Simplicial structures on model categories and functors

with Charles Rezk and Stefan Schwede

ABSTRACT. A simplicial structure on a model category provides higher order structure such as composable mapping spaces and simplifies the definition of homotopy invariant colimits. We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer-Kan and answers a question of Hovey; that is, we show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. Another application shows that a simplicial model category structure is unique up to simplicial Quillen equivalence for any fixed underlying model category.