The category of bisimplicial presheaves carries a model structure for which the weak equivalences are defined by the diagonal functor and the cofibrations are monomorphisms. This model structure has the most cofibrations of a large family of model structures with weak equivalences defined by the diagonal. The diagonal structure for bisimplicial presheaves specializes to a diagonal model structure for bisimplicial sets, for which the fibrations are the Kan fibrations.
Keywords: bisimplicial presheaves, diagonal functor, model structure
2010 MSC: Primary 18G30; Secondary 18F20, 55U35
Theory and Applications of Categories, Vol. 28, 2013, No. 10, pp 250-268.