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Equivalences between homotopy theories of diagrams. (English) Zbl 0706.55015
Algebraic topology and algebraic K-theory, Proc. Conf., Princeton, NJ (USA), Ann. Math. Stud. 113, 180-205 (1987).
[For the entire collection see Zbl 0694.00022.]
Let \({\mathcal S}\) denote the (simplicial) category of simplicial sets, let \({\mathcal A}\) be a small simplicial category. Then it has been shown by A. K. Bousfield and D. M. Kan [Homotopy limits, completions and localizations, Lect. Notes Math. 304 (1972; Zbl 0259.55004)] that the resulting diagram category \({\mathcal S}^{{\mathcal A}}\) of simplicial functors \({\mathcal A}\to {\mathcal S}\) has a rich homotopy theory in the sense of D. G. Quillen [Homotopical algebra, Lect. Notes Math. 43 (1967; Zbl 0168.209)]. The object of this paper is to characterize those maps between small simplicial categories inducing equivalences of homotopy theories. One of the main results is the following.
Theorem. Let f: \({\mathcal A}\to {\mathcal B}\) be a functor between small simplicial categories. Then the induced functor \(f^*: {\mathcal S}^{{\mathcal B}}\to {\mathcal S}^{{\mathcal A}}\) is an equivalence of homotopy theories if and only if f is a weak r-equivalence, i.e. f satisfies (a) and (b):
(a) for every two objects \(A_ 1,A_ 2\in {\mathcal A}\), the induced map \(\hom (A_ 1,A_ 2)\to \hom (fA_ 1,fA_ 2)\) is a weak equivalence,
(b) every object in the category of components \(\pi_ 0{\mathcal B}\) is a retract of an object in the image of \(\pi_ 0f.\)
Furthermore a relative version of the theorem is proved. Applications are given, amongst other things, to problems of realization of homotopy commutative diagrams and delocalization of small simplicial categories.