## Homotopy theory of modules over operads in symmetric spectra (2009)

Citations: | 4 - 2 self |

### BibTeX

@MISC{Harper09homotopytheory,

author = {John E. Harper},

title = {Homotopy theory of modules over operads in symmetric spectra},

year = {2009}

}

### OpenURL

### Abstract

We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.

### Citations

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Citation Context ...do homotopy theory, and in particular, provide a framework for constructing and calculating derived functors. A useful introduction to model categories is given in [4]; see also the original articles =-=[33, 34]-=- and the more recent [15, 19, 20]. When we refer to the extra structure of a monoidal model category, we are using [38, Definition 3.1]; an additional condition involving the unit is assumed in [25, D... |

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Citation Context ...c spectra. Remark 1.2. For ease of notation purposes, we have followed Schwede [37] in using the term flat (e.g., flat stable model structure) for what is called S (e.g., stable S-model structure) in =-=[21, 36, 39]-=-. The theorem remains true when the positive flat stable model structure on symmetric spectra is replaced by the positive stable model structure. This follows immediately from the proof of Theorem 1.1... |

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Citation Context ...e inherited in an appropriate sense from the stable weak equivalences and the positive flat stable fibrations in symmetric spectra. Remark 1.2. For ease of notation purposes, we have followed Schwede =-=[37]-=- in using the term flat (e.g., flat stable model structure) for what is called S (e.g., stable S-model structure) in [21, 36, 39]. The theorem remains true when the positive flat stable model structur... |

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Citation Context ... and Shipley [38] is that the category of monoids in symmetric spectra has a natural model structure inherited from the (flat) stable model structure on symmetric spectra. This result was improved in =-=[16]-=- to algebras and left modules over any non-Σ operad O in symmetric spectra. One of the theorems of Shipley [39] (resp. Mandell, May, Schwede, and Shipley [28]) is that the category of commutative mono... |

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