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
- Domains of maps in J are small relative to cofibrations
- Cofibrations are effective monomorphisms, i.e. any cofibration $f:X\to Y$ is the equalizer of the two obvious maps $Y\to Y \coprod_X Y$
- 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)?