Detecting isomorphisms in the homotopy category

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.