Realizing diagrams in the homotopy category by means of diagrams of simplicial sets. (English) Zbl 0514.55020

Dwyer, W. G.; Kan, D. M.

Proc. Am. Math. Soc. 91, 456-460 (1984).

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

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MSC:

55U35	Abstract and axiomatic homotopy theory in algebraic topology
55U10	Simplicial sets and complexes in algebraic topology
55S99	Operations and obstructions in algebraic topology

Keywords:

realizing diagrams in the homotopy category; diagrams of simplicial sets; homotopy inverse limit of simplicial sets; obstruction theory

References:

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