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Homework Help: Functor categories

  1. May 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Let [tex]\mathbb N[/tex] be the discrete category of natural numbers. Describe the functor category [tex]\mathrm{Ab}^{\mathbb N}[/tex] (commonly known as the category of graded abelian groups).

    2. Relevant equations

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

    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 [tex]\otimes\colon V^{\otimes i}\times V^{\otimes j}\to V^{\otimes i+j}[/tex] in the case of the tensor algebra)
  2. jcsd
  3. May 24, 2010 #2
    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.
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