# Generators in formal deformations of categories

## TL;DR

This paper applies Lurie's formal moduli theory to analyze deformations of $k$-linear $$-categories, showing under certain conditions that deformations have zero curvature and possess compact generators.

## Contribution

It establishes conditions under which formal deformations of $k$-linear $$-categories have zero curvature and include compact generators, extending deformation theory in higher categories.

## Key findings

- Deformations have zero curvature under specified conditions.
- Deformations admit compact generators.
- The theory connects formal moduli problems with categorical deformations.

## Abstract

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a $k$-linear $\infty$-category for a field $k$. Our main result states that if $\mathcal{C}$ is a $k$-linear $\infty$-category which has a compact generator whose groups of self extensions vanish for sufficiently high positive degrees, then every formal deformation of $\mathcal{C}$ has zero curvature and moreover admits a compact generator.