# Do graded modules form an additive category?

I got two conflicting references. On page 466 of Shastri's Algebraic Topology, he claims that graded modules with homogeneous morphisms form an additive cat, whereas Hilton and Stammbach say on page 75 of their "Course in Homological Algebra" that this is true only if we restrict to morphisms of degree 0. I worked on it for a while and it seems to me that neither one is completely correct. I'd say that graded modules form an additive cat iff we restrict to morphisms of some degree k, not necessarily equal to 0.

1) If we do not restrict to some degree k, then I can't seem to find what would be the zero object. I think it doesn't exist, because if we take the family ${0_i}$ then we still have all possible shifts of the zero map (from or to this zero object).

2) Products exist even if we don't restrict to homogeneous morphisms of some degree.

3) the hom-sets are abelian and have a bilinear composition in any of the three cases as well.

• Dear Rodrigo, you might want to have a look at Section 3 (Graded categories) of Chapter 2 of Saul Lubkin's book Cohomology of completions (North-Holland Mathematics Studies 42, 1980). – Fred Rohrer Oct 24 '13 at 6:25