A Model 2-Category of Enriched Combinatorial Premodel Categories

Abstract

Quillen equivalences induce equivalences of homotopy theories and therefore form a natural choice for the "weak equivalences" between model categories. In [21], Hovey asked whether the 2-category Mod of model categories has a "model 2-category structure" with these weak equivalences. We give an example showing that Mod does not have pullbacks, so cannot be a model 2-category. We can try to repair this lack of limits by generalizing the notion of model category. The lack of limits in Mod is due to the two-out-of-three axiom, so we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a 2-category CPM with excellent algebraic properties: CPM has all limits and colimits and is equipped with a tensor product (representing Quillen bifunctors) which is adjoint to an internal Hom. The homotopy theory of a model category depends in an essential way on the weak equivalences, so it does not extend directly to premodel categories. We build a substitute homotopy theory under an additional axiom on the premodel category, which holds automatically for a premodel category enriched in a monoidal model category V. The 2-category of combinatorial V-premodel categories VCPM is simply the 2-category of modules over the monoid object V, so VCPM inherits the algebraic structure of CPM. We construct a model 2-category structure on VCPM for V a tractable symmetric monoidal model category, by adapting Szumiło's construction of a fibration category of cofibration categories [35]. For set-theoretic reasons, constructing factorizations for this model 2-category structure requires a technical variant of the small object argument which relies on an analysis of the rank of combinatoriality of a premodel category.