What are Model Categories? (Homotopy Theories pt 1/4)

11 Jul 2022 - Tags: homotopy-theories

I’m a TA at the HoTTEST Summer 2022, a summer school about Homotopy Type Theory, and while I feel quite comfortable with the basics of HoTT, there’s a ton of things that I should really know better, so I’ve been doing a lot of reading to prepare. One thing that I didn’t know anything about was $\infty$-categories, and the closely related model categories. I knew they had something to do with homotopy theory, but I didn’t really know how. Well, after lots of reading, I’ve finally figured it out, and I would love to share with you ^_^.

This was originally going to be one post, but it ended up having a lot of tangentially related stuff all mashed together, and it felt very disorganized and unfocused¹. So I’ve decided to split it into three four posts, with each post introducing a new, more abstract option, and hopefully saying how it solves problems present in the more concrete settings.

Let’s get to it!

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First of all, what even is a “homotopy theory”? Let’s look at the primordial example:

But there’s another motivating example, which we also call “homotopy”:

Obviously these are related – after all, from a topological space we can get an associated “singular cochain” of $R$-modules. Then a homotopy of spaces induces a homotopy of cochains, and indeed the cohomology of the cochain complex agrees with the cohomology of the space we started with.

More abstractly, what links these situations? Well, we have some objects that we want to consider “the same up to homotopy”, and we capture this (as the category inclined are liable to do) by picking out some special arrows. These are the “homotopy equivalences” – and they’re maps that we want to think of as isomorphisms… but which might not actually be.

So, in $\mathsf{Top}$, we have the class of weak homotopy equivalences², which we want to turn into isomorphisms. And in $\mathsf{Ch}(R\text{-mod})$, the category of chains of $R$-modules, we want to turn the quasi-isomorphisms into isomorphisms.

With these examples in mind, what should a homotopy theory be?

Following our examples, we want to think of the arrows in $\mathcal{W}$ as being morally isomorphisms, even though they might not actually be isomorphisms.

Now, a (small) category is an algebraic structure. It’s just a set with some operations defined on it, and axioms those operations satisfy. So there’s nothing stopping us from just… adding in new arrows, plus relations saying that they’re inverses for the arrows we wanted to be isomorphisms. By analogy with ring theory, we call this new category the Localization $\mathcal{C}[\mathcal{W}^{-1}]$. This is also called the Homotopy Category of $(\mathcal{C}, \mathcal{W})$.

For example, if we localize $\mathsf{Top}$ at the weak homotopy equivalences, we get the classical homotopy category $\mathsf{hTop}$. If we invert the quasi-isomorphisms of chains of $R$ modules, we get the derived category⁵ of $R$ modules $\mathcal{D}(R\text{-mod})$.

There are many important examples of two relative categories presenting the same homotopy theory. To start, let’s consider the category $s\mathsf{Set}$ of simplical sets, equipped with a notion of weak equivalence. It turns out that this presents the same homotopy theory as $\mathsf{Top}$ with weak homotopy equivalences!

This means that if we have a question about topological spaces up to homotopy, we can study simplicial sets instead, with no loss of information! This is fantastic, since simplicial sets are purely combinatorial objects, so (in addition to other benefits) it’s much easier to tell a computer how to work with them⁷!

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This is great, but there’s one hitch…

The homotopy category $\mathcal{C}[\mathcal{W}^{-1}]$ might be terribly behaved. For instance, even if $\mathcal{C}$ is (co)complete, the homotopy category almost never is⁸! Even worse, it’s possible that we end up with a proper class of arrows between two objects, even if $\mathcal{C}$ started out locally small. Lastly, it’s difficult to tell when two relative categories present the same homotopy theory.

This is all basically because arrows in $\mathcal{C}[\mathcal{W}^{-1}]$ are really hard to understand! After all, a general arrow from $A$ to $B$ in $\mathcal{C}[\mathcal{W}^{-1}]$ looks like this:

\[A \overset{\sim}{\leftarrow} C_0 \to C_1 \overset{\sim}{\leftarrow} \cdots C_k \to B\]

where all the $\overset{\sim}{\leftarrow}$s are in $\mathcal{W}$. After we invert $\mathcal{W}$, these arrows all acquire right-facing inverses, which we can compose in this chain to get an honest arrow $A \to B$.

