van Kampen theorems for toposes
type: Item
instance of: scholarly article
title: van Kampen theorems for toposes
zbMATH Open document ID: 1030.18003
DOI: 10.1016/S0001-8708(03)00010-0
author: Stephen Lack
author: Marta Bunge
published in: Advances in Mathematics
publication date: 2003-11-16T00:00:00Z
review text: \textit{R. Brown} and \textit{G. Janelidze} [J. Pure Appl. Algebra 119, 255-263 (1997; Zbl 0882.18005)] proved a `van Kampen theorem' for lextensive categories: in this context the theorem becomes the assertion that, under suitable hypotheses, pseudofunctors defined on the duals of such categories map certain pushouts to pullbacks. The authors of the present paper first introduce a notion of `extensive \(2\)-category' to which the Brown-Janelidze result can be extended. Then they show that the \(2\)-categories of toposes over a base, and of locally connected toposes over a base, satisfy their definition of extensiveness, which enables them to recover the `classical' van Kampen theorem for the fundamental groupoids of toposes, defined in terms of coverings. \(\{\)There is a minor error in Proposition 2.4: in the statement of this Proposition, the words `fully faithful' should be replaced by `pseudomonic'. However, this does not affect any of the other results in the paper\(\}\).
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18F15
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 55Q20
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 2003840
zbMATH Keywords: lextensive categories
zbMATH Keywords: extensive 2-category
zbMATH Keywords: locally connected toposes
zbMATH Keywords: van Kampen theorem
reviewed by: Peter Johnstone
MaRDI profile type: Publication
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full work available at URL: s0001-8708(03)00010-0
OpenAlex ID: W2027866234
description: scientific article; zbMATH DE number 2003840