http://www.ncatlab.org/nlab/show/homotopy+coherent+diagram
http://mathoverflow.net/questions/81721/a-homotopy-commutative-diagram-that-cannot-be-strictified
From Goerss-Jardine intro to chapter VIII: “The Dwyer-Kan theorem immediately leads to realization theorems for ho- motopy coherent diagrams in cases where the simplicial categories A model ho- motopy coherence phenomena. A realization of a homotopy coherent diagram X:A→S is a functor Y :I→S which is weakly equivalent to X in a strong sense. Insofar as the information arising from X typically consists of simplicial set maps X(α) : X(i) → X(j), one for each morphism α of I which only respect the composition laws of I up to some system of higher homotopies, a realization Y is a replacement of X, up to weak equivalence, by a collection of maps Y (α) : Y (i) → Y (j) which define a functor on the nose. Diagrams of spaces which are not quite functorial are really very common: the machines which produce the algebraic K-theory spaces, for example, are not functors (on scheme categories in particular), but they have homotopy coherent output for categorical reasons [49]. Approaches to homotopy coherence for simplicial set diagrams arising from some specific resolution constructions are discussed in Section 3; traditional ho- motopy coherence (in the sense of [90]) is one of the examples. But more generally, we take the point of view that a homotopy coherent dia- gram on a fixed index category I is a simplicial functor X : A → S defined on any simplicial resolution A of I. We can further ask for realization results concerning homotopy coherent diagrams A → M taking values in more general simplicial model categories M. This is the subject of Section 4. We derive, in particular, realization theorems for homotopy coherent diagrams taking values in pointed simplicial sets, spectra and simplicial abelian groups (aka. chain complexes).”
nLab page on Homotopy coherence