This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense.
[Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is the homotopy category of topological spaces a concrete category (in the sense, say, of Kurosh and Freyd)?". You may still want to read what follows, for a bit of motivation]
Historically, I'd bet people started to look at concrete metric spaces $(X,d)$ before exploring the utility of the abstraction given by general topological spaces $(X,\tau)$ ($\tau$ is a topology). The heuristic idea captured by the concept of a topological space is endowing a set $X$ with a "geometry" that forgets the rigidity of a hypothetical metric structure $(X,d)$, though retaining the "qualitative" aspects of the geometry given by the metric.
Of course there are non-metrizable topological spaces, but let stick to metrizable ones for the moment. I think it should be possible to see a topological space $(X,\tau)$ as an equivalence class of metric spaces: $(X,[d])$, where $[d]$ is the class of all metrics on the set $X$ that give rise to the same topology. So, an $(X,\tau)$ just has several "rigid" models $(X,d)$, and a morphism of topological spaces $f:(X,\tau) \rightarrow (Y,\tau')$ is given by taking any map $(X,d) \rightarrow (Y,d^{'})$ of "representatives" which is continuous according to the "metric ball" definition.
[Please correct the above picture if it is imprecise or even just wrong!!]
The (perhaps naive) way I have always thought about homotopy is that it is an even "further step" in making the geometry more "qualitative" and less rigid: you can "collapse positive dimensional appendices" of a space as far as they are contractible, and so forth. When trying to make this formal, you consider "homotopy types", which are equivalence classes $[(X,\tau)]$ of topological spaces, where $(X,\tau) \sim (X',\tau')$ if there is a homotopy equivalence $\varphi:(X,\tau) \rightarrow (X',\tau')$. What are morphisms in the homotopy category? Just morphisms $f$ between "representatives", but now you have also to consider them up to homotopy, i.e. you take $[f]$ where $f \sim f'$ if there's a homotopy $\alpha: f \rightarrow f'$.
It's ugly to think of topological spaces as $[(X,d)]$ (or, rather, $(X,[d])$): it's better to use the simpler and more expressive abstraction $(X,\tau)$.
So, the question is:
- Is there some kind of "homotopology" $h$ (whatever it is) that one can put on sets $S$ so that each homotopy type $[(X,\tau)]$ is fully described by a "homotopical space" (whatever it is) $(S,h)$ and homotopy classes of morphisms $[f]$ correspond to "$h$-compatible" (whatever it means) set-theoretic maps $F:S \rightarrow S'$?
(I don't even dare asking about the existence of "non-topologizable homotopical spaces" because the above question is already by far too vague!)
Best Answer
No.