Simplicial functors and homotopy coherence

Abstract

Suppose that A is a simplicial category. The main objects of study in this chapter are the functors X: A → S taking values in simplicial sets, and which respect the simplicial structure of A. In applications, the simplicial category A is typically a resolution of a category I, and the simplicial functor X describes a homotopy coherent diagram. The main result of this chapter (due to Dwyer and Kan, Theorem 2.13 below) is a generalization of the assertion that simplicial functors of the form X: A → S are equivalent to diagrams of the form I → S in the case where A is a resolution of the category I. The proof of this theorem uses simplicial model structures for categories of simplicial functors, given in Section 1, and then the result itself is proved in Section 2.