Subobject classifier: Difference between revisions

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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> For the topos of [[sheaf (mathematics)|sheaves]] of sets on a [[topological space]] ''X'', it can be described in these terms: For any [[open set]] ''U'' of ''X'', <math>\Omega(U)</math> is the set of all open subsets of ''U''. 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. A [[quasitopos]] has an object that is almost a subobject classifier; it only classifies strong subobjects.

For a small category <math>C</math>, the '''subobject classifer''' in the '''topos of presheaves''' <math>\mathrm{Set}^{C^{op}}</math> is giventhe asfunctor follows. Forsending any <math>c \in C</math>, <math>\Omega(c)</math> isto the set of [[Sieve (category theory)|sieves]] on <math>c</math>.

== Notes ==