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I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) in Deligne: Catégories tannakiennes.

My question is similar to this one.

Given a cocomplete tensor category $\mathcal{C}$ the paper shows that for every object $X\in \mathcal{C}$ there is a commutative ring $A_X\in \mathcal{C}$ such that $A_X\otimes X\cong A_X^{\oplus\text{dim}X}$, where this is an isomorphism of $A_X$-modules.

However, in the final stage of the proof, it is claimed that there is a commutative ring $B\in \mathcal{C}$ such that $X\otimes B \cong B^{\oplus \text{dim} X}$ holds for all $X\in \mathcal{C}$.

I fail to see how the first statement implies the second. The only idea I have is to consider the coproduct $\bigoplus_{X\in \mathcal{C}} A_X$ over all the $A_X$. But I am not sure if this would even be a commutative ring in $\mathcal{C}$.

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  • $\begingroup$ You want to take a set of generators of $\mathcal C$ and then take the colimit of all tensor products of $A_X$ for finitely many $X$ in $\mathcal C$. This involves using the unit of the algebras to define the maps in this colimit. $\endgroup$
    – Will Sawin
    Commented Nov 17, 2019 at 0:47
  • $\begingroup$ Thank you for answering. Do you mean the following: Let $S$ be a set of tensor generators of $\mathcal{C}$. Then set $A := \text{colim}_{J\subseteq S,~J~\text{finite}} \bigotimes_{X'\in J} A_X'$?. $\endgroup$
    – Aaron Wild
    Commented Nov 18, 2019 at 14:56
  • $\begingroup$ Yes, that's what I'm thinking. $\endgroup$
    – Will Sawin
    Commented Nov 18, 2019 at 16:12

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