Sur les A-infini cat\'egories

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 $H^0 tw A$, where $tw A$ is the A-infinity-category of twisted objets of a certain A-infinity-category A.