Splitting idempotents in a fibered setting
Constructions producing a Cauchy complete category with respect to a class of idempotents from any category are well-known [\textit{P. Freyd}, Repr. Theory Appl. Categ. 2003, No. 3, xxiii, 1--164 (2003; Zbl 1041.18001); Abelian categories. An introduction to the theory of functors. New York-Evanston-London: Harper and Row, Publishers (1964; Zbl 0121.02103); \textit{P. J. Freyd} and \textit{A. Scedrov}, Categories, allegories. Amsterdam etc.: North-Holland (1990; Zbl 0698.18002)]. Assuming that the reader is already familiar with the basics of ordinary and fibered category theory, this paper provides an explicit elementary description of the Cauchy completion of the category \(\mathbf{Fib}(\mathbb{B})\) of fibrations with a fixed base category \(\mathbb{B}\) and of the category \(\mathbf{Fib}\) of fibrations with any base category. The author gives two universal constructions: The first arises by splitting reindexing stable vertical idempotent morphisms in the total category of fibrations, while the second arises by simultaneously splitting idempotents in the total and base categories of fibrations.
scientific article; zbMATH DE number 7001425