Function complexes for diagrams of simplicial sets

Abstract

Let S be the category of simplicial sets, let D be a small category and let S D denote the category of D-diagrams of simplicial sets. Then S D admits a closed simplicial model category structure and the aim of this note is to show that, for every cofibrant diagram X ϵ S D and every fibrant diagram Y ϵS D, the homotopy type of the function complex hom( X, Y) can be computed as a homotopy inverse limit involving function complexes in S between the simplicial sets that appear in X and Y.