# Do we implicitly consider model categories to be locally small?

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 ?

Thanks

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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

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