Every homotopy theory of simplicial algebras admits a proper model.

*(English)*Zbl 0994.18008To 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.

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)

##### MSC:

18G55 | Nonabelian homotopical algebra (MSC2010) |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

18C10 | Theories (e.g., algebraic theories), structure, and semantics |

##### Keywords:

closed model categories; categories of simplicial algebras; algebraic theory; simplicial model category structure; localization model category
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DOI

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