Detecting isomorphisms in the homotopy category

Arlin, Kevin; Christensen, J Daniel

doi:10.2140/agt.2023.23.2975

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

Detecting isomorphisms in the homotopy category

Kevin Arlin and J Daniel Christensen

Algebraic & Geometric Topology 23 (2023) 2975–2991

DOI: 10.2140/agt.2023.23.2975

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

Open Access made possible by participating institutions via Subscribe to Open.

Volume 23, issue 7 (2023)