Deformations of triangulated categories with t-structures via derived injectives

TL;DR

This paper advances the deformation theory of pretriangulated dg-categories with t-structures by establishing equivalences between different deformation problems and extending t-structures to derived categories.

Contribution

It introduces a method to extend t-structures to dg-derived categories and proves a deformation equivalence involving derived injective objects, enriching the deformation theory framework.

Findings

Extended t-structures to dg-derived categories.

Established deformation equivalence between bounded t-deformations and dg-deformations of derived injectives.

Avoided curvature issues by focusing on cohomologically concentrated dg-categories.

Abstract

This paper provides the final ingredient in the development of the deformation theory of pretriangulated dg-categories endowed with a nice t-structure, which was initiated by the authors and is modeled after the previously developed deformation theory of abelian categories. We show how to extend a t-structure on a pretriangulated dg-category to its dg-derived category so that the Yoneda embedding becomes t-exact. We construct several equivalences between deformation problems; in particular, we prove a deformation equivalence between the bounded t-deformations of a bounded t-dg-category on the one hand, and dg-deformations of the dg-category of derived injective ind-dg-objects on the other hand. Since this latter dg-category is cohomologically concentrated in nonpositive degrees, we do not encounter curvature.