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I know that in Ring Category we have:

-Objects: Rings.

-Arrows: Ring homomorphisms.

I do not know which are the objects and arrows in Graded Ring Category. In general, which is the definition of Graded Ring Category?

P.S. I need it in order to see why it makes a difference to take inverse limit in this two categories. (I am working in symmetric functions vs symmetric polynomials. Do not see the difference between this two concepts.)

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A graded ring can mean a few different things, but it's usually a sequence of abelian groups $R_i$ together with multiplication maps $R_i\otimes R_j\to R_{i+j}$ satisfying associativity, with a unit element $1\in R_0$. A morphism of graded rings is a sequence of abelian group maps respecting multiplication and unit.

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