*Abstract:*

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the different ways in which they can be constructed.

Realizability toposes and presheaf toposes are two important
classes of toposes. All of the former and many of the latter arise
by adding ``good'' quotients of equivalence relations to a simple
category with finite limits. This construction is called the
*exact completion* of the original category. Exact completions
are not always toposes and it was not known, not even in the
realizability and presheaf cases, when or why toposes arise in this
way.

Exact completions can be obtained as the composition of two
related constructions. The first one assigns to a category with
finite limits, the ``best'' regular category (called its *regular
completion*) that embeds it. The second assigns to a regular
category the ``best'' exact category (called its *ex/reg
completion*) that embeds it. These two constructions are of
independent interest. There are quasi-toposes that arise as regular
completions and toposes, such as those of sheaves on a locale, that
arise as ex/reg completions but which are not exact
completions.

We give a characterization of the categories with finite limits whose exact completions are toposes. This provides a very simple way of presenting realizability toposes, it allows us to give a very simple characterization of the presheaf toposes whose exact completions are themselves toposes and also to find new examples of toposes arising as exact completions.

We also characterize universal closure operators in exact completions in terms of topologies, in a way analogous to the case of presheaf toposes and Grothendieck topologies. We then identify two ``extreme'' topologies in our sense and give simple conditions which ensure that the regular completion of a category is the category of separated objects for one of these topologies. This connection allows us to derive good properties of regular completions such as local cartesian closure. This, in turn, is part of our study of when a regular completion is a quasi-topos.

The second extreme topology gives rise, as its category of
sheaves, to the category of what we call *complete*
equivalence relations. We then characterize the locally cartesian
closed regular categories whose associated category of complete
equivalence relations is a topos. Moreover, we observe that in this
case the topos is nothing but the ex/reg completion of the original
category.

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