No set of spaces detects isomorphisms in the homotopy category
Published 2019-10-09Version 1
We show that the homotopy category of spaces admits no set of objects jointly reflecting isomorphisms. This was claimed by Heller, but his argument relied on the statement that for every set of spaces, long enough sequential diagrams admit weak colimits which are privileged with respect to the given set. We show that this statement is false, by showing 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.