  We 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.
