Postnikov towers, k-invariants and obstruction theory for DG categories

TL;DR

This paper develops Postnikov towers, k-invariants, and obstruction theory for dg categories, enabling the determination of when a dg functor can be lifted based on obstruction classes.

Contribution

It introduces a new framework for Postnikov towers and obstruction theory in dg categories, inspired by Drinfeld's DG quotient, and proves a rigidification theorem.

Findings

Obstruction classes determine liftability of dg functors.

Established a rigidification theorem for homologically connective dg categories.

Provided tools for analyzing dg categories via Postnikov towers.

Abstract

By inspiring ourselves in Drinfeld's DG quotient, we develop Postnikov towers, k-invariants and an obstruction theory for dg categories. As an application, we obtain the following `rigidification' theorem: let A be a homologically connective dg category and F:B -> H0(A) a dg functor to its homotopy category. If the family of obstruction classes vanishes, then a lift for F exists.