Operads, Algebras and Modules in General Model Categories

By Markus Spitzweck. 2001

98 citations recorded by the source corpus.

In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. In a second part we develop the thoery of S-modules of [EKMM]., which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fe33a6e0304852d18207a6c9873d53735cf2307d

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Citations in the corpus

THE BOARDMAN-VOGT RESOLUTION OF OPERADS IN MONOIDAL MODEL CATEGORIES (2005, unknown): “E-m ail: cberger@ m ath.unice.fr”
Uniqueness of the multiplicative cyclotomic trace (2011, result): “…maps (for each active map (cid:2) Since this map clearly preserves weak equivalences, it factors as a map Since O is coﬁbrant, the results of [48, §4] produce a J -semi model category structure on the category Alg O ( C ) The proof of [36, Theorem 1.3.4.20] goes through in this context to…”
Equivalences of monoidal model categories (2002, methodology): “agraph of [SS00, 6.2] show that jis a pointwise trivial coﬁbration, and hence a (pointwise) weak equivalence of CI-categories. Remark 6.4. We actually believe that CI-Catis a semi-model category; see [Sp] or [Hov98, 3.3] for a deﬁnition. The above statements are the only parts of this structure that”
Homotopy theory of modules over operads in symmetric spectra (2007, methodology): “The reader may contrast this with a path object approach explored in [2], which amounts to verifying the conditions of [38, Lemma 2.3(2)]; compare also [17, 41 ].”
RESOLUTION OF COLOURED OPERADS AND RECTIFICATION OF HOMOTOPY ALGEBRAS (2005, methodology): “erent theory of such resolutions is provided by a Quillen homotopy theory ofoperads, as established by Hinich [16] for operadsin chain complexes, by Rezk [26] for simplicial operads, and by Spitzweck [29] and the authors [3] for operads in general monoidal model categories. When such a homotopy theo”
Operads and PROPs (2006, background): “ras and dg Lie algebras [91, 103]. Operads, in a reasonable monoidal model category, themselves form a model category [7, 28] such that algebras over coﬁbrant operads are homotopy invariant, see also [100]. Minimal operads mentioned in the OPERADS AND PROPS 3 previous item are particular cases of co”