# Differential graded bocses and $A_{\infty}$-modules

## TL;DR

This paper develops the theory of twisted modules over triangular differential graded bocses, establishing their categorical properties and linking them to $A_{
abla}$-algebra modules, thus advancing the understanding of homotopical and algebraic structures.

## Contribution

It introduces the category of twisted modules over triangular differential graded bocses and proves their categorical properties, connecting them to $A_{
abla}$-algebra modules.

## Key findings

- Idempotents split in the category of twisted modules.
- The category admits a Frobenius structure.
- Homotopically trivial modules correspond to acyclic complexes.

## Abstract

We introduce and study the category of twisted modules over a triangular differential graded bocs. We show that in this category idempotents split, that it admits a natural structure of a Frobenius category, that a twisted module is homotopically trivial if and only if its underlying complex is acyclic, and that any homotopy equivalence of differential graded bocses determines an equivalence of the corresponding homotopy categories of twisted modules. The category of modules over an $A_{\infty}$-algebra is equivalent to the category of twisted modules over a triangular differential graded bocs, so all preceding statements lift to the former category.