Okay so what's a model category? First of all, I'm talking about Quillen model categories, which are things that show up in algebraic topology and homotopy theory. Sorry to disappoint the growing cadre on model theorists who have been hanging around here lately.

The basic idea is that a model category is a category with extra structure that makes it a suitable place to do homotopy theory (and it's quite a bit of structure). One really essential piece of structure however is the idea of a weak equivalence.

Among several other pieces of structure, a model category C comes equipped with some distinguished class of morphisms inside C that we've designated "weak equivalences." These need not be topological in nature, it's just a designation (with its origins in topology).

The sort of canonical examples are the category of topological spaces, where weak equivalences are the continuous maps which induce isomorphisms on all homotopy groups, and chain complexes over a ring R, where the weak equivalences are the quasi-isomorphisms.

Just a reminder that quasi-isomorphisms are chain maps that induce isomorphisms on all homology groups, so they're sort of like the "algebraic" version of the weak equivalences in topology. From now on I'll just write w.e.'s instead of "weak equivalences."

Given a category C in which you've chosen some collection of w.e.'s W in Mor(C), you can just formally invert all of the w.e.'s. Just take the smallest category containing C in which every morphism in W has an inverse. I'll call this Ho(C), for "homotopy category of C."

Again, the classical examples are Top, with classical w.e.'s, and Ch(R), with quasi-isos. You get THE homotopy category, and THE derived category (of R) respectively.

I'm hoping these two examples are enough motivation for you to think that this is a useful general construction to have. We can do a lot with the homotopy category and the derived category, and they're SIMPLER than the full categories of spaces and chain complexes.

I also want to point out that we COULD just stop here. There is a notion of something called a "relative category," which is nothing more than a category with a choice of "weak equivalences," and you can definitely do stuff with them.

But by requiring more structure, we can get better control over the homotopy category. In particular, we can actually compute homotopy classes of maps between objects (sometimes), and we can choose specific MODELS of a given homotopy type, or of a homotopy class of maps.

In other words, if you're only working in the homotopy/derived category, you only ever know what you're working with "up to homotopy," and sometimes that's just not enough to prove what you want to prove.

A model category structure will let you retain "homotopical control" over what you're doing (i.e. make sure any constructions you do are homotopy invariant) while giving you actual models (not just equivalence classes) of whatever object you want to look at.

So here's the structure of a model category (it's a lot):
- a category C with all limits and colimits,
- three distinguished classes of morphisms in C, called weak equivalences, fibrations, and cofibrations, which we'll denote WE, FIB, and COF

- (notation) a map in WE⋂FIB is called an acyclic fibration (and we'll write AFIB for these maps) and a map in WE⋂COF is called an acyclic cofibration (and we'll write ACOF) for these

- every morphism in C, f:X→Y factors as i:X→Z followed by p:Z→Y where i is a cofibration and p is an acyclic fibration

- every morphism in C, f:X→Y factors as j:X→W followed by q:W→Z, where j is an acyclic cofibration and q is a fibration

If you have a commutative square like in this picture (without the dotted arrow), i∈COF, p∈FIB and EITHER i or p is also a weak equivalence, then there exists a dotted arrow (here labeled h) making that square commute:

And these ones, which I don't anticipate figuring into this thread so much:

- the maps in WE all satisfy the "2-out-of-3" property ( https://ncatlab.org/nlab/show/two-out-of-three)
- WE, COF and FIB are all closed under retracts (in the arrow category, as described here: https://ncatlab.org/nlab/show/retract#in_arrow_categories)

So okay, that's a lot of data. How are we supposed to think about all this stuff?

Again, the weak equivalences are the things you want to invert to get at the "underlying homotopy theory" of the category, i.e. you want to kill of all the data except homotopy equivalence classes (of objects, and maps between them) for some version of the word "homotopy."

You're maybe supposed to think about the fibrations as something like surjections or covers, but I think it's really useful to understand where they're coming from in a (somewhat) historical sense.

And the relevant historical idea here is the "homotopy lifting property." Basically, when you've got a nice enough map of spaces p:X→Y and a homotopy between two maps f,g:Z→Y such that one end of the homotopy can be "lifted" to a map Z→X, you can lift the entire homotopy.

It's pretty useful to be able to do this, for instance if you've got a covering space you'll want to be able to lift a path in the base space to a path in the covering space, perhaps for reasons of controlling the fundamental group.

But it turns out lots of maps have the "homotopy lifting property," even ones which aren't covering spaces. Maps p:X→Y that have the homotopy lifting property with respect to all pairs of maps f,g:Z→Y are called Hurewicz fibrations.

And, indeed, there is a model structure on Top in which the fibrations are precisely the Hurewicz fibrations. So FIB in a model category is supposed to be something like "maps along which we can lift homotopies."

But hold on, we've already seen a "lifting property" that maps in FIB are supposed to satisfy, namely, we're supposed to be able to insert a lift along a fibration whenever the left hand side of the square is an acyclic cofibration. Here's the picture again to remind you:

(I'm very sorry for these pictures, I'm just taking them from Wikipedia and then they're getting blown up and they're awful)