Quasitoposes as elementary quotient completions

TL;DR

This paper characterizes when elementary quotient completions form quasitoposes or toposes, linking categorical constructions with logical and topos-theoretic properties, and unifies several existing results in the field.

Contribution

It provides a comprehensive characterization of elementary quotient completions as quasitoposes or toposes, extending previous work and connecting various categorical frameworks.

Findings

Elementary quotient completion is a quasitopos under certain conditions.

Elementary quotient completion is an elementary topos in specific cases.

The results unify and generalize existing characterizations of exact completions.

Abstract

The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category with finite products and weak equalisers. In this paper we characterise when an elementary quotient completion is a quasi-topos. We obtain as a corollary a complete characterisation of when an elementary quotient completions is an elementary topos. As a byproduct we determine also when the elementary quotient completion of a tripos is equivalent to the doctrine obtained via the tripos-to-topos construction. Our results are reminiscent of other works regarding exact completions and put those under a common scheme: in particular, Carboni and Vitale's characterisation of exact completions in terms of their projective objects, Carboni and Rosolini's characterisation of locally cartesian closed exact completions, also in the revision by Emmenegger, and Menni's characterisation of the exact completions which are elementary toposes.