I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition in Hovey's or Hirschhorn's book, i.e. there are sets of maps I and J who detect the trivial fibrations and fibrations by lifting and whose domains are small relative to I-cell (resp. J-cell). There is a weaker notion used by Emily Riehl, which I will ask about at the very end.

When you want to do Bousfield localization it's common to assume $M$ is in addition either combinatorial or cellular. Combinatorial means cofibrantly generated and locally presentable as a category. Cellular is more mysterious. Morally, it's asking for good control over cell complexes (i.e. things built from the maps in I via gluing). Formally, it's asking for

  1. Domains of maps in J are small relative to cofibrations
  2. Cofibrations are effective monomorphisms, i.e. any cofibration $f:X\to Y$ is the equalizer of the two obvious maps $Y\to Y \coprod_X Y$
  3. Domains and codomains of I are compact (in the sense of Hirschhorn, not Hovey) relative to I-cell.

Condition 1 is standard and usually easy to verify (for example, it comes for free if $M$ is locally presentable). Condition 2 is basically asking that the intersection of the two copies of $Y$ in $Y\coprod_X Y$ is equal to $X$, so it's also not that hard to check in categories where cofibrations are inclusions of some kind. Condition 3 is a pain to check, because it's really asking for a cardinal $\kappa$ such that for any presentation of a map $f:A\to B$ as a transfinite composition of pushouts along a chosen set of cells then any map from a (co)domain $X$ of a map in $I$ to $B$ factors through some subcomplex of size $\leq \kappa$. Again, if $M$ is locally presentable then this should be true automatically, at least if the collection of subcomplexes is filtered (since you know that all objects are small relative to filtered colimits).

It's clear that not every cellular model category is combinatorial. For example, Top is not (Hovey proves on page 49 of his book that the two-point Sierpinski space is not small). It's clear that not every combinatorial model category is cellular, because condition 2 can fail. For example, Hirschhorn provides such an example as 12.1.7. Another example is in Finnur Larusson's paper "The Homotopy Theory of Equivalence Relations." Another is the category of small categories Cat, as I learned today from a preprint of Amrani Ilias called "Stabilization of the category of simplicial objects in Cat." Similarly, if you take spectra valued in Cat it's not cellular (but it is combinatorial) as Deb Vicinsky has shown in her thesis. There are also examples of model categories which are not cofibrantly generated. My favorite is the Strom model structure on Top, and the proof is in Raptis's paper on Homotopy Theory for Posets. It seems to me that all the other conditions of cellularity are true here (suitably interpreted) except for cofibrantly generated, but perhaps I am wrong. Lastly, there is a model structure on a locally presentable category which is provably not cofibrantly generated, and it's given by Boris Chorny's example from the paper "The Model Category of Maps of Spaces is not Cofibrantly Generated." The category in question is simply the arrow category valued in sSet. When I tell these examples to people in the infinity category community I always get the same question, so now I'm putting it out to the MathOverflow world:

(1) Is there an example of a complete and cocomplete infinity category which does not admit any cofibrantly generated model?

This is probably a hard question. Chorny's example doesn't work, because every presentable infinity category gives rise to a combinatorial model category (which in this case will have the same weak equivalences but different cofibrations), obtained by embedding into simplicial presheaves then taking a Bousfield localization. A related question is

(2) Is there an example of a complete and cocomplete infinity category which does not admit a cellular model, but does admit a cofibrantly generated one?

This seems much more likely to be true to me, since from a model category perspective there really is a difference between cellularity and combinatoriality. But since all statements required for cellularity are about cofibrations, perhaps there's a sneaky way to always choose a nice set of cofibrations when you pass from a presentable infinity category to a model category. Perhaps you could shrink the cofibrations via Bousfield colocalizations, for example.

(3) Are there examples which have arisen in practice of non presentable infinity categories?

Lastly, I want to better understand where Riehl's definition of cofibrantly generated fits into this story. In a combinatorial model category the smallness hypotheses are automatic, so it seems Riehl's definition matches the usual one. Similarly, I suppose (1) and (3) in the definition of cellularity force Riehl's definition and the usual one to agree, though perhaps there is a way to weaken cellularity to algebraic cellularity in the sense of Riehl's algebraic wfs. Riehl also has notions of enriched weak factorization systems which allow you to do things like saying the Strom model structure is cofibrantly generated in her enriched sense. I am less certain about Chorny's examples. Anyway, for now my main question regarding Riehl's definition is:

(4) Is there an infinity category which admits the required colimits to form a factorization system but which does not admit any model that has a factorization system in the sense of Riehl (either simply an algebraic wfs or an enriched wfs)?