There’s a zoo of techniques for working with homotopy categories, but they all come down to trying to tame this unwieldy definition of arrow⁹. In this post, we’ll be focusing on a very flexible approach by endowing $(\mathcal{C}, \mathcal{W})$ with a Model Structure.

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Roughly, to put a model structure on $(\mathcal{C}, \mathcal{W})$ (which we now assume to be complete and cocomplete) is to choose two new subfamilies of arrows: the fibrations and the cofibrations.

From these, we define some “nice” classes of objects:

• $X$ is called fibrant if the unique arrow to the terminal object is a fibration
• $X$ is called cofibrant if the unique arrow from the initial object is a cofibration
• $X$ is called bifibrant if it is both fibrant and cofibrant

Let’s see some examples:

A fibration $f$ is an arrow that’s easy to “lift” into from cofibrant objects $A$:

For instance, covering spaces and bundles are examples of fibrations in topology. If we restrict attention to the CW-complexes then every object is cofibrant, and this statement is basically the homotopy lifting property for covering spaces.

Algebraically, there’s a model structure where a map of chains of $R$-modules $f : A_\bullet \to B_\bullet$ is a fibration if each $f_n$ is a surjection. The cofibrant objects in this model structure are exactly the levelwise projective complexes, so this lifting property becomes the usual lifting property for projective modules.

Dually, a cofibration $i$ is an arrow that’s easy to “extend” out of provided our target $Y$ is fibrant:

We should think of a cofibration as being a subspace inclusion $A \hookrightarrow X$ where $A$ “sits nicely” inside of $X$.

For instance, the inclusion arrow of a “good pair” is a cofibration. Thus inclusion maps of subcomplexes of a simplicial/CW/etc. complex are cofibrations. More generally, any subspace inclusion $A \hookrightarrow X$ where $A$ is a nieghborhood deformation retract in $X$ will be a cofibration.

Algebraically, a map $f : A_\bullet \to B_\bullet$ is a cofibration exactly when each $f_n$ is an injection whose cokernel is projective¹⁰. This is a somewhat subtle condition, which basically says that each $B_n \cong A_n \oplus P_n$ where $P_n$ is projective. It should be intuitive that given a map $A_n \to Y_n$, it’s easy to extend this to a map $B_n \to Y_n$ under these hypotheses.

Precisely, these triangles are really special cases of squares. In the fibration case, the left side of the square is the unique map from the initial object to $A$. Dually, in the cofibration case the right hand side of the square should really be the unique map from $Y$ to the terminal object.

This lets us unify these diagrams into a single axiom, which says that a square of the form

has a lift (the dotted map $X \to E$) whenever $i$ is a cofibration, $f$ is a fibration, and one of $i$ or $f$ is a weak equivalence.

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The model category axioms¹¹ imply that every object is weakly equivalent to a bifibrant object. Since, after localizing, our weak equivalences become isomorphisms, this means we can restrict attention to the bifibrant objects… But why bother?

Well in any model category we have a notion of “homotopy” between maps $f,g : A \to B$ which is entirely analogous to the topological notion. Then by studying homotopy-classes of maps, we’ll be able to get a great handle on the arrows in $\mathcal{C}[\mathcal{W}^{-1}]$!

Precisely, each object $A$ is weakly equivalent to a cylinder object $A \times I$ which acts like $A \times [0,1]$ in topology¹². In particular, it has two inclusions $\iota_0 : A \to A \times I$ and $\iota_1 : A \to A \times I$.

Now if $f, g : A \to B$, then a homotopy between $f$ and $g$ is a map $H : A \times I \to B$ so that the following triangle commutes:

If there is a homotopy between $f$ and $g$, we say that $f$ and $g$ are homotopic or homotopy equivalent, often written $f \sim g$.

