Why use $\infty$-categories over model categories?

Question

Let's say that I know (roughly) how derived categories help us solve problems. After all, we want to consider chain complexes up to homotopy equivalence, and the derived category literally lets us do that. Moreover, since the category of modules embeds into the category of (bounded) chain complexes, the (injective or projective) model structures on the derived category let us compute $\text{Ext}$ and $\text{Tor}$ (for instance), and I'm definitely sold on these being interesting and useful.

But I hear a lot of people talking about how, instead of localizing fully, we should really be keeping track of how our weak equivalences invert. This is packaged together into the $\infty$-category (properly $(\infty,1)$-category) associated to a model category (more generally a relative category).

I understand (roughly) how this localization process works, but I don't yet understand why I should care. What problems are made easier by considering the $\infty$-category $L(\mathcal{C}, \mathcal{W})$ rather than the localization $\mathcal{C}[\mathcal{W}^{-1}]$? I'm familiar with the analogy that "model categories are like the coordinate charts of $\infty$-categories", so maybe there aren't a lot of computational benefits to using this language.

If it doesn't solve problems directly, though, are there conceptual reasons to use the language of $\infty$-categories? I would be perfectly willing to believe that it's easier to build "$\infty$-functor categories", for instance, than it is to build a "model functor category between model categories"... But if that's the case, I would want a good reason to care about $\infty$-functor categories included in the answer. I have an intuitive sense of why we might care (since I care about 1-category theory) but seeing an example would be good for me.

So then:

What are some examples, as concrete as possible, which showcase how $L(\mathcal{C},\mathcal{W})$ is easier to work with (either computationally or conceptually) than $\mathcal{C}[\mathcal{W}^{-1}]$?

Additionally, if there are any references which showcase explicit computations with $\infty$-categories, ideally to solve problems or give insight that a naked model category couldn't do, I would love to hear about those as well.

Also, while my question is coming from the point of view of derived categories in algebra, I would (of course) be interested in examples from the topological side too.

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Thanks in advance! ^_^

Answer 1

An important construction in a derived category $\mathcal{D}$ is the mapping cone $cone(f)$ of a morphism $f:X \to Y$ in $\mathcal{D}$. Namely, to each $f$ as above, we'd like to associate an exact triangle

$$X \longrightarrow Y \longrightarrow cone(f) \longrightarrow X[1]$$

with the property that the composition $X \to cone(f)$ is $0$. Moreover, we ask that whenever there is a map $Y \to Z$ such that the composition $X \to Y \to Z$ is zero, there is a map $cone(f) \to Z$ making the appropriate diagram commute. Fortunately, such a thing always exists, and actually every exact triangle turns out to be isomorphic to one of these.

Notice that $cone(f)$ behaves a bit like a cofiber or quotient object. In fact, if we could show that the morphism $cone(f) \to Z$ above always existed $\textit{uniquely}$, then the cone of $f$ would literally be its cokernel. Unfortunately this is not the case, and hence there is no functorial way of assigning to a morphism its mapping cone.

The non-functoriality of the mapping cone is one the problems with derived categories that the notion of infinity categories allows us to circumvent. The relaxed notion of colimits in infinity categories means that we no longer demand universal objects to be uniquely defined on the nose, but only up to a contractible space of choices. As a result, in the infinity categorical analogue of a derived category (i.e. a "stable" $\infty$-category), the "mapping cone" of a morphism is just its cofiber - a construction which is fully functorial.

Answer 2

There are quite a few reasons listed on the nLab: https://ncatlab.org/nlab/show/homotopy%20theory%20FAQ#what_is_the_homotopy_category_of_an_1category_what_are_its_limitations

Brief summary:

• In the homotopy category, limits and colimits (with the except of simple cases like products and coproducts) typically do not exist. In contrast, the setting of model categories and/or quasicategories allows for a correct handling of limits and colimits.

• Even worse, there is no way to talk about diagrams in the homotopy category, other than the trivial cases. For example, we cannot talk about homotopy commutative squares using homotopy categories, since the required data of a homotopy between two compositions is not recorded in the homotopy category.

• Most other constructions relying on (co)limits and diagrams fail for similar reasons.

Answer 3

There are at least two distinct aspects to the question : one is internal to $\infty$/model categories, while the other one is external, i.e. about the category of categories.

The internal aspect is quite often discussed (and also vey important) : as mentioned in other answers, all kinds of statements/constructions involving diagrams just fail in the homotopy category (functoriality of cones, of homotopy co/limits, homotoyp coherent diagrams , ...) or are not conceptually clear in model categories (the universal property of a single homotopy co/limit).

While these are important, and matter conceptually, I feel like they do not necessarily show why $\infty$-categories are better suited than model categories : for instance, most (even if not all) $\infty$-categories where one meaningfully computes homotopy co/limits are co/complete, and so you can satisfy yourself with a "global" universal property. There are, of course, also internal aspects where $\infty$-categories are more convenient, but I want to focus on the next part of my answer.

This is about the external aspects, namely when one talks about categories, as opposed to a category. It is "folklore knowledge" that category theory becomes really useful when one applies it to several categories and functors between them. In this regard, model categories fail, not only because you cannot consider a category of functors between two such, but also because diagrams of model categories are a complicated thing to understand conceptually. On the other hand, diagrams of (small or large) $\infty$-categories make perfect sense and are really useful. For instance one can take homotopy co/limits of $\infty$-categories and easily obtain meaningful results and constructions, while taking homotopy co/limits of model categories looks like a mess (I don't even know if it can be made sense of).

For example, the idea that quasicoherent sheaves on a scheme are a local notion is completely built in the statement that $QCoh$, as a functor from (the opposite category of) schemes to $\infty$-categories satisfies Zariski descent: $QCoh(X) \simeq \lim_{Spec(R)\to X} QCoh(Spec(R))$.

Similarly, you can phrase Galois descent as the statement that $Mod_A \simeq (Mod_B)^{hG}$ for $A\to B$ a faithful Galois extension.

This kind of tool means you can also apply category theory to itself which is one of the places where it really shines. For instance, you prove things about algebra objects in $C$ and apply it to $C= Cat_\infty$ or $Pr^L$. This is also extremely well suited to define highly-structured objects, such as norms in motivic homotopy theory.

In my mind, the latter, external aspect, is what really distinguishes the two points of view. You can make cones functorial from a model-category theoretic perspective, and you can talk (to some extent) about homotopy co/limits. But you can not apply model category theory to itself.