Weak equivalence of internal categories (Q1840735)

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)].