Quasi-toposes as elementary quotient 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.The paper contains results presented by the authors at several international meetings in the past years, in particular at Logic Colloquium 2016 and Category Theory 2017, and during the Trimester devoted to Types, Homotopy Type Theory and Verification at the Hausdorff research Institute for Mathematics in 2018. We would like to thank the organisers of these events who gave us the opportunity to present our results.