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
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