From types to sets

Abstract

The aim is to construct the free topos generated by a category. Up to equivalence, one may assume that each topos is equipped with a canonical choice of representative subobjects satisfying certain obvious conditions, and one insists that morphisms between toposes preserve these exactly. Between the category of small categories and that of toposes one inserts the category of “dogmas” (Volger's closed logical categories). Dog is known to be equational over Cat, and it is here shown to be a reflective subcategory of Top. A dogma is a category with canonical finite products, it has a Heyting algebra object Ω which admits arbitrary objects as exponents such that, for each object A, the canonical morphism Ω → Ω A has a left adjoint ∃ A and a right adjoint ∀ A , and it satisfies the usual axiom of extensionality (interpreted in the obvious way). The construction of the topos generated by a dogma follows Volger: its objects are “sets” 1 → Ω A and its morphisms are “relations” between sets which are universally defined and single valued. Inasmuch as a topos consists of sets, a dogma consists of types, and we find here much of traditional type theory in a categorical setting, which incorporates both Frege's process of set abstraction and something like Russell's theory of description.