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Morita homotopy theory for (\(\infty\),1)-categories and \(\infty\)-operads. (English) Zbl 1422.55039

Morita model structures are model structures, in the sense of Quillen, where the weak equivalences are the Morita equivalences, a notion suitably defined for the category of algebraic objects under consideration. For instance, a map of simplicial categories is a Morita weak equivalence if it is homotopically fully faithful and homotopically essentially surjective up to retracts.
The main object of this paper is to prove the existence of Morita model structures on the categories of simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of \((\infty,1)\)-categories and \(\infty\)-operads. In the cases of simplicial categories and simplicial operads, the authors show that these Morita model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories [J. E. Bergner, Trans. Am. Math. Soc. 359, No. 5, 2043–2058 (2007; Zbl 1114.18006)] and the Cisinski-Moerdijk model structure on simplicial operads [D.-C. Cisinski and I. Moerdijk, J. Topol. 6, No. 3, 705–756 (2013; Zbl 1291.55005)], respectively.

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18D50 Operads (MSC2010)
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References:

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