Categories in which all strong generators are dense
This paper addresses the question of which total categories have all small strong generators dense. The problem is studied in the context of enriched categories. Let \({\mathcal V}\) be a symmetric, monoidal, closed category which is small complete and cocomplete and has intersections of all families of monics. A \({\mathcal V}\)-category \({\mathfrak A}\) is total iff for every \({\mathcal V}\)-functor \({\mathfrak A}^{op}\to^{t}{\mathcal V}\), the identity \(1_{{\mathfrak A}}: {\mathfrak A}\to {\mathfrak A}\) admits a colimit z(t) indexed by t. [For more on total categories, see \textit{G. M. Kelly}, Cah. Topologie Géom. Différ. Catégoriques 27, 109-132 (1986; Zbl 0593.18007).] Let y(a) denote the representable \({\mathcal V}\)-functor \({\mathfrak A}(\_,a): {\mathfrak A}^{op}\to {\mathcal V}\). If \({\mathfrak A}\) is total and z takes strong monics \(r\rightarrowtail y(a)\) to monics in \({\mathfrak A}\), then small strong generators are dense, that is if \(m: {\mathfrak C}\to {\mathfrak A}\) is a fully faithful \({\mathcal V}\)-functor with \({\mathfrak C}\) small and if m is strongly generating, then it is dense. This result is applied to several examples taken from topos theory and the theory of abelian categories.
scientific article; zbMATH DE number 3989614
scientific article