The coshape invariant and continuous extensions of functors
type: Item
instance of: scholarly article
title: The coshape invariant and continuous extensions of functors
zbMATH Open document ID: 1223.54022
author: Vladimir Baladze
published in: Topology and its Applications
publication date: 2011-08-04T00:00:00Z
review text: Let \(\mathfrak T\) be a category and \(\mathfrak P \subset \mathfrak T\) be a full subcategory. Using the category of direct systems \(\pmb{dir}-\mathfrak T\) resp. a quotient category \(\pmb {dir}-\mathfrak T/\sim =\pmb{inj}-\mathfrak T\), the author defines the notion of a \(\mathfrak P\)-coexpansion \(\pmb p: \pmb X \to (X)\) of an object \(X \in \mathfrak T\) (\((X)\) denoting the trivial direct system of one single object), analogous to the concept of an \(ANR\)- resolution of a space, and he calls the category \(\mathfrak P\) to be codense in \(\mathfrak T\), whenever every object of \(\mathfrak T\) admits such a \(\mathfrak P\)-coexpansion. He constructs in this case a coshape category \(\pmb {CSH}_{(\mathfrak T, \mathfrak P)}\). Since he is primarily interested in the case that \(\mathfrak T\) is the homotopy category of (based or unbased) pairs of topological spaces and \(\mathfrak P \) the subcategory of finite CW-spaces, his main theorem asserts, that in this case the category \(\mathfrak P\) is codense in the category \(\mathfrak T\). In the last section the paper deals with the problem of extending functors \(T: \mathfrak P \to \pmb {Gr}\) =\,category of groups over \(\mathfrak T\), using the coshape functor.
reviewed by: Friedrich Wilhelm Bauer
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 55N05
Mathematics Subject Classification ID: 55N40
Mathematics Subject Classification ID: 55P55
zbMATH DE Number: 5935887
zbMATH Keywords: extension of functors
zbMATH Keywords: coexpansion
zbMATH Keywords: codense subcategory
zbMATH Keywords: coshape category
MaRDI profile type: Publication
full work available at URL: j.topol.2011.05.014
OpenAlex ID: W1985692857
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DOI: 10.1016/J.TOPOL.2011.05.014
description: scientific article; zbMATH DE number 5935887