What is the motivation behind the Factorization Systems of Model Categories?

Question

I am currently trying to get a grasp of higher category theory, being promised to get a nice framework to do homotopy theory (which I currently understand to be a theory of dealing with categories with weak equivalences generated by / interacting with some notion of interval/path-object/cylinder-object).

When reading about quasicategories it is unavoidable to learn about model categories, especially when following the approach of Prof. Cisinski's Higher Categories and Homotopical Algebra. I do appreciate the fact that model categories give very nice computational tools (e.g. for computing the homotopy category and some notions of homotopy limits), but the motivation behind their definition is elusive to me. Let me make precise what I mean:

Consider the following settings of homotopy theory we might try to put into a common framework:

1. For topological spaces the standard interval $I=[0,1]$ lets us generate a notion of weak equivalence called homotopy equivalence and we would like to study spaces up to homotopy equivalence.
2. Introducing homotopy groups, we might want to classify topological spaces up to their homotopy groups and the notion of weak homotopy equivalence is born. Yet for good spaces (e.g. CW-complexes) the notion of weak homotopy equivalence coincides with the notion of homotopy equivalence in (1), so the interval comes into play again.
3. In homological algebra we consider the category of chain complexes and there is a notion of homotopy of maps of complexes, which again can be generated by some interval object. We want to study chain complexes up to their homology groups, so again the notion of homotopy equivalence of chain complexes is to broad and we need to use quasiisomorphisms as weak equivalences.
4. Category theory usually works up to equivalence of categories, which is strictly weaker than isomorphisms of categories. One can show that this notion of weak equivalence is yet again generated by the interval object given by the free category with an isomorphism.

So it becomes clear that a nice 1-categorical framework should incorporate a well behaved notion of weak equivalence and (maybe) a notion of interval generating or at least being related to this.

But where do the weak factorization systems come from?

In the case of (1) one might have the idea that having the homotopy lifting property (Hurewicz-fibration) or the homotopy extension property (Hurewicz-cofibration) is a useful thing to have. To me it is already slightly less clear, why one might have the idea in (2) to consider Serre-fibrations, and completely unclear, where Serre-cofibrations arise. Similarly it is unclear to me why I should have the idea to look at similar lifting problems in (3) or (4) and to expect them to behave as in the axioms of a model category.

It is likely that there is no general pattern to notice here and that model categories are modelled after (1) and then were observed to work in most other settings as well. Or they may have evolved slowly by picking up on tricks and facts used here and there. But I still have hope for a good general reason to consider these lifting properties, similar to how trying to figure out what associativity of $n$ morphisms up to homotopy leads to something simplicial.

TLDR: What is the intuitive (besides they get the job done) reason to consider weak factorization systems as used in the definition of a model category?

As always thank you very much for your time, considerations and answers!

Answer 1

One thing you get from studying (the homotopy theory of) topological spaces, simplicial sets or chain complexes, is that not all maps behave well with respect to homotopical notions.

For instance, one very "annoying" thing is that pullback along arbitrary maps might not preserve weak equivalences. But you notice that some maps behave well with respect to homotopical notions, for instance pullback along certain maps do preserve weak equivalences.

There are two very different situations in which maps behave well : when they look like very neat inclusions, or very neat surjections. Of course the meaning of "very neat" varies from one context to another, but these are the kinds of maps that behave correctly with respect to homotopy theory. With hindsight, it's pretty reasonable : if a map is a very neat inclusion, you have some "wiggle room" for your homotopies (in fact this is exactly what happens in the homotopy theory of spaces)

Then you also notice that in your classical settings, every map can be factored as a very neat inclusion followed by a very neat surjection (and in fact you can choose one of them to be a weak equivalence) - this is motivated by the very classical constructions of cones and path spaces in the homotopy theory of spaces, which existed beforehand.

What this tells you in particular, is that you can reduce many things to these nicely behaved maps, and since they are nicely behaved, you can do lots of things even with arbitrary maps - in particular it is clear how this factorization property is desirable. If these maps are to behave well, it's good to know that up to homotopy, any map looks like one that behaves well !

One thing that's important to spot here is that these notions came from the practice of homotopy theory - Quillen (and others) saw how people did homotopy theory and the commonalities in those contexts, and what they were used for. Now this idea of approximating arbitrary maps by nice maps not only appeared in the contexts he looked at , but also seemed essential and pretty practical.

Note, in particular, that it's not at all the case that you need a whole factorization system in general. The main point is the approximation of maps by nice maps - for instance since you're reading Cisinski's book, you'll see later that he has chapters about ($\infty$-)categories with weak equivalences and fibrations. In that setting you don't have a whole factorization system, but you still have those nice maps (the fibrations) that carry your computations and also help you control your localization - since not every map is nice, it's good to have nice maps that approximate things.

tldr : Factorization systems aren't the conceptual thing. The conceptual thing underlying these notions is that of having maps that behave nicely with respect to your "homotopy theory"; and being able to approximate arbitrary maps by nicely behaved maps.

In the classical settings, there are two ways to be well-behaved : being a neat inclusion, or being a neat surjection. In many settings, you have both, and you can factor your maps in the way you probably know, and this is a huge technical convenience. In some settings you only have one kind - this is still very good, as long as you can approximate arbitrary maps by nice map, but it's less technically convenient.