  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.
