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Singular functors and realization functors. (English) Zbl 0555.55019
D. G. Quillen [Homotopical Algebra, Lect. Notes Math. 43 (1967; Zbl 0168.209)] showed that the singular functor and the realization functor have certain properties which imply the equivalence of the weak homotopy theory of topological spaces with the homotopy theory of simplicial sets. In this note the authors generalize this result to show that one can, in essentially the same manner, establish the equivalence of other homotopy theories (e.g. equivariant homotopy theories) with the homotopy theories of simplicial diagrams of simplicial sets.
Let M be a simplicial category which satisfies axioms MO and SMO of Quillen. Let \(\{O_ e)_{e\in E}\) be a set of objects of M satisfying four direct limit axioms which permit among other things the use of Quillen’s small object argument; such a set is called a ”set of orbits” for M. With these assumptions the authors prove that M admits a closed simplicial model category structure in which the simplicial structure is the given one and in which a map \(X\to Y\) in M is a weak equivalence or a fibration iff for every \(e\in E\) the induced map of function complexes \(\hom (O_ e,X)\to \hom (O_ e,Y)\) is a weak equivalence or fibration of simplicial sets. They then prove that the homotopy theory of M is equivalent to the homotopy theory of \(S^ C\) of C-diagrams of simplicial sets (for a suitable choice of small simplicial category C) in the sense that these two categories have the same simplicial homotopy categories.