Homological Epimorphisms and Hochschild-Mitchell Cohomology

In this work, we study the Hochschild-Mitchell Cohomology of triangular matrix categories. Given a triangular matrix category $\mathrm{\Lambda }=\left[\begin{array}{cc}\mathcal{T}& 0\\ M& \mathcal{U}\end{array}\right]$, we investigate the relationship of the Hochschild-Mitchell cohomologies $H^i(\mathrm{\Lambda })$ and $H^i(\mathcal{U})$ of $\mathrm{\Lambda }$ 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].