Van Kampen theorems for categories of covering morphisms in lextensive categories
type: Item
instance of: scholarly article
title: Van Kampen theorems for categories of covering morphisms in lextensive categories
zbMATH Open document ID: 0882.18005
DOI: 10.1016/S0022-4049(96)00028-X
author: Ronald Brown
author: George Janelidze
published in: Journal of Pure and Applied Algebra
publication date: 1998-02-11T00:00:00Z
review text: The authors use a formulation of the Van Kampen theorem in terms of fundamental groupoids and covering spaces to describe an appropriate categorical setting for this type of result. The key ingredients in their formulation are the notions of ``lextensive'' category (i.e., a category with finite limits and finite coproducts that are disjoint and universal) and of effective descent morphism. The ``open covering'' being of effective descent turns out to be equivalent to the generalized Van Kampen theorem. This is a considerable improvement over the classical situation, where only sufficient homotopical conditions for the theorem were known. This very elegant result can then be used not just for topological coverings, but also in algebraic settings. A number of examples are supplied.
Mathematics Subject Classification ID: 18B99
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 1038604
zbMATH Keywords: effective descent morphism
zbMATH Keywords: lextensive category
zbMATH Keywords: Van Kampen theorem
reviewed by: J. Koslowski
MaRDI profile type: Publication
cites work: Q3926769
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description: scientific article; zbMATH DE number 1038604