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Browsing the model category literature, I’ve found a proliferation of slightly different definitions of a model category, some requiring finite limits/colimits, some requiring small limits/colimits (or U-small limits/colimits), some requiring that the factorization of morphisms be functorial (presumably by analogy with the mapping cylinder).
How do these lead to different theories of model categories, and are the differences significant?
Not very. I’ve only ever heard of a couple examples where the factorizations can’t be made functorial, and they aren’t exactly commonly used examples. Some things are easier to define when things are functorial, but usually you can reach the same conclusions in a nonfunctorial case with a bit more work. But hardly anyone bothers since almost everything is functorial in practice.
Whether you can make the factorizations into enriched functors in an enriched model category is a much deeper and harder question, but this doesn’t ever affect the definition of model category.
The basic theory of model categories requires only finite limits and colimits. I think the main reason people ask for arbitrary small (limits and) colimits is to be able to apply the small object argument, which is important for the construction of model categories, but not for the basic theory once you have a model category. That and the fact that most model categories people study in practice are complete and cocomplete.
On an unrelated note, quick question: Is there any significance to showing that a U-small model category has a U-small localization at weak equivalences if we accept the axiom of universes?
If I understand you correctly, then any U-small category with weak equivalences has a U-small localization, so no, there’s no significance to that. If you mean locally U-small, then there is the same significance as in the usual proof that a model category has a locally small homotopy category.
Yes, I meant locally U-small. I guess the question about significance is that without universes, it’s not clear that the localization even exists at all (significant!). I guess I don’t see why the size will even be important if we have proven existence.
The whole point of model cats is that it allows the construction, for large model categories, of the localisation as a locally small category, that is, the proof that a locally small localisation exists. Knowing that Ho(sSet) = Ho(Top) = Ho(CW) is incredibly useful, where the latter has a nice presentation by a (2,1)-category, allowing for CW approximations and so on. Just knowing the existence of some localisation is a little bit trickier to work with.
If you want to think about localisations a bit, look up Dorette Pronk’s 1996 article in JPAA. She constructs a localisation of a 2-category of internal groupoids, but one can’t tell that it is locally essentially small - it’s so complicated. However, the 2-category of anafunctors is another (equivalent) localisation, and this is easy, using WISC, to prove locally essentially small.
The localization has a universal property. Since all categories are U-small for some U (when using universes, we require that Ob and Arr are honest sets). Using universes, the existence of the localization is clear (this follows pretty readily from the axiom of universes). Since absolute size isn’t an obstruction to existence, do we need to worry about relative size in the case of model categories? That is, if we just characterize the localization by its universal property, then prove existence, do we actually need to know anything about it?
Whether or not a category is locally small is a serious question, whether or not you have universes. In particular, most mathematics happens in the world of U-large but U-locally-small categories that have U-small limits and colimits, but not U-large ones. The matching of size between homsets and limits is the important thing; for instance this is essential in the adjoint functor theorem. Freyd’s theorem that a U-small category with U-small limits is a poset shows that you can’t work all at the same level all the time.
Also, even without universes I think there is no problem constructing a localization as a “locally large” category, if you’re willing to play some set-theoretic tricks.
There we go, that’s the answer I was looking for!
A general comment on the “proliferation of definitions of model categories”:
to some extent it is no wonder that one can and should consider model cats with different properties (for instance with finite vs all limits, or different sizes) as there are also (oo,1)-categories with these different properties differing. The limits in a model cat give rise to the existence of homotopy limits wich model (oo,1)-limits, and accordingly it makes sense to have these or those assumptions on them.
Maybe more interesting is the opposite question: even in view of a moderate “proliferation of definitions”, the definition of a model category is remarkably constrained, with all the pieces depending delicately on each other. So another interesting question would be: which kinds of (oo,1)-categories precisely may be modeled by model categories? A precise answer to that is currently available for combinatorial model categories. I am not aware of more general statements.
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