Category ring

Modules over a category ring

Let be a category with finite . Then a module over the the category ring is equivalent to a functor , rep and we have an equivalence of categories¹

Proof

Let be a functor, Let and thus construct the -graded vector space

and for a morphism and homogenous vector define

and extend this definition linearly first for nonhomogenous vectors and then general . Clearly this defines a -module.

Now suppose is a natural transformation with components . Then

defines an -graded linear map. Moreover, by naturality of , for a morphism and homogenous vector

so by linearity is a -module isomorphism. Therefore is a functor.

Conversely, suppose is a -module. We define a functor as follows:

• for ;
• for and .

Now suppose is a -module homomorphism. We define a transformation with components

which is well-defined since is -graded. Moreover, for any and

so the following diagram commutes

whence is natural and is a functor.

It is not difficult to see the natural equivalences required to make this an equivalence.

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Footnotes

1. assuming the Axiom of Choice. ↩