The homotopy category doesn’t have all colimits. I just learned a nice explicit example of this phenomenon in Emily Riehl’s class on categorical homotopy theory at Harvard. The example (7.21 in the notes), attributed to Michael Andrews and Markus Hausmann, is for the coequalizer of the identity map and reflection maps from the circle to itself. This colimit doesn’t exist in the homotopy category of spaces.
Mapping into another object, such as K(A,n), converts colimits to limits and so we can compute that
- The reduced integral cohomology vanishes
- The reduced mod 2 cohomology is zero unless n=1, in which it is .
From this and the universal coefficients theorem the first integral homology is a group A with Hom(A, Z/2) = 0 and Hom(A, Z) = Ext(A, Z/2) = Ext(A, Z) = 0. But such a group doesn’t exist (look at the long exact Hom-Ext sequence).