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) ?

  • $\begingroup$ I don't agree with much of this. I'm curious to what kind of answers this will get from more seasoned pros. $\endgroup$ – Randall Aug 8 '18 at 19:13
  • $\begingroup$ @Randall : please tell me what you don't agree with, I have to admit it's mostly what I felt from what I read; perhaps I didn't read enough $\endgroup$ – Maxime Ramzi Aug 8 '18 at 19:26
  • $\begingroup$ I'm actually finding it hard to formulate. $\endgroup$ – Randall Aug 8 '18 at 19:26
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    $\begingroup$ I think the key thing to keep in mind is that it's not about "this is the right thing that 'homotopy' should mean" but rather "this is the thing that actually comes up in applications". Abstract homotopy theory was not invented because people wanted to abstract away homotopy theory, but because they needed it to deal with actual categories that came up in practice which were formally similar to classical homotopy theory. $\endgroup$ – Eric Wofsey Aug 8 '18 at 20:13
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    $\begingroup$ I'd venture to say that weak equivalences don't really describe homotopy theory better than homotopy equivalences. What they describe is the homotopy category of CW complexes, which is equivalent to the category obtained from $Top$ by inverting exactly the weak equivalences. If this is the category you wish to study then certainly weak equivalence is the way to go - and this is almost tautological - but in general I would not agree that weak equivalences are superior in any way. $\endgroup$ – Tyrone Aug 8 '18 at 20:56

The focus on weak equivalences instead of homotopies is largely a consequence of Grothendieck's slogan to work in a nice category with bad (overly general) objects, rather than working in a bad category that has only the good objects. Typically, there is a good notion of homotopies between maps that is well-behaved, but only on the "good objects". If we worked with a category consisting of only the good objects, then we wouldn't need weak equivalences, but we also would be sad because our category probably wouldn't have things like limits and colimits, and would generally be difficult to work with. So instead we enlarge our category to allow objects which are "bad" and which don't directly relate to the homotopy theory we really want to study. To do homotopy theory with the bad objects, we introduce a notion of weak equivalence which lets us say every bad object is actually equivalent to some good object, as far as our homotopy theory is concerned.

A basic example of this is simplicial sets and Kan complexes. Simplicial sets form a really really nice category that is easy to work with combinatorially or algebraically. However, on their own, they are awful for the purposes of homotopy theory. If you model some nice topological spaces as the geometric realizations of some simplicial sets, then most continuous maps between your spaces will not come from maps between the simplicial sets, even up to homotopy. We can define a notion of homotopy between maps of simplicial sets, but it is really poorly behaved (it's not even in equivalence relation, though you could take the equivalence relation it generates).

Now, there is a very special type of simplicial set which is really good for modeling homotopy theory, namely Kan complexes. The singular set of any topological space is a Kan complex. Homotopy classes of maps between two Kan complexes are naturally in bijection with homotopy classes of maps between their geometric realizations. So we have this great theory of Kan complexes which models the classical homotopy theory of spaces and has the advantage that our objects are more combinatorial and we don't have to deal with the pathologies of pointset topology.

However, despite all the nice things about Kan complexes, they don't form a particularly nice category. They aren't just the category of presheaves on a simple little category like simplicial sets are, and don't even have colimits. We can't work with them combinatorially nearly as easily as we can general simplicial sets.

So, we'd really like to use the entire category of simplicial sets and not just Kan complexes. But this is awkward, because we don't have a good notion of homotopy for simplicial sets, and don't even have "enough" maps between most simplicial sets to model what we want them to model. The solution is that we do still have a good notion of weak equivalence which works for all simplicial sets, and after inverting weak equivalences we get the homotopy category we want. Every simplicial set is weak equivalent to a Kan complex, and when working with just Kan complexes, weak equivalences give the same homotopy theory as homotopies between maps would.

Let me end with a more down-to-earth observation. A homotopy between maps $f,g:X\to Y$ is defined as a map $H:X\times I\to Y$ such that $Hi_0=f$ and $Hi_1=g$. Here $i_0:X\to X\times I$ is defined by $i_0(x)=(x,0)$ and $i_1$ is $i_1(x)=(x,1)$.

