The monoidal bicategory of (pre-)model categories
Quillen's model categories provide a setting for abstract homotopy theory. The totality of all Quillen model categories constitute the objects of a nice 2-category, wherein morphisms are left Quillen functors. The study of the structure of this 2-category might fancifully be called "the model theory of model categories".
In this talk, we explain how the 2-category of combinatorial model categories is a symmetric monoidal closed 2-category under the tensor product which classified left Quillen bifunctors. (Though actually this is not quite true; I have to work with a weaker notion of "combinatorial pre-model category"; all this will be explained). This allows us to recover all sorts of familiar constructions with zero further effort: injective and projective model structures, monoidal model structures and enrichment, the fact that various weighted limit and colimit functors are right or left Quillen, and so on. Most intriguingly, we get a nice characterisation of Reedy categories: they are non-trivial examples of dualisable (= nuclear) objects.
Dr Richard Garner, Macquarie University