# Functor categories

1. May 23, 2010

### error792

1. The problem statement, all variables and given/known data
Let $$\mathbb N$$ be the discrete category of natural numbers. Describe the functor category $$\mathrm{Ab}^{\mathbb N}$$ (commonly known as the category of graded abelian groups).

2. Relevant equations

3. The attempt at a solution
Since $$\mathbb N$$ is discrete, a functor $$\mathbb N\xrightarrow A\mathrm{Ab}$$ is simply a sequence $$(A_n) = A_0,A_1,\dots$$ of abelian groups; an arrow $$(A_n)\xrightarrow{\sigma} (B_n)$$ is given by arrows $$A_0\xrightarrow{\sigma_0}B_0, A_1\xrightarrow{\sigma_1}B_1,\dots$$.

This looks right, but seems too simple to me. I don't know very much about grading, but I thought there had to be some way of "going up the A's" (like $$\otimes\colon V^{\otimes i}\times V^{\otimes j}\to V^{\otimes i+j}$$ in the case of the tensor algebra)

2. May 24, 2010

### eok20

For a graded ring you need to have an i thing times a j thing be an i+j thing. Since this problem is just with abelian groups, there is no such condition.