Now let $p:X\times I\to X$ denote the first projection. Observe that $pi_0=pi_1=1_X$. So, if we formally adjoin an inverse to $p$, $i_0$ and $i_1$ will become equal (both equal to $p^{-1}$), and consequently $Hi_0=f$ and $Hi_1=g$ will become equal.

In other words, imposing the homotopy equivalence relation on maps is essentially the same thing as considering all of the projection maps $p:X\times I\to X$ to be "weak equivalences". In this way, the classical equivalence relation on morphisms approach to homotopy is really just a special case of using weak equivalences. But weak equivalences are more general and flexible, and can be used in settings (like simplicial sets as discussed above) where an equivalence relation on morphisms would not do what you want.

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  • $\begingroup$ I like the idea about good categories of bad objetcs rather than bad categories of good objects. Do you have anything to say about the related question that I mention at the end ? $\endgroup$ – Maxime Ramzi Aug 9 '18 at 6:04
  • $\begingroup$ Not really, and it's a good question. In particular, I believe (but may be wrong) that working merely with a category with an equivalence relation on the morphisms does not yield the same suprisingly rich theory that weak equivalences yield as described in Hurkyl's answer. From a historical perspective, though, that doesn't explain why people use weak equivalences, since people started doing that long before that rich story was discovered. (And for all I know maybe there is a different but similarly rich story you can get using equivalence relations on morphisms.) $\endgroup$ – Eric Wofsey Aug 9 '18 at 14:00
  • $\begingroup$ If I remember it right, Quillen did Not know how powerful this notion could be. Colimits and limits are powerful ways to approximate objects. It would be sad if we cannot use these operations. During my reading, my feeling is that sometimes we don’t want to distinguish some morphisms if they are closed enough, namely linked by a chain of ‘weak equivalences’. Thus we quotient these morphisms and form a new category. However this category is out of control: it’s hard to compute the Hom space. A model structure helps us doing this: just compute Hom space between good objects, which is homotopy. $\endgroup$ – user12580 Aug 13 '18 at 15:10
  • $\begingroup$ I appreciated your answer for providing a good motivation to study this theory. I want to quote a word from May(not precisely): people should do some serious computations before they immerse into this quite abstract thing. $\endgroup$ – user12580 Aug 13 '18 at 15:16

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.

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  • $\begingroup$ Is your first "For whatever reason" because you don't know, because it's technical, or just that it's what exeperience taught us ? And secondly, about your last bit, doesn't the fact that $Cat$ is Quillen equivalent to $Top$ and $sSet$ suggest that we might be interested in modelling homotopy by $2$-categories ? (Perhaps $(2,1)$-categories more specifically, but not necessarily). I'll have to look more at homotopy types, what you mentioned is very interesting (thank you for your answer !) $\endgroup$ – Maxime Ramzi Aug 9 '18 at 6:20
  • $\begingroup$ @Max: My first is indeed because I don't know; I've never really considered the issue beyond noticing it exists. And secondly, yes! You commented as I was writing that very addendum! $\endgroup$ – user14972 Aug 9 '18 at 6:21
  • $\begingroup$ Thank you for that addendum ! Great answer ! $\endgroup$ – Maxime Ramzi Aug 9 '18 at 6:27

Here's an effort at an "answer." The structure you mention--weak equivalences, fibrations, cofibrations--are of course the reason for the study of model categories. Importantly, these notions, correctly axiomatized as a model category, allow for the abstract construction of homotopies and many of their general properties, no matter the setting. You start by defining cylinders, path objects, the whole nine yards, getting it all axiomatically. So in a sense, weak equivalences give birth to the notion of homotopy. The (co)fibrations are then there to help you with the calculations. And the theory of model categories has been very successful, so momentum has kept things this way.

I'm not sure how easy it would be to go the other way. If you abstract homotopies, maybe you can say what a weak equivalence should be (although there would be a problem there for objects that aren't co/fibrant). But then how do you get (co)fibrations to handle homotopy (co)limits and the like?

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  • $\begingroup$ Homotopy (co)limits are well-defined with just the notion of weak equivalences, you don't need (co)fibrations. (Of course, if you have them it helps for the computation, as you say in your first paragraph.) $\endgroup$ – Pece Aug 9 '18 at 6:27

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