This brings us to the big punchline:

Thus, a very common way we use model structures to perform computations is by first replacing the objects we want to compute with by weakly equivalent bifibrant ones. For instance, we might replace a module by a projective resolution¹³. Then maps in the homotopy category $\mathcal{C}[\mathcal{W}^{-1}]$ are just maps in $\mathcal{C}$ up to homotopy¹⁴!

Notice this, off the bat, solves one of the problems with homotopy categories. Maybe $\mathcal{C}[\mathcal{W}^{-1}]$ isn’t locally small, but it’s equivalent to something locally small.

Moreover, model structures give us a very flexible way to tell when two relative categories have the same homotopy theory. Indeed, say we we have a pair of adjoint functors $L \dashv R$ between $\mathcal{C}_1$ and $\mathcal{C}_2$ that respect the weak equivalences in the sense that $f : c_1 \to R(c_2)$ is in $\mathcal{W}_1$ if and only if its adjoint $\tilde{f} : L(c_1) \to c_2$ is in $\mathcal{W}_2$. Then $\mathcal{C}_1[\mathcal{W}_1^{-1}] \simeq \mathcal{C}_2[\mathcal{W}_2^{-1}]$, and moreover, this equivalence can be computed from the adjunction $L \dashv R$.

This is called a Quillen Equivalence between \((\mathcal{C}_1, \mathcal{W}_1)\) and \((\mathcal{C}_2, \mathcal{W}_2)\)¹⁵.

Moreover again, even if $\mathcal{C}[\mathcal{W}^{-1}]$ doesn’t have (co)limits, we can always construct homotopy (co)limits, which we can compute using the same cylinder objects from before. For instance, the homotopy pushout of

will be the colimit of the related diagram:

Geometrically, rather than gluing $X$ to $Y$ along $A$ directly, we’re adding a path from $f(a)$ to $g(a)$. This is the same up to homotopy, but is better behaved. We do this by taking disjoint copies of $X$ and $Y$, and then gluing one side of the cylinder $A \times I$ to $X$ along $f$, and gluing the other side of $A \times I$ to $Y$ along $g$.

Something like this will always work, but knowing how to modify our diagram (and why) can be quite involved¹⁶. Thus we’ve succeeded in computationally solving the (co)limit issue, but it would be nice to have a more conceptual framework, in which it’s obvious why this is the “right thing to do”…

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We’ve seen that a model structure on a relative category helps to make computations in the localized category $\mathcal{C}[\mathcal{W}^{-1}]$ effective (or even possible). But model categories have a fair number of problems themselves.

For one, there’s no great notion of a functor between model categories. In the case that a functor $F : (\mathcal{C}_1, \mathcal{W}_1) \to (\mathcal{C}_2, \mathcal{W}_2)$ comes from and adjoint pair, then we can derive it to get a functor on the homotopy categories, but this is too restrictive to be the general notion of functor between model categories.

Related to functors, if $\mathcal{C}$ has a model structure and $\mathcal{I}$ is an indexing category, then $\mathcal{C}^\mathcal{I}$, the category of $\mathcal{I}$-diagrams in $\mathcal{C}$ (with the pointwise weak equivalences) may not have a model structure. See here, for instance¹⁷.

With this in mind, we would like to have a version of the theory of model categories which has better “formal properties”. While working with a single model category is often quite easy to do, as soon as one looks for relationships between model categories, we’re frequently out of luck. In fact, since (co)limits come from functors, as soon as we’re interested in (co)limits we already run into this problem! This is one philosophical reason for the complexity of homotopy (co)limits.

The solution lies upwards, in the land of $\infty$-categories. These are categories with homotopy theoretic structure (in the classical sense of topological spaces) built in from the start. Miraculously, these will solve all the above problems and more – while the category of model categories is a terrible place to live (if we can define it at all…) the category of $\infty$-categories¹⁸ is extremely similar to the category of categories. Then, since every model category presents an $\infty$-category, we’ll be able to use this machinery to solve our formal problems with model categories!

How exactly does this work? You’ll have to read more in part 2!

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