On closed model categories: standard arguments and fibrantly cogenerated categories

Question

Some not very clever questions on closed model categories.

1. For a (left or right) Quillen functor $F:C\to D$ what arguments does one usually use for proving that $Ho F$ is fully faithful when restricted to a full subcategory $C'$ of $Ho C$? I suspect that there should exist some very 'classical' reasonings of this sort.

2. I would like to study the duals of motivic homotopy categories, so my categories are fibrantly cogenerated. Where could I find some information on the properties of such categories (in particular, on the existence of Bousfield localizations)? A related question: which notions of the model category theory are self-dual (even if they do not 'look to be so')?

3. How could one prove that a fibrantly cogenerated model category is a spectral one in the sense of the paper: Schwede, Stefan(D-MUNS-GFM); Shipley, Brooke(1-PURD) Stable model categories are categories of modules. (English summary) Topology 42 (2003), no. 1, 103–153?

Answer 1

1. It may be easier to prove that the map induced on the mapping spaces between fibrant and cofibrant objects is a weak equivalence, provided that $C$ and $D$ are enriched over some closed symmetric monoidal model category and $F$ is continuous. Of course this is not a necessary condition, but it may be considered a "homotopy coherent way" to say that $F$ is fully faithful on the level of the homotopy categories.

2. The definition of a model category is self-dual. So are all the general theorems, therefore there is no need to write a separate exposition of the theory of fibranly generated model categories. But the set-theoretical properties of the underlying category are not dualizable. More specific, if the original category is locally presentable, then its dual is almost never locally presentable (except for complete latices, see Adamek-Rosicky's book "Locally presentable and accessible categories"). In particular $\lambda$-presentable objects in a locally presentable model category are $\lambda$-copresentable in the dual category, but need not be presentable for any cardinal. Dualizing the standard theorems on Bousfield localization you obtain that in a dual of a combinatorial model categories there exist right Bousfield localizations (colocalizations) with respect to any set of maps and left Bousfield localizations with respect to any set of objects.

3. An enrichment over a closed sym. mon. model category is a self-dual concept, the roles of the tensor product and the cotensor product are switched in the opposite category.