An important concept in the study of model categories is that of "cofibrantly generated model categories". These are nice because all morphisms can be obtained from a small subset of them and in general these specific categories are often easier to work with.
Now I was wondering why I can find almost nothing about "fibrantly generated model categories". Since duality is everything in category theory I expected to find at least something about them.
I probably don't know enough about these structures to realise some obvious stuff, but I do find it curious. Is there a solid reason for their non-existence?
Best Answer
The most important cofibrantly generated model categories are the combinatorial ones, which are also locally presentable; by Dugger’s theorem these are all Quillen equivalent to left Bousfield localizations of the projective model structure on some category of simplicial presheaves. Such categories are a lot like the classical model category of simplicial sets: you build things out of cell complexes, most notably.
Certainly the opposite of a cofibrantly generated model category is fibrantly generated, but this corresponds roughly to taking presheaves valued in the opposite category of simplicial sets, which is not as important. This is a homotopy theory analogue of the fact that locally presentable categories are more practically important than their opposites, though the notions are formally equivalent by duality. A particular problem with getting fibrant generation is the shortage of cosmall objects in most “natural” categories.