2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103587We define $H$-Galois extensions for $k$-linear categories and a Hopf algebra $H$ and prove the existence of a Grothendieck spectral sequence for Hochschild-Mitchell cohomology, related to this situation. This spectral sequence is multiplicative and for a $H$ a group algebra decomposes as a direct sum indexed by the set of conjugacy classes of the group. We also compute some Hochschild-Mitchell cohomology groups of categories with infinite associated quiver.21 pagesK-Theory and HomologyRings and Algebras16W50; 18E05; 16W30; 16S40; 16D90text