Graded direct sums

A graded (preadditive) category is a preadditive category such that the hom groups have a Z-grading compatible with composition. In Heller’s paper of 1958 he talks about direct sums in graded categories: one requires the projections and the coprojections to be homogeneous (I would also require them to have degree 0 but Heller doesn’t require this).

Today seems to be the day for silly questions, because I was wondering if a graded category which has direct sums as an additive category (i.e., ignoring the grading) necessarily has direct sums as a graded category.

The answer is no (please stop reading here; it won’t get any clearer from here on out). For example, start with a semi-simple abelian category A generated by two non-isomorphic simple objects X and Y. Then consider the graded category Grgr(A) of graded objects of A (see Tag 09MM). Let’s denote [n] the shift functors on graded objects. Then consider the subcategory B of Grgr(A) containing 0, containing arbitrary finite direct sums of shifts of copies of K = X ⊕ Y and containing arbitrary shifts of L = X ⊕ Y[1] and M = X ⊕ Y[2]. Then, forgetting the grading, we see that K ⊕ K is the direct sum of L and M. But, even with the definition in Heller, K ⊕ K is not the graded direct sum of L and M in this category. In fact, the direct sum L ⊕ M in Grgr(A) is not isomorphic to any object of B, but B, viewed as an preadditive category has direct sums.