Exact completions and toposes

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 quasitoposes 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.