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Homotopy theories of unital algebras and operads

Abstract : This thesis deals with the homotopical properties of algebras over an operad, of operads themselves andof colored operads, in the framework of chain complexes. We introduce a new bar-cobar adjunctionbetween unital operads and curved conilpotent cooperads. This allows us to endow the latter with aDépôt de thèseDonnées complémentairesmodel structure induced by the projective model structure on operads along this adjunction, which thenbecomes a Quillen-equivalence. This result allows us to study the homotopy theory of operads in theworld of cooperads which is more powerful: for instance, fibrant-cofibrant objects can be described interms of operads up to homotopy. We then apply the same strategy to algebras over an operad. Morespecifically, we endow the category of coalgebras over the Koszul dual cooperad with a model structureinduced by that of the category of algebras along their bar-cobar adjunction, which becomes a Quillenequivalence.This allows us to describe explicitly for the first time some homotopy properties of algebrasover a not necessarily augmented operad. In the last part, we introduce the notion of homotopy coloredoperad that we compare to Moerdijk--Weiss' infinity-operads by means of a functor: the dendroidalnerve. We show that it extends existing constructions due to Lurie and Faonte and we study itshomotopical properties. In particular, we show that its restriction to colored operads is a right Quillenfunctor. All this allows us to connect explicitly two different worlds of higher operads
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Brice Le Grignou. Homotopy theories of unital algebras and operads. General Mathematics [math.GM]. Université Côte d'Azur, 2016. English. ⟨NNT : 2016AZUR4058⟩. ⟨tel-01375927v2⟩



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