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: For any open set U of X, 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.
For a small category, the subobject classifer in the topos of presheaves is given as follows. For any, is the set of sieves on .
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