Categories in which all strong generators are dense
type: Item
instance of: scholarly article
title: Categories in which all strong generators are dense
zbMATH Open document ID: 0612.18004
DOI: 10.1016/0022-4049(86)90065-4
author: B. George
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 3989614
zbMATH Keywords: dense functor
zbMATH Keywords: total categories
zbMATH Keywords: enriched categories
zbMATH Keywords: small strong generators
zbMATH Keywords: topos theory
zbMATH Keywords: abelian categories
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
Wikidata QID: Q56687255
full work available at URL: 0022-4049(86)90065-4
OpenAlex ID: W2055146012
cites work: Q4124990
cites work: Q5616191
cites work: Q5586461
cites work: Q3491793
cites work: Two-dimensional sheaf theory
cites work: Cosmoi of Internal Categories
cites work: Notions of topos
cites work: The Family Approach to Total Cocompleteness and Toposes
cites work: Yoneda structures on 2-categories
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description: scientific article; zbMATH DE number 3989614