  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.
