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Do we implicitly consider model categories to be locally small?

I have the impression (but am not sure) that many references on model categories assume that all the categories are locally small, but not all of them redefines what a category is, and there is nowhere a mention of the categories being locally small. For example in Hovey's or Hirschhorn's books, that I find excellent books by the way.

Dwyer, Hirschhorn and Kan in Model Categories and More General Abstract Homotopy Theory, fix a universe $U$ and allow the objects to be classes but the hom-sets to be $U$-sets.

I think the locally small condition is necessary in the construction of the homotopy category. Almost all references (the smaller ones too) say something like "the hom-sets in the homotopy category are the quotient sets $\mathcal{M}(RQ X, RQ Y) / \sim$".

Of course, most of the model categories $\mathcal{Top}, \mathcal{sSet}$, chain complexes, simplicial presheaves on small sites, etc are locally small. So my question is

Is it a standard convention to assume that model structure are only on locally small categories ?


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Why is this tagged "logic"? The Wikipedia article on model categories does not state that they have anything in particular to do with the logical concept of a model. –  Henning Makholm Jun 13 '12 at 21:04
Maybe it is tagged wrong, the tag logic was here for the set/class concepts of logic. Yeah I'll change that to category theory or homotopy theory. –  Bogdan Jun 13 '12 at 21:22
The title is meant to be an elevator pitch line; not the contain the actual question (or at least not to hold it instead of the body). I also removed the [set-theory] question, because while it is a set theoretical question in its essence, the answer is not really about set theory, but rather about the foundations of the theory of model categories. –  Asaf Karagila Oct 22 '12 at 23:32

1 Answer 1

Have a look on Joyal's page on model categories.

In particular Corollary 1 shows that if the model category $E$ is locally small than the associated homotopy category is also locally small.

I guess, as you say, in practice this is not of much concern since the catgeories we are usually concerned with are locally small

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Thanks. Yeah, that's just because a set quotiented by an equivalence relation is again a set. But what about having an equivalence relation on a proper class ? Can we as easily take the quotient by looking at the partition (which could be a proper class) of equivalence classes ? Seems right –  Bogdan Jun 29 '12 at 9:39
Hmmm, I'm not sure! Here is a question - where is the motivation for this coming from? Is there a particular category you are interested in? –  Juan S Jun 30 '12 at 2:21
There was no particular example. In fact, I was writing my master thesis I would have preferred to just assume that all categories are locally small (all the categories in the project are), and wanted to know if the other references do as well. They seemed to do it because most of them say "quotient set", but maybe set means set in a bigger universe and thus maybe class... –  Bogdan Jun 30 '12 at 9:32

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