G''(''V'')).

Roughly speaking an assertion inside this topos is variably true or false, and its truth value from the viewpoint of an open subset ''U'' is the open subset of ''U'' where the assertion is true.

=== Presheaves ===
Given a small category <math>C</math>, the category of [[presheaves]] <math>\mathrm{Set}^{C^{op}}</math> (i.e. the [[functor category]] consisting of all contravariant functors from  <math>C</math> to <math>\mathrm{Set}</math>) has a subobject classifer given by the functor sending any <math>c \in C</math> to the set of [[Sieve (category theory)|sieves]] on <math>c</math>. The classifying morphisms are constructed quite similarly to the ones in the sheaves-of-sets example above.

=== Elementary topoi ===
Both examples above are subsumed by the following general fact: every [[elementary topos]], defined as a category with finite [[Limit (category theory)|limits]] and [[power object]]s, necessarily has a subobject classifier.<ref>Pedicchio & Tholen (2004) p.8</ref> The two examples above are [[Topos|Grothendieck topoi]], and every Grothendieck topos is an elementary topos.