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
–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
