# $ L_\infty $-spectral sequences for Hochschild cohomology

## TL;DR

This paper introduces a method to preserve higher algebraic structures in Hochschild cohomology computations using spectral sequences, especially for minimal A_infinity-categories, and applies it to wrapped Fukaya categories.

## Contribution

It presents a new strategy to retain higher structures in spectral sequences for Hochschild cohomology, particularly via twisting by Maurer-Cartan elements in A_infinity contexts.

## Key findings

- Recovered formality results for wrapped Fukaya categories
- Established an L_infinity quasi-isomorphism between Hochschild complex and cohomology
- Demonstrated the effectiveness of the twisting approach in higher structure preservation

## Abstract

Spectral sequences are a common tool to compute cohomology spaces, but higher structure is often lost on the way. In this article we exhibit a strategy to retain the higher structure on the cohomology, which works in case the chain complex with higher structure is presented as the twisting of a simpler chain complex by a Maurer-Cartan element. This works particularly well in case of the Hochschild complex of a minimal $ A_\infty $-category, where the Hochschild complex can be seen as the twisting of the Hochschild complex of the underlying associative algebra or category. As an application, we recover the existing formality result for Hochschild cohomology of wrapped Fukaya categories of punctured surfaces, and provide an $ L_\infty $-quasi-isomorphism between the Hochschild complex and its cohomology.