*Semantics Club 20 10 2000*

# Closure operators in exact completions

### Abstract

Realizability toposes are very interesting. In particular, because of their connection with the semantics of programming languages. It is
then important to understand how they can be constructed. Around ten years after their introduction, it was observed that they arise
as solutions to a universal problem. Namely, that of adding quotients to a regular category in order to make it exact. That is,
realizability toposes are examples of ex/reg completions (beware, this is a different problem than that of exact completions discussed
in a previous semantics club). We want to understand how this happens and it turns out that an important insight is gained if one
studies universal closure operators in exact completions. It is well known that universal closure operators in the topos of presheaves
on a small category C correspond to Grothendieck topologies on C. We give an analogous characterization of closure operators in the
exact completion of a (not necessarily small) category with finite limits. We then characterize the separated objects and sheaves with
respect to certain "canonical" topologies. Finally, we explain the relation to the problem characterizing ex/reg completions that are
toposes.