학술논문
More exact completions that are toposes.
Document Type
Journal
Author
Menni, Matías (4-EDIN-FCI) AMS Author Profile
Source
Subject
03 Mathematical logic and foundations -- 03D Computability and recursion theory
03D99None of the above, but in this section
03Mathematical logic and foundations -- 03G Algebraic logic
03G30Categorical logic, topoi
18Category theory; homological algebra -- 18A General theory of categories and functors
18A35Categories admitting limits
03D99
03
03G30
18
18A35
Language
English
Abstract
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 [``A characterization of the left exact categories whose exact completions are toposes'', 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.