Diagonal model structures

The original purpose of this note was to display a model structure for the category sSet of bisimplicial sets whose cofibrations are the monomorphisms and whose weak equivalences are the diagonal weak equivalences, and then show that it is cofibrantly generated in a very precise way. The project grew to include analogous model structures on categories of bisimplicial presheaves. These model structures are the diagonal model structures of the title. The fibrations for the diagonal model structure on bisimplicial sets are the Kan fibrations, which are defined by a lifting property with respect to the bisimplicial analogues of inclusions of horns in simplices. A horn can be viewed as the part of boundary ∂∆ of a bisimplex ∆ that one gets by removing a single cell of maximal total degree — the inclusions of all such horns are simple examples of anodyne extensions of bisimplicial sets. It is relatively painless to show that the diagonal model structures exist for all categories s Pre(C) of bisimplicial presheaves — this result is Theorem 4. It is also easy to show that the diagonal functor and its left adjoint d∗ define a Quillen equivalence d∗ : sPre(C) s Pre(C) : d