toposes

Toposes are categories that generalize the category of sheaves on a space, providing a unifying framework for geometry and logic. They come in two closely related flavors: elementary toposes and Grothendieck toposes. An elementary topos is a category with finite limits, exponentials (making it cartesian closed), and a subobject classifier, an object that internalizes truth values and classifies subobjects of any object. This structure endows the topos with an internal language: a higher-order intuitionistic logic in which one can interpret sets, functions, and logical propositions inside the category itself.

A Grothendieck topos is a topos that is equivalent to the category of sheaves Sh(C, J) on a site, where C is a small category and J is a Grothendieck topology. Presheaf categories Set^(C^op) form a fundamental example of a topos (a presheaf topos, corresponding to a site with the trivial topology). The category of sheaves on a topological space X, Sh(X), is a classic Grothendieck topos and serves as a canonical example of a topos modeling geometric spaces. The base topos Set is the simplest example, illustrating how toposes generalize ordinary set theory.

Key structural features include a subobject classifier Ω, which classifies monomorphisms, and exponentials that provide internal function objects. Morphisms between toposes, called geometric morphisms, consist of an inverse image functor f^* that preserves finite limits and has a right adjoint f_*; essential geometric morphisms may also have a left adjoint. Topos theory thus offers a flexible language for both geometry (as generalized spaces) and logic (as internal languages).