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-infintity-quasi-isomorphisms. We then adapt these methods to describe the derived category 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-infini-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 $H^{0}twA$, where $twA$ is the A-infinity-category of twisted objets of a certain A-infinity-category A.

230 pages, 11 figures, ph. D. thesis in french

Cited by in corpus (30)

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• Deformation theory of objects in homotopy and derived categories II: pro-representability of the deformation functor
• Deformation theory of objects in homotopy and derived categories III: abelian categories
• DG-algebras and derived A-infinity algebras
• Simplicial nerve of an A-infinity category
• Minimal models for monomial algebras
• Sheaves on local Calabi-Yau varieties
• Generating the superpotential on a D-brane category: I
• The uniqueness problem of dg-lifts and Fourier-Mukai kernels
• Categorical Koszul duality
• ${\mathbb{P}}^n$-functors
• The derived contraction algebra
• Triangle-generation in topological D-brane categories
• The derived deformation theory of a point
• A-infinity functors and homotopy theory of dg-categories
• Donaldson-Thomas invariants of length 2 flops
• Bar category of modules and homotopy adjunction for tensor functors
• Exact Lefschetz fibrations associated with dimer models
• Koszul A-infinity algebras and free loop space homology
• Curved A-infinity-categories: adjunction and homotopy
• Invariants of knotted surfaces from link homology and bridge trisections
• Koszul duality for Iwasawa algebras modulo p
• Transferring algebra structures on complexes
• Batanin's category of pruned trees is Koszul
• Cylinders for non-symmetric DG-operads via homological perturbation theory
• Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)