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Every homotopy theory of simplicial algebras admits a proper model. (English) Zbl 0994.18008
To axiomatize the notion of a “homotopy theory” D. G. Quillen introduced closed model categories [“Homotopical Algebra”, Lect. Notes Math. 43 (1967; Zbl 0168.20903)], and produced a number of examples of such, one class of which are categories of simplicial algebras. In this paper is examined the case of simplicial algebras, i.e., simplicial objects in a category of algebras associated to an algebraic theory in the sense of F. W. Lawvere [“Functorial semantics of algebraic theories”, Proc. Natl. Acad. Sci. USA 50, 869-872 (1963; Zbl 0119.25901)], and more generally the case of simplicial algebras over a multi-sorted, simplicial theory. This class of examples includes simplicial groups, rings etc. The author shows that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category.
Theorem B. Let \(T\) be a (possibly simplicial, possibly multy-sorted) theory, and let \(T\)-alg be the corresponding category of simplicial \(T\)-algebras, equipped with a simplicial model category structure in which a map is a weak equivalence of fibrations if it is a weak equivalence or fibration of the underlying simplicial sets.
Then there exists a morphism \(S\to T\) of simplicial theories such that: (1) the induced adjoint pair \(S\)-alg\(\leftrightarrows T\)-alg is a Quillen equivalence of model categories, and
(2) \(S\)-alg is a proper simplicial closed model category.
A corollary of this theorem is:
Theorem A. The homotopy theory of a category of simplicial algebras always admits a proper model.
The author remarks that “Whether any reasonable homotopy theory (e.g., one associated to a model category) admits a proper model is an open question”.
Theorem C. Given the hypotheses of Theorem B, suppose that in addition \(T\)-alg is a pointed category. Then \(S\) can be chosen as in Theorem B so that \(S\)-alg is also a pointed category.
Theorem D. Given the hypotheses of Theorem B (respectively of Theorem C), the theory \(S\) can be chosen as in Theorem B (or Theorem C) so that \(S\)-alg is a cellular model category in the sense of P. Hirschhorn [“Model categories and their localizations”, Preprint, http://www-math.mit.edu/~psh].
Corollary. For any set of maps in \(S\)-alg there is a localization model category structure with respect to this set.
Reviewer: Ioan Pop (Iaşi)

18G55 Nonabelian homotopical algebra (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18C10 Theories (e.g., algebraic theories), structure, and semantics
Full Text: DOI
[1] B. Badzioch, Algebraic theories in homotopy theory, Preprint, 2000 · Zbl 1028.18001
[2] Bousfield, A.K.; Friedlander, E.M., Homotopy theory of γ-spaces, spectra, and bisimplicial sets, (), 80-130 · Zbl 0405.55021
[3] Boardman, J.M., Homotopy structures and the language of trees, (), 37-58 · Zbl 0242.55012
[4] Borceux, F., Categories and structures, ()
[5] Bousfield, A.K., The localization of spaces with respect to homology, Topology, 14, 133-150, (1975) · Zbl 0309.55013
[6] Boardman, J.M.; Vogt, R.M., Homotopy invariant algebraic structures on topological spaces, Lecture notes in math., 347, (1973), Springer Berlin · Zbl 0285.55012
[7] W.G. Dwyer, P. Hirschhorn, D.M. Kan, General abstract homotopy theory, in preparation
[8] Dwyer, W.G.; Kan, D.M., Simplicial localizations of categories, J. pure appl. algebra, 17, 3, 267-284, (1980) · Zbl 0485.18012
[9] P.G. Goerss, M.J. Hopkins, Resolutions in model categories, Preprint
[10] P.G. Goerss, M.J. Hopkins, Andre-Quillen (co-)homology for simplicial algebras over simplicial operads, Preprint · Zbl 0999.18009
[11] Getzler, E.; Jones, J.D.S., Operads, homotopy algebra, and iterated integrals for double loop spaces, Preprint
[12] Goerss, P.G.; Jardine, J.F., Simplicial homotopy theory, (1999), Birkhäuser Basel · Zbl 0949.55001
[13] P. Hirschhorn, Localization in model categories, Preprint, http://www-math.mit.edu/ psh
[14] Hovey, M., Model categories, (1999), American Mathematical Society Providence, RI · Zbl 0909.55001
[15] Kan, D.M., On c.s.s. complexes, Amer. J. math., 79, 449-476, (1957) · Zbl 0078.36901
[16] Lawvere, F.W., Functorial semantics of algebraic theories, Proc. nat. acad. sci. USA, 50, 869-872, (1963) · Zbl 0119.25901
[17] Quillen, D.G., Homotopical algebra, Lecture notes in math., 43, (1967), Springer Berlin · Zbl 0168.20903
[18] Quillen, D.G., Rational homotopy theory, Ann. of math., 90, 65-87, (1969)
[19] C. Rezk, Spaces of algebra structures and cohomology of operads, Ph.D. dissertation, MIT, Cambridge, MA, 1996
[20] S. Schwede, Stable homotopy of algebraic theories, Preprint · Zbl 0897.55008
[21] J. Smith, Combinatorial model categories, in preparation
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