Differential graded triangular matrix categories

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 $Λ:= \left( \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} \right)$. 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].