topos

Topos (plural topoi) is a concept in category theory that generalizes the category of sets. It is a category that behaves like a universe of sets, possessing finite limits, exponentials, and a subobject classifier. In a topos, one can form products and equalizers, there are function objects (cartesian closed structure), and there is a distinguished object that classifies subobjects, playing the role of truth values.

There are two common notions. An elementary topos is a category with finite limits, exponentials, and a subobject classifier. A Grothendieck topos is a category equivalent to the category of sheaves on a Grothendieck site, denoted Sh(C, J). This connection to sheaves makes Grothendieck topoi a natural generalization of categories of sheaves on topological spaces.

Examples include the category of sets, which is a topos with a two-element subobject classifier; the category of presheaves Set^C^op, which is a topos for any small category C; and the category of sheaves on a topological space, which is a Grothendieck topos.

Key features of topoi include internal higher-order intuitionistic logic, encoded by the subobject classifier and the exponential objects. Morphisms between topoi, called geometric morphisms, preserve the logical and geometric structure. Grothendieck topoi satisfy exactness and have specified generators, in line with Giraud’s axioms.

Topos theory provides a unifying framework for geometry and logic, offering a flexible setting for doing mathematics with an intrinsic language and internal logic. It originated with work of Lawvere and Tierney on elementary topoi and Grothendieck’s development of sheaf theory in algebraic geometry.