Topoi

Topoi, plural of topos, are categories that generalize the category of sets and the category of sheaves on a space. They provide a setting in which one can do mathematics with a built‑in notion of logic and truth values. There are two widely used perspectives: Grothendieck topoi and elementary topoi.

Grothendieck topoi are defined as categories equivalent to categories of sheaves Sh(C, J) on a small site (C, J). They satisfy Giraud’s axioms: they have finite limits, arbitrary small colimits, a generator, and are exact; they are cocomplete and have a subobject classifier. In particular, they are cartesian closed, so they support internal exponentiation and function spaces.

Elementary topoi, introduced by Lawvere and Tierney, are defined axiomatically as finitely complete, cartesian closed categories equipped with a subobject classifier. This formulation does not require any site; it captures the essential logical and structural features of sheaf toposes and set‑like categories. The internal logic of a topos is intuitionistic higher‑order logic, with truth values given by the object of subobjects Ω.

Examples include the category of sets, Set, which is the prototypical topos; the category Sh(X) of sheaves on a topological space X; and the presheaf topos Set^C for any small category C.

Morphisms between topoi are geometric morphisms, consisting of an adjoint pair (f^*, f_*) with f^* left exact. Geometric morphisms preserve the topos structure and relate models of theories internal to topoi. Topoi provide a unifying framework for geometry, logic, and their interconnections, including classifying toposes for geometric theories.