  • $\begingroup$ I should advertise that a related question is: mathoverflow.net/questions/189301/… $\endgroup$ – David White Jun 19 '15 at 22:37
  • $\begingroup$ Dear David, I have a strong suspicion that you and the people you are talking to intend to be asking for more, eg complete and cocomplete infinity categories. Otherwise there are certainly many, many examples. $\endgroup$ – Tyler Lawson Jun 20 '15 at 3:14
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    $\begingroup$ Dear Tyler: yes, I think it would be good to assume that for the purpose of this question. Of course, if it's lacking one can still ask for a model structure in the sense of Hovey's book rather than a model category but I don't feel the need to get into that right now. So I'll edit to make it clear I want things to be bicomplete, and thanks for pointing this out $\endgroup$ – David White Jun 20 '15 at 6:49

If you believe Vopěnka's principle then any cofibrantly generated model category is Quillen equivalent to a combinatorial one and thus its underlying $\infty$-category is presentable. It follows that any non-presentable complete and cocomplete $\infty$-category would give a positive answer to (1). The pro-category of a small finitely complete $\infty$-category is not presentable but it is co-presentable (its opposite category is presentable). However, the pro category of a large cocomplete and finitely complete $\infty$-category is complete and cocomplete but neither presentable nor copresentable. In "Higher Topos Theory" Lurie considers the pro category of spaces (see Definition It thus seems that this category gives a positive answer to (1) and (3). This example was also considered in the world of model categories: It is Isaksen's strict model structure on pro-simplicial sets. If you are interested in a cocombinatorial model category that arose naturally you have Morel's (resp. Quick's) model structure on simplicial pro-finite sets that models the $\infty$-category of p-pro-finite (resp. pro-finite) spaces.

Edit: It turns out that there is a mistake in Raptis's paper showing that under Vopenka's principle any cofibrantly generated model category is Quillen equivalent to a combinatorial one. Raptis and Rosicky posted a fixed proof, in which they had to assume that the domains of the generating (acyclic) cofibrations (in the cofibrantly generated model category) are small with respect to certain types of filtered colimits of cofibrations, more general than just chains of cofibrations. See also the remarks of Tim Campion below.

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    $\begingroup$ There is a mistake in that paper. $\endgroup$ – Ilan Barnea Jul 14 '15 at 12:15
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    $\begingroup$ From going over the proof of $(i) \Rightarrow (ii)$ in Theorem 14 in Simpson's paper I think the problem is that he claims that the domains and codomains of the generating cofibrations are small in the $\infty$-categorical sense. First, the smallness condition is only demanded on the domains of the generating cofibrations in a cofibrantly generated model category (see for example Hovey's book Model Categories). $\endgroup$ – Ilan Barnea Jul 18 '15 at 22:36
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    $\begingroup$ But more importantly I think is that the smallness condition demanded in the definition of a cofibrantly generated model category only speaks about $\kappa$-filtered ordinal colimits while the smallness condition demanded in the definition of a presentable $\infty$-category speaks about all $\kappa$-filtered colimits (this is the same for $\kappa=\omega$ but stronger in the general case). $\endgroup$ – Ilan Barnea Jul 18 '15 at 22:36
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    $\begingroup$ There is also a mistake in Raptis's paper showing that under Vopenka's principle every cofibrantly generated model category is Quillen equivalent to a combinatorial one. The functor $\kappa_\mathcal{S}$ discussed in Prop. 3.1 does not necessarily land in $\mathcal{C}_\mathcal{S}$ as claimed (e.g. take $\mathcal{C}$ to be abelian groups and $\mathcal{S} = \{\mathbb{Z}\}$), and so the category $\mathcal{C}_\mathcal{S}$, which is supposed to carry the combinatorial model structure, need not be cocomplete. $\endgroup$ – Tim Campion Jul 19 '16 at 12:18
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    $\begingroup$ I've spoken to him and apparently he is aware, and has made progress on an alternative approach. $\endgroup$ – Tim Campion Aug 13 '16 at 5:33

Maybe I should record these as answers:

(1) If you add "left proper" to the requirement then the answer is that there are lots of examples. A theorem of Dugger says that any left proper, cofibrantly generated model category admits a set W of cofibrant objects which detect weak equivalences (via the derived mapping space out of them). There are lots of examples of ordinary categories that don't admit a set of generators... take their nerve and get a counterexample. The category of schemes is a nice example. I imagine someone who knows more about these things can get rid of the left-proper assumption.

In general since ordinary category theory embeds into $\infty$-category theory you could probably cook up lots of silly counterexamples using stuff we know about ordinary categories. I mean, if the underlying $\infty$-category of some model category has only discrete mapping spaces, it probably wasn't a very interesting model category... Maybe that even implies the weak equivalences had to be isomorphisms? I don't know.

(2) follows from (1)

(3) Pro-categories are an example, or opposite categories of interesting presentable $\infty$-categories.

I don't know anything about (4).

  • $\begingroup$ Hi Dylan. Thanks for your answer. Can you give a precise reference for (1)? If I ever learned that result before I must have forgotten it. Regarding (3), I like the pro example. Can you think of a specific instance when the opposite of a presentable $\infty$-category was needed? I can't seem to think of a time anyone needed the opposite of a combinatorial model category. $\endgroup$ – David White Jun 20 '15 at 16:17
  • $\begingroup$ Pro-categories are examples of opposites of presentable things... Indeed: something is accessible if and only if it is "Ind" on a small category, so pro guys are precisely the opposites of accessible categories. When the small category you started with has finite (co)limits then you get presentable categories and their opposites for Ind and Pro. $\endgroup$ – Dylan Wilson Jun 20 '15 at 17:19
  • $\begingroup$ And I'll add the reference when I get back- about to board a plane :) $\endgroup$ – Dylan Wilson Jun 20 '15 at 17:20
  • $\begingroup$ I found a place where the dual came up naturally: mathoverflow.net/questions/117267/… $\endgroup$ – David White Jun 23 '15 at 20:43
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    $\begingroup$ A.5 here: hopf.math.purdue.edu/Dugger/smod.pdf $\endgroup$ – Dylan Wilson Jun 29 '15 at 11:19

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