Realizing diagrams in the homotopy category by means of diagrams of simplicial sets

Abstract:

Given a small category ${\mathbf {D}}$, we show that a ${\mathbf {D}}$-diagram $\bar X$ in the homotopy category can be realized by a ${\mathbf {D}}$-diagram of simplicial sets iff a certain simplicial set $r\bar X$ is nonempty. Moreover, this simplicial set $r\bar X$ can be expressed as the homotopy inverse limit of simplicial sets whose homtopy types are quite well understood. There is also an associated obstruction theory. In the special case that ${\mathbf {D}}$ is a group (i.e. ${\mathbf {D}}$ has only one object and all its maps are invertible) these results reduce to the ones of G. Cooke.