Topos theory

contravariant functor to the category of sets and functions

In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion of localization. The Grothendieck topoi find applications in algebraic geometry, and more general elementary topoi are used in logic.

structure on a category C which makes the objects of C act like the open sets of a topological space

in category theory