Hochschild-Mitchell cohomology of a locally bounded category

Abstract

We consider k-linear categories A which are locally bounded, that is, for every object x, the sums Sigma(y) dim(k) A(x,y) and Sigma(y) dim(k) A(y,x) are finite. Let x(1), x(2),..., be an enumeration of the objects of A such that for each n is an element of N, the full subcategory A((n)) of A formed by the objects x(1), x(2),...,x(n) is convex in A. We compute the Hochschild-Mitchell cohomology of A as the inverse limit of the cohomologies of the categories A((n)).