Triangular matrix categories over quasi-hereditary categories

In this paper, we prove that the lower triangular matrix category $\Lambda =\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right ]$ , where $\mathcal{T}$ and $\mathcal{U}$ are $\textrm{Hom}$ -finite, Krull–Schmidt $K$ -quasi-hereditary categories and $M$ is an $\mathcal{U}\otimes _K \mathcal{T}^{op}$ -module that satisfies suitable conditions, is quasi-hereditary. This result generalizes the work of B. Zhu in his study on triangular matrix algebras over quasi-hereditary algebras. Moreover, we obtain a characterization of the category of the $_\Lambda \Delta$ -filtered $\Lambda$ -modules.