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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?

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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.

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  • $\begingroup$ Is there a reason why small objects are so "much more" common than cosmall objects? $\endgroup$
    – NDewolf
    Jul 21, 2020 at 14:18
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    $\begingroup$ @NDewolf It's essentially because this is what happens in sets and most categories of interest are eventually built on sets. A $\kappa$-cosmall set $S$ is one such that, in particular, any function $\prod_{i\in I} A_i \to S$ depends on only $\kappa$ coordinates in the domain, which never happens for $|S|>1$. $\endgroup$ Jul 21, 2020 at 18:03

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