Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I

TL;DR

This paper introduces a geometric framework for A-infinity algebras and categories using formal schemes, clarifying key concepts and connecting to non-commutative geometry, Hochschild complexes, and topological field theories.

Contribution

It develops a geometric approach to A-infinity structures, explores Hochschild complexes geometrically, and generalizes several conjectures including Deligne's conjecture.

Findings

Geometric interpretation of Hochschild complexes.

Homological versions of properness and smoothness.

Action of the PROP of 2D surfaces on Hochschild chains.

Abstract

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne's conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product. This action is essentially equivalent to the structure of 2-dimensional Topological Field Theory associated with a Calabi-Yau category.