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Detecting isomorphisms in the homotopy category

Kevin Arlin and J Daniel Christensen

Algebraic & Geometric Topology 23 (2023) 2975–2991
Abstract

We show that no generalization of Whitehead’s theorem holds for unpointed spaces. More precisely, we show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms. We give an explicit counterexample involving infinite symmetric groups. In contrast, we prove that the spheres do jointly reflect equivalences in the homotopy 2–category of spaces. We also show that homotopy colimits of transfinite sequential diagrams of spaces are not generally weak colimits in the homotopy category, and furthermore exhibit such a diagram with the property that none of its weak colimits is privileged, which means, roughly, that it sees the spheres as compact objects. The nonexistence of a set jointly reflecting isomorphisms in the homotopy category was originally claimed by Heller, but our results on weak colimits show that his argument had an inescapable gap, leading to the need for the new proof given here.

Keywords
homotopy category of spaces, privileged weak colimit, conservative, generator, Brown representability, graph of groups, fundamental groupoid
Mathematical Subject Classification 2010
Primary: 18A30, 55P65, 55U35
References
Publication
Received: 17 February 2020
Revised: 10 January 2022
Accepted: 21 January 2022
Published: 26 September 2023
Authors
Kevin Arlin
Department of Mathematics
UCLA
Los Angeles, CA
United States
J Daniel Christensen
Department of Mathematics
University of Western Ontario
London, ON
Canada

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