Differential graded triangular matrix categories

TL;DR

This paper extends the concept of triangular matrix algebras to differential-graded categories, establishing an equivalence between certain dg-module categories and dg-comma categories, thus generalizing classical algebra results.

Contribution

It introduces a dg-category analogue of triangular matrix algebras and proves an equivalence with dg-module categories, extending classical algebraic results to the dg setting.

Findings

Established an equivalence of dg-categories involving dg-modules and dg-comma categories.

Constructed the dg-triangular matrix category from two dg-categories and a bimodule.

Generalized a classical result for Artin algebras to the dg-category context.

Abstract

This paper focuses on defining an analog of differential-graded triangular matrix algebra in the context of differential-graded categories. Given two dg-categories $\mathcal{U}$ and $\mathcal{T}$ and $M \in \text{DgMod}(\mathcal{U} \otimes \mathcal{T}^{\text{op}})$, we construct the differential graded triangular matrix category $\Lambda := \left( \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} \right)$. Our main result is that there is an equivalence of dg-categories between the dg-comma category $(\text{DgMod}(\mathcal{T}),\text{GDgMod}(\mathcal{U}))$ and the category $\text{DgMod}\left( \left( \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} \right)\right)$. This result is an extension of a well-known result for Artin algebras (see, for example, [2,III.2].