fibrant generation of $sSet_{Quillen}$?

Question

I wonder if someone has proved that $sSet_{Quillen}$ is not a fibrantly generated model structure ? Do we know something about the possible fibrant generation of $sSet_{Quillen}$ ?

Thanks

Answer 1

sSet is not fibrantly generated. I originally thought the issue would be with the lack of cosmall objects, but it's even worse than that. In fact, there is no set of maps in sSet that can detect the acyclic cofibrations via lifting. This fact is due to Bill Dwyer, and I learned it from Remark 5.7 in this excellent paper by Dan Isaksen: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.235.7756&rep=rep1&type=pdf

It's worth remarking that other than Pro categories it's very hard for a model category to be fibrantly generated because there are often very few cosmall objects (in order to be fibrantly generated, the domains and codomains of the generating fibrations would have to be cosmall). On page 34 of Hovey's book, right after Proposition 2.1.18, it is proven that the only cosmall objects in Set are the empty set and the one point set, so this should give you a good idea of why fibrant generation fails.

Last year Hess and Shipley wrote a paper about coalgebras over a comonad and they needed a result about right-to-left transfer of model structure (i.e. along a right adjoint rather than a left adjoint). They invented the notion of a Postnikov Presentation to get around the lack of fibrant generation. This requires less but is still sufficient to build a model structure and it does transfer nicely. So I recommend reading that paper and seeing if their results are enough for whatever application you have in mind.