# Isomorphism that is not a bijection in underlying sets?

The question is pretty much the same as >here<. This wikipedia article claims that in "Homotopy category of CW complexes" there is an isomorphism which is not bijective in the underlying sets, while this category should admit underlying set (e.g. faithful functor to Set).

I am not really familiar with CW complexes or Homotopy theory really. Can someone elaborate on what is going on? In the linked post, there is a proof that whenever $$f$$ is iso in cat. $$\mathcal{C}$$, then its image $$F(f)$$ under any functor $$F$$ must be iso, because $$F(f^{-1})$$ is its inverse.

• I would say it's very arguable whether objects in the homotopy category "admit underlying sets" or not. If you insist they do, it's in a vastly weaker sense than admitting a faithful functor to sets-there's not even a function on isomorphism classes of objects. I've edited the Wikipedia article to be (hopefully) less confusing. – Kevin Carlson Aug 20 '19 at 20:17

The condition for an arrow $$g:y\to x$$ in a category $$\mathbf C$$ to be the inverse of $$f:x\to y$$ is equational: $$fg=\mathrm{id}_y$$ and $$gf=\mathrm{id}_x$$. Hence it will be preserved by any functor, by which I mean that $$F(g)$$ is a inverse for $$F(f)$$ for any functor $$F$$ with domain $$\mathbf C$$: indeed, $$F(g)F(f) = F(gf) = F(\mathrm{id}_x) = \mathrm{id}_{F(x)}$$ and similarly for the other equation.
Hence, if you have any functor from $$\mathbf C \to \mathsf{Set}$$, it will map isomorphism to bijections. In the Wikipedia article, it is just said "every object admits an underlying set", with no indication that this mapping should be a functor (or even extended to maps): indeed, in the category $$\mathsf{Hot}$$, the objects are CW-complexes and the morphisms are the continuous functions modulo homotopy between them, so you can define a mapping $$\mathsf{Hot}\rightsquigarrow \mathsf{Set}$$ on objects by mapping a CW-complex to its underlying set (just forget the topology), but in no way can you complete that into a functor.
"Underlying set" is simply not a functor on the homotopy category. The closest thing is $$\text{Hom}(\bullet, -)$$ which returns $$\pi_0$$, and which sends homotopy equivalences to bijections as expected.