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created cellular model category.
I still need a discussion on what exactly one can say about the relation to combinatorial model category. There must be some good statement, but I am not sure yet.
Some kind soul wrote me an email with the following content
The first condition in a cellular model category is that the domains and codomains of the generating cofibrations are compact. My remark is that in my understanding what Hirschhorn calls a compact object (namely Definition 10.8.1 in his book) is very far from the definition given in the link in nLab. The definition of compact object given in nLab is what Hirschhorn calls $\omega$-small object, and I don’t see any relation between the two. This can be very misleading.
Perhaps a related issue, in the entries on combinatorial model categories and on Bousfield localization it is mentioned a few times that the notion of a cellular model category is more general then that of a combinatorial model category. Can you please point me to a reference to this fact. I find it a bit hard to believe since for example the right Bousfield localization theorem (Theorem 5.1.1 in Hirschhorn’s book) holds for cellular model categories while for combinatorial model categories it is an open problem according to what [some expert] told me.
Probably something indeed needs to be fixed here. However, right now I have no chance to look into any of this, let alone make edits. I’ll leave this message here either for me to come back to later, or else for whoever feels motivated to look into this.
Hirschhorn’s definition of “compact” is what we might call small, i.e. $\kappa$-compact for large-enough regular cardinals $\kappa$.
$\mathbf{Top}$ (in Hirschhorn’s sense, so e.g. the category of compactly generated weakly Hausdorff spaces) is a cellular model category that is not combinatorial. I suppose that was the motivating example. This example is already on the page.
Thanks. I am quasi-offline. If you wanted to do me a favor (me and all other nLab users, that is) you could briefly add whatever comment/link the page might need for clarification. Thanks!
Hmmm. Actually, on further reflection, I’m not so confident that “compact relative to $I$” is equivalent to “small relative to $I$”. The forward implication appears as Proposition 10.8.7, but I cannot find the reverse implication. It should at least be true in the case where $I$ is the class of all morphisms (which we usually get for free in a combinatorial model category), but there is still a non-trivial check involved.
I added a warning about the meaning of “compact”.
I guess you are taking for granted that the cofibrations are effective monomorphisms; otherwise there are trivial counterexamples.
It is well known that every object in a locally presentable category is small (relative to the class of all morphisms). It should also be true that every object is compact, at least under the following hypothesis: for any large-enough regular cardinal $\kappa$ and any presented relative cell complex, the poset of subcomplexes of size $\lt \kappa$ is $\kappa$-filtered. But perhaps that hypothesis is not always satisfied.
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