More exact completions that are toposes
It has been known for some time that the realizability topos constructed from a partial combinatory algebra is equivalent to the exact completion of its full subcategory of `partitioned assemblies' (i.e., it is the free (Barr-)exact category generated by the latter as a finitely complete category). In a previous paper (J.\ Pure Appl.\ Algebra, to appear), the author obtained conditions on a finitely complete category which are equivalent to its exact completion being a topos. The present paper presents a variant of these conditions in which the original category is assumed to have a full mono-localizing subcategory of `chaotic objects' which is a topos, as is the case for the category of partitioned assemblies. Under this assumption, the conditions can be simplified, which enables the author to give some interesting examples of categories whose exact completions are, or are not, toposes.
scientific article; zbMATH DE number 1758490