Detecting isomorphisms in the homotopy category

TL;DR

This paper investigates the limitations of detecting isomorphisms in the homotopy category of spaces, providing counterexamples and contrasting behaviors between the homotopy category and the homotopy 2-category.

Contribution

It demonstrates that no set of objects can jointly reflect isomorphisms in the homotopy category, refutes a previous claim, and contrasts this with the spheres' reflection in the homotopy 2-category.

Findings

No set of objects reflects isomorphisms in the homotopy category.

Spheres jointly reflect equivalences in the homotopy 2-category.

Counterexamples involve large symmetric groups and transfinite diagrams.

Abstract

We show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms by giving an explicit counterexample involving large symmetric groups. We also show that, in contrast, the spheres jointly reflect equivalences in the homotopy 2-category of spaces. The non-existence of such a set in the homotopy category was originally claimed by Heller, but his argument relied on the statement that for every set of spaces, long enough transfinite sequential diagrams admit weak colimits which are privileged with respect to the given set. Using the theory of graphs of groups, we show that this statement is false, by proving that for every ordinal with uncountable cofinality, there is a diagram indexed by that ordinal which admits no weak colimit that is privileged with respect to the spheres.