TITLE: Why is it that homotopy is better described by weak equivalences than by homotopies?
QUESTION [17 upvotes]: I've been reading about (abstract or not) homotopy theory, and I seem to have understood (correct me if I'm wrong) that weak equivalences describe homotopy better than homotopies, in the following sense :
Intuitively, if I wanted to abstract away from classical homotopy theory, my first guess wouldn't be to say that we should consider categories with distinguished classes of morphisms; it would probably be to say we should consider $2$-categories, or perhaps categories with some congruence on the morphisms, or categories with a given object that "classifies homotopies" (playing the role of $I$), or something in this direction, i.e. I would try to give an abstract description of homotopies, not weak equivalences. That's probably connected to my lack of practice in classical homotopy theory, but at least from a beginner's perspective, that's what I would do.
Now, as this idea is very intuitive and probably naive, it must have occurred to some mathematicians who decided for some reason that this wasn't the way to go, and that actually, equivalences, fibrations and cofibrations were the thing to study. My question is : what's that reason ? How does one go from homotopies to weak equivalences ?
Could you give an intuitive reason/heuristic why, or is an answer necessarily technical (in which case I probably couldn't follow all of it, but I would be happy to know that it is) ? 
Another very related question is : are any of the approaches I mentioned interesting in that regard ($2$-categories, or categories with a congruence- they're interesting for other reasons, I wonder if they're interesting for homotopy theory, especially $2$-categories) ?

REPLY [4 votes]: I have two intuitions on this that may be helpful.

The first is to ponder a place to do "abstract homotopy theory" as a black box — I'll call such a thing an $\infty$-category.
If we have such a thing $\mathcal{X}$, in order to work with it we need a way to actually be able to specify objects and arrows and how they compose. We would like to organize this data into an ordinary category $C$ so that we can actually work with it. And we will need some mysterious thing (which I will call a "functor") that is a mapping $C \to \mathcal{X}$ that interprets the elements of $C$ as being elements of $\mathcal{X}$.
Another thing we might imagine is that we could throw away all of the higher homotopical information; by looking just at the equivalence classes of morphisms, we would expect to get an ordinary category, which is something we work with.
So, we demand there is a category $h \mathcal{X}$, which we call the "homotopy category" of $\mathcal{X}$, together with mysterious mapping $\mathcal{X} \to h \mathcal{X}$ (another "functor") that carries out the above transformation.
The nice thing, now, is that the composite $C \to \mathcal{X} \to h \mathcal{X}$ is just an ordinary functor between ordinary categories — everything in the diagram $C \to h \mathcal{X}$ we understand and can work with, and can use this as a substitute for working in the mysterious $\mathcal{X}$.
For whatever reason, the nicest situations are when $C \to h\mathcal{X}$ is actually a localization of ordinary categories — that is, there is a subcategory $W \subseteq C$ (e.g. the subcategory of everything that maps to an isomorphism in $h\mathcal{X}$) such that $C \to h\mathcal{X}$ identifies $h \mathcal{X}$ with $C[W^{-1}]$.
For whatever reason, the data we need is traditionally expressed via the pair $(C,W)$ rather than via the functor $C \to h \mathcal{X}$.
It turns out in $\infty$-categories that the $\mathcal{X}$ we express via the pair $(C,W)$ turns out to be precisely the $\infty$-category you get by taking the localization of $\infty$-categories rather than of ordinary categories.

The second intuition is that there is a model structure on Cat, the Thomason model structure, that is Quillen equivalent to the usual model structures on sSet and Top; that is, categories can serve as models for homotopy types just as topological spaces or simplicial sets do.
The neat thing is that when we consider a pair $(C,W)$ (called a relative category) consisting of a category and its subcategory of weak equivalences, you can interpret it as follows:

$C$ can be viewed in the ordinary way as an ordinary category
$W$ can be viewed as a model for a homotopy type 

so this gives a way to blend the notions of category and of homotopy type together.
The "invertible" nature of the arrows from $W$ comes from the fact homotopy types have fundamental groupoids, not fundamental categories, so the structure we model with $W$ has inverses, it's just that $W$ itself lacks them. Which is why we call them weak equivalences rather than merely equivalences.

To give precise statements to some of the things I said above, in $(\infty,1)$ category $\mathrm{Cat}_{(\infty,1)}$ of small $(\infty,1)$-categories, the $(\infty,1)$-category $\mathcal{X}$ presented by the relative category $(C,W)$ can be constructed as a pushout
$$ \require{AMScd} \begin{CD}
W @>>> C
\\ @VVV @VVV
\\ \mathrm{Grpd}_\infty(W) @>>> \mathcal{X}
\end{CD} $$
where $\mathrm{Grpd}_\infty(W)$ is the $\infty$-groupoid generated by $W$. Furthermore, it satisfies a universal property: for any other $(\infty,1)$-category $\mathcal{Y}$, the $(\infty,1)$-functor category $\mathrm{Funct}(\mathcal{X}, \mathcal{Y})$ is equivalent to the full ($\infty-$)subcategory of $\mathrm{Funct}(C, \mathcal{Y})$ spanned by the functors that send every morphism in $W$ to an equivalence in $\mathcal{Y}$.

Another big idea about doing abstract homotopy theory is that your categories shouldn't have a set of morphisms between objects; you should have a whole homotopy type of morphisms!
So we want an enriched category in a suitable sense.
It turns out that, while we might want to weaken associativity so it only holds up to equivalence, in the models we use for homotopy types it is safe to actually require composition to be strictly associative.
Above, I mentioned that, in the Thomason model structure, categories can serve as models for homotopy types. But if we (strictly) enrich in Cat... that's the same thing as a strict 2-category!
So we can do abstract homotopy theory in strict 2-categories as you suggest... although maybe this is kind of a cheat, because we still have weak equivalences, namely all of the 2-morphisms.