A model 2-category of enriched combinatorial premodel categories

Abstract

In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have pullbacks and so cannot form a model 2-category. This lack of pullbacks can be traced to the two-out-of-three axiom on the weak equivalences of a model category. Accordingly, we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a complete and cocomplete closed symmetric monoidal 2-category $\mathbf{CPM}$ whose tensor product represents Quillen bifunctors. For a monoidal combinatorial premodel category $V$, the 2-category $V\mathbf{CPM}$ of $V$-enriched combinatorial premodel categories is simply the category of modules over $V$ (viewed as a monoid object of $\mathbf{CPM}$), and therefore inherits the algebraic structure of $\mathbf{CPM}$. The homotopy theory of a model category depends in an essential way on the weak equivalences, so it does not extend directly to a general premodel category. We develop a substitute homotopy theory for premodel categories satisfying an additional property which holds automatically for model categories and also for premodel categories enriched in a monoidal model category. In particular, for a monoidal model category $V$, we obtain a notion of Quillen equivalence of $V$-premodel categories which extends the one for $V$-model categories. When $V$ is a tractable symmetric monoidal model category, we construct a model 2-category structure on $V\mathbf{CPM}$ with these Quillen equivalences as the weak equivalences, by adapting Szumi\l{}o's construction of a fibration category of cofibration categories.