In traditional mathematics and set theory, the category Set of all sets is the archetypical topos. However, in predicative mathematics with its notion of sets, these do not form a topos since they do not satisfy the powerset axiom. In constructive mathematics, we may be weakly predicative and almost form a topos; the resulting weaker structure which Bishop sets (for example) form is a predicative topos (van den Berg).
There are various notions of predicative topos in the literature (see references below); here we discuss the notion advocated by Benno van den Berg.
In other words, the axiom of multiple choice appears in different strengths in the literature; see at WISC. Accordingly there is the notion of strong/weak predicative topos if it satisfies the strong/weak version of this axiom; van den Berg’s notion of predicative topos is the weak version.
The goal of formulating a notion of predicative topos was stated and pursued in a series of articles containing
The definition then appears in
A non-example constructed via the internal logic of an unbounded -topos is given in