Simplicial structures on model categories and functors
with Charles Rezk and Stefan Schwede
A simplicial structure on a model category provides higher order structure
such as composable mapping spaces and simplifies the definition of homotopy
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.