Weak equivalence of internal categories
type: Item
instance of: scholarly article
title: Weak equivalence of internal categories
zbMATH Open document ID: 0989.18004
DOI: 10.1023/A:1008732224512
author: Renato Betti
published in: Applied Categorical Structures
publication date: 2002-07-22T00:00:00Z
review text: By a weak equivalence of two internal categories \({\mathcal C}\) and \({\mathcal D}\) in a topos \(E\), the author means what is usually referred to as their being Morita equivalent, in the sense that the corresponding categories of modules (or presheaf toposes) are equivalent. In particular, the notion of weak equivalence employed in this paper is different from that due to the reviewer and Pare [\textit{M. Bunge} and \textit{R. Paré}, Cah. Topol. Géom. Différ. Catégoriques 20, 373-399 (1979; Zbl 0432.18003)]. This choice of terminology may in fact lead to confusion precisely because both notions are referred to in the paper. Whereas the former has associated with it the Cauchy completion of a category \({\mathcal C}\), the latter also involves the stack completion. A problem with the latter, however, is that it is not stable under ``weak equivalences'' in the sense of this paper, and the author seeks a notion of completion which is at once strictly contained in the Cauchy completion and also stable under weak equivalence (in his sense). Such a notion of ``an intermediate category of \({\mathcal C}\)'' which is also stable in the sense indicated is isolated by means of a new concept of ``Cauchy generators''. The advantage over the usual Cauchy completion is that it seems closer in spirit to the original criterion of \textit{K. Morita} himself [Trans. Am. Math. Soc. 103, 451-469 (1962; Zbl 0113.03003)].
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1563349
zbMATH Keywords: Morita equivalence
zbMATH Keywords: weak equivalence
zbMATH Keywords: internal categories
zbMATH Keywords: Cauchy completion
zbMATH Keywords: stack completion
zbMATH Keywords: Cauchy generators
reviewed by: Marta Bunge
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1563349