Homological Epimorphisms and Hochschild-Mitchell Cohomology

In this work, we study the Hochschild-Mitchell Cohomology of triangular matrix categories. Given a triangular matrix category $\Lambda=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$, we investigate the relationship of the Hochschild-Mitchell cohomologies $H^{i}(\Lambda)$ and $H^{i}(\mathcal{U})$ of $\Lambda$ and $\mathcal{U}$ respectively; and we show that they can be connected by a long exact sequence. This result extend the well-known result of Michelana-Platzeck given in [S. Michelena, M. I. Platzeck. {\it{Hochschild cohomology of triangular matrix algebras}}. J. Algebra 233, (2000) 502-525].
Comment: In this new version, the title of the article has been changed and a new section has been added