On a fat small object argument

Jiri Rosicky
Friday, July 26, 2013 - from 14:50 to 15:30
Consider a finitely combinatorial model category $\mathcal K$; it means that $\mathcal K$ is locally finitely presentable and cofibrations and trivial cofibrations are cofibrantly generated by morphisms between finitely presentable objects. Then the usual small object argument yields that any cofibrant object is an $\omega_1$-filtered colimit of $\omega_1$-presentable objects. The reason is that cofibrant replacements of finitely presentable objects are countably presentable. But this can be improved if we replace transfinite compositions, which are thin and long, by fat and short compositions which are good colimits in the sense of Lurie. In this way we get that any cofibrant object is a filtered colimit of finitely presentable cofibrant objects. As a consequence we get the result of Joyal and Wraith that any acyclic simplicial set is a filtered colimit of finitely presentable acyclic simplical sets. But, working more, we can even show that any weak equivalence of simplicial sets is a filtered colimit of weak equivalences which are finitely presentable in the category of morphisms of simplicial sets. Hence the category of weak equivalences of simplicial sets is finitely accessible. Again, the usual argument only gives that it is $\omega_1$-accessible. This can be extended to finitely combinatorial model categories $Set^\mathcal C$ where the class of cofibrations is the class of monomorphisms. The fat small object argument can be extended to general combinatorial model categories. Good colimits can be used for proving that combinatorial categories, i.e., locally presentable categories equipped with a class of morphisms cofibrantly generated by a set of morphisms, are closed under appropriate 2-limits, in particular under pseudopullbacks. But we do not know whether combinatorial categories (with left Quillen functors) are closed under pseudopullbacks. Joint work with M. Makkai, G. Raptis and L. Vok\v r\'\i{}nek.