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Sur les A-infini-catégories

Abstract : We study (not necessarily connected) Z-graded A-infinity-algebras and their A-infinity-modules. Using the cobar and the bar construction and Quillen's homotopical algebra, we describe the localisation of the category of A-infinity-algebras with respect to A-infinity-quasi-isomorphisms. We then adapt these methods to describe the derived category DA of an augmented A-infinity-algebra A. The case where A is not endowed with an augmentation is treated differently. Nevertheless, when A is strictly unital, its derived category can be described in the same way as in the augmented case. Next, we compare two different notions of A-infinity-unitarity : strict unitarity and homological unitarity. We show that, up to homotopy, there is no difference between these two notions. We then establish a formalism which allows us to view A-infinity-categories as A-infinity-algebras in suitable monoidal categories. We generalize the fundamental constructions of category theory to this setting : Yoneda embeddings, categories of functors, equivalences of categories... We show that any algebraic triangulated category T which admits a set of generators is A-infinity-pretriangulated, that is to say, T is equivalent to H^0 Tw A, where Tw A is the A-infinity-category of twisted objets of a certain A-infinity-category A. Thus we give detailed proofs of a part of the results on homological algebra which M. Kontsevich stated in his course ``Triangulated categories and geometry'' (ENS Ulm, Paris, 1998).
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Contributor : Kenji Lefèvre-Hasegawa <>
Submitted on : Wednesday, December 15, 2004 - 12:40:08 PM
Last modification on : Friday, April 10, 2020 - 5:24:22 PM
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  • HAL Id : tel-00007761, version 1


Kenji Lefèvre-Hasegawa. Sur les A-infini-catégories. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2003. Français. ⟨tel-00007761⟩



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