Curved A-infinity-categories: adjunction and homotopy

TL;DR

This paper develops a unified theory of curved A-infinity-categories, extending classical uncurved theories, and introduces functors that establish adjunctions and homotopy equivalences, applicable to Fukaya categories and matrix factorizations.

Contribution

It introduces a general framework for curved A-infinity-categories using adjoint algebra and Q functors, unifying various categorical theories and proving key homotopy and adjunction results.

Findings

Established adjunction and homotopy theorems for functors U_e and Q_*

Unified treatment of curved and uncurved A-infinity-categories

Proved the Positselski-Kontsevich vanishing result

Abstract

We develop a theory of curved A-infinity-categories around equivalences of their module categories. This allows for a uniform treatment of curved and uncurved A-infinity-categories which generalizes the classical theory of uncurved A-infinity algebras. Furthermore, the theory is sufficiently general to treat both Fukaya categories and categories of matrix factorizations, as well as to provide a context in which unitification and categorification of pre-categories can be carried out. Our theory is built around two functors: the adjoint algebra functor U_e and the functor Q_*. The bulk of the paper is dedicated to proving crucial adjunction and homotopy theorems about these functors. In addition, we explore the non-vanishing of the module categories and give a precise statement and proof the result known as "Positselski-Kontsevich vanishing".