Pushouts in category of topological spaces modulo homotopy

Question

Let $HTOP$ denote the category whose objects are topological spaces and whose morphisms are equivalence classes of continuous maps modulo homotopy equivalence. I m wondering what are the properties of this category. Does it have products, coproducts, pushouts ? I tried searching Google but it seems I don't know the right keywords

Answer 1

This category is often called simply “the homotopy category.” It has effectively no limits and colimits, other than products and coproducts which are constructed as in TOP. It was the first natural example of a non-concrete category, as proven by Freyd in “homotopy is not concrete.”

More commonly studied is its full subcategory spanned by the CW complexes, which is equivalent to the category of all spaces localized at the weak homotopy equivalences. This continues to lack any good categorical properties. It doesn’t even have a generating set, though the category of pointed connected CW complexes modulo homotopy famously does—namely, the spheres, by Whitehead’s theorem.

These various homotopy categories are the ur-examples of homotopy categories of model categories, which were invented in large part to get around the non-existence of categorical constructions in homotopy categories by allowing the construction of homotopically meaningful limits and colimits. For instance, a homotopy pushout of $A\leftarrow C\to B$ represents maps from $A,B$, not which are equal on $C$ like a pushout in Top, nor which are equal on $C$ up to homotopy like a (non-existent) pushout in the homotopy category, but maps from $A$ and $B$ together with a chosen homotopy between them on $C$. This is the double mapping cylinder. The more recent theory of $\infty$-categories is also designed in large part to make homotopy limits and colimits behave more like ordinary limits and colimits.