# Grothendieck derivators vs $\infty$-categories

I have some questions on derivators and $$(\infty,1)$$-categories, I would be grateful if someone could help me.

• Is there some problems that $$(\infty,1)$$-categories/derivators can resolve but derivators/$$(\infty,1)$$-categories cannot resolve?

• Why do so many people prefer $$(\infty,1)$$-categories than Grothendieck derivators?

• Is there a good place to learn about $$(\infty,1)$$-categories than Grothendieck derivators but with a historical and comparing point of view?

• Can one take a limit of a diagram of derivators? – Lennart Meier Jul 27 at 11:11
• The following paper shows interesting relations between both: Arlin, Kevin. 2020. “On the $\infty$-Categorical Whitehead Theorem and the Embedding of Quasicategories in Prederivators.” ArXiv:1612.06980 [Math], February. arxiv.org/abs/1612.06980. – Fernando Muro Jul 27 at 16:59
• Another version of Lennart's comment: can you talk about sheaves of derivators? One of the great strenghts of ∞-cats is that they work very well in families (so, for example, you can rephrase faithfully flat descent as "$\mathrm{QCoh}(-)$ is a sheaf") – Denis Nardin Jul 27 at 17:46
• I don't know that I have a really compelling answer here, but to Denis and Lennart's points the answer is: at best only in terms of homotopy limits in a model structure, which is a major advantage of $\infty$-categories. Derivators are better suited to working within a single homotopy theory at a time. Regarding your second question, well, since Lurie began writing there has been vastly more machinery developed for $\infty$-categories, and some things (see above) have been done only in that framework. – Kevin Arlin Jul 28 at 6:33
• Very roughly speaking, by [Ren09] the $2$-category of derivators is equivalent to: 1. Take the $(\infty,1)$-category of $(\infty,1)$-categories. 2. Truncate it to a $(2,1)$-category. 3. Perform a $2$-categorical localization inverting those $1$-morphisms which induce an equivalence on homotopy categories. [Ren09] Renaudin, Olivier. 2009. “Plongement de Certaines Théories Homotopiques de Quillen Dans Les Dérivateurs.” Journal of Pure and Applied Algebra 213 (10): 1916–1935. doi.org/10.1016/j.jpaa.2009.02.014. – Fernando Muro Jul 29 at 18:01

The short answer is that $$(\infty,1)$$-categories are the "real" object of interest. Derivators are a tool for working with them that is sometimes (for some people) easier to use, but doesn't remember all the information and hence is not always applicable.

When you work with $$(\infty,1)$$-categories, you have to deal explicitly with higher-dimensional coherence all the time. Everything is determined only up to equivalence. This can be kind of a pain, so it's convenient to have 1-categorical structures that neverthless carry $$\infty$$-categorical information.

The classical kind of 1-categorical structure used for this is a Quillen model category. This carries "too much" information, in that two objects can be equivalent in an $$(\infty,1)$$-category but not isomorphic in the model category it presents, and similarly two model categories can present equivalent $$(\infty,1)$$-categories but not be equivalent categories. Thus a model category needs a notion of "weak equivalence" between its objects, and similarly we have a notion of "Quillen equivalence" between model categories. But a model category contains all the information of an $$(\infty,1)$$-category, and so we can work with all the higher coherences as necessary.

A derivator is sort of a "dual" to a model category: it carries "too little" information. An advantage is that it is not subject to the issues of weak equivalence: two objects of an $$(\infty,1)$$-category are equivalent if and only if they are isomorphic in the corresponding derivator, and similarly two (locally presentable) $$(\infty,1)$$-categories are equivalent if and only if their corresponding derivators are equivalent (in a 1-categorical sense: derivators are the objects of a 2-category just like 1-categories are, and here we mean equivalence internal to that 2-category).

But a derivator doesn't have enough information to do everything we might want to with higher coherences. It's essentially an enhancement of the homotopy 1-category (the quotient by homotopies) that remembers the notion of homotopy-coherent diagram, and therefore also of homotopy limit and colimit (and homotopy Kan extension). This is sufficient for a surprising amount of homotopy coherence, at least when working only within a single $$(\infty,1)$$-category. But there are some things it can't do (or not very well), notably those that involve diagrams of $$(\infty,1)$$-categories and things like functor $$(\infty,1)$$-categories.

So the answer to your first question is yes, there are things you can do with $$(\infty,1)$$-categories but not with derivators; but no, anything you can do with a derivator can also be done with an $$(\infty,1)$$-category. And I think this mostly answers your second question as well: given that derivators are not good enough for everything, even if they make certain things easier, it's understandable that many people prefer not to learn two different languages and stick to $$(\infty,1)$$-categories even if they happen to be doing something that might be a bit easier with derivators. (On the other hand, I personally feel that it is easier to be sloppy with $$(\infty,1)$$-categories --- partly because being precise is so much work --- and easier to be precise with derivators, which is one reason that I still use the latter sometimes.)

Historically, derivators were introduced (by Heller, Grothendieck, and Franke, fairly independently) before practical notions of $$(\infty,1)$$-category were available. So at the time they were the only way of getting rid of the "weak equivalences"; but even with that advantage they never really caught on. I'm not quite sure why not; perhaps the requisite 2-category theory was also offputting at the time?

As for references, I don't have any suggestions other than those on the nLab page. I particularly like the expository aspect of Moritz Groth's work. You may also be interested in this blog post of mine.

• Mike, you got my like, for two reasons: 1) because you had the guts to post an answer (a very good one, but that is beside the point) rather than "dancing around" 2) the second one is that it helps me personally to understand the context. So: even though I used the inappropriate (but not totally inappropriate, I will explain later why) term "invariant", whereas I should have said "real", th bottom line is this: infinity cats are (at least at the current level of understanding) THE homotopy object, whereas both Quillen and associates and Derivators are TOOLS. – Mirco A. Mannucci Jul 29 at 17:49
• As for your reconstruction of the history and the role of weak equivalences, I have some reservations, on which I will comment over the weekend. Meanwhile I invite everyone to like it and the poster to accept it. BRAVO – Mirco A. Mannucci Jul 29 at 17:50
• My humble suggestion about why derivators never caught on is because working with them is really a lot harder than you seem to think, or at least this has been my experience. It might be that they are very natural for people well versed in 2-category theory though (which I am very much not). – Denis Nardin Jul 29 at 18:15
• @DenisNardin I've (co-)written four papers using derivators. My experience is that it takes a bit of getting used to to think of a "coherent diagram" as a primitive object rather than something "put together" out of objects, morphisms, and (perhaps) homotopies, but once that barrier is surmounted the technology is quite pleasant, especially in that one doesn't worry about things with names like left-quasi-marked-fibrations of simplicial sets but can just use standard 1-categorical ideas like adjunctions. Perhaps, as you say, some of that is due to my familiarity with 2-categories. – Mike Shulman Jul 29 at 19:27
• @MircoA.Mannucci The only statement along those lines that I know of is the result of Renaudin, that the 2-categories of locally presentable derivators and of locally presentable $(\infty,1)$-categories (which he constructs using the 2-category of combinatorial model categories and Quillen adjunctions) are equivalent. – Mike Shulman 2 days ago