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I'm trying to follow the proof of proposition 7.22.7 from

Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2024-6/hbk). xvi, 343 p. (2015). ZBL1365.18001.

I think I understand everything but the assignment $J$ which takes an endomorphism in $C^{\boxtimes n+1}$ of $A(n)$ to an endomorphism of $\otimes^{n}$. Meaning that for any $V_{1},\dots, V_{n}$ we need to get an endomorphism of $V_{1}\otimes\dots\otimes V_{n}$.

Here $\boxtimes$ is the Deligne tensor product, and $A(n)$ is the object in $C^{\boxtimes n+1}$ representing the functor $Hom_{C}(\otimes^{n+1}(\_),1):C^{\boxtimes n+1}\to Vect$.

If I for instance follow the example 7.22.8 in the middle of the proof, I agree that $A(n)=\bigotimes_{V_{1},\dots, V_{n}\in \mathcal{O}(C)} (V_{1}\otimes\dots\otimes V_{n})\boxtimes V_{1}^{\ast}\boxtimes\dots \boxtimes V_{n}^{\ast}$ indexed over isomorphism classes of simple objects.

But already for $n=1$ and for $C$ a fusion category as in the example, if $f\in End_{C\boxtimes C}(\bigoplus_{V\in \mathcal{O}(C)} V\boxtimes V^{\ast})$, then f is meant to be assigned, for any object $V_{1}\in C$, to the endomorphism

$id_{C}\boxtimes Hom_{C}(1,\_\otimes V_{1})(\bigoplus_{V\in \mathcal{O}(C)} V\boxtimes V^{\ast})(f)$

The authors claim in particular that $id_{C}\boxtimes Hom_{C}(1,\_\otimes V_{1})(\bigoplus_{V\in \mathcal{O}(C)} V\boxtimes V^{\ast})\cong V_{1}$ and so this assignment indeed is an endomorphism of, in this case, $V_{1}$.

However I do not see this isomorphism and in fact as $Hom_{C}(1,V^{\ast}\otimes V_{1})\in Vect$, I'm struggling to see how this object in $C\boxtimes Vect$ is being identified with one in $C$. I might be doing something silly but my calculation yields only $\bigoplus V\boxtimes Hom(1,V^{\ast}\otimes V_{1})$ which I do not see how to reduce to $V_{1}$.

Any help or reference would be appreciated.

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The category $$\mathsf{Vect}$$ behaves like a unit with respect to the Deligne tensor $$\boxtimes$$. I think the technical way to say it is that there is a canonical 2-natural equivalence $$\mathcal{C}\boxtimes\mathsf{Vect}\simeq\mathcal{C}$$. If we use this equivalence to identify $$\mathcal{C}$$ with $$\mathcal{C}\boxtimes\mathsf{Vect}\subseteq\mathcal{C}\boxtimes\mathcal{C}$$ then the isomorphism follows from the fact that $$\mathsf{Hom}(\mathbf 1 , V^*\otimes V_i)\cong\mathsf{Hom}(V , V_i)\cong \delta_{V,V_i}\cdot k$$.

Technically I think that should be a left dual $$^*V$$, instead of the way it is written, but I will need to go back and read the proof.

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  • $\begingroup$ Thanks! I think this does work for the case where C is a fusion category and I can write A(n) as above, but do you happen to know how to reason for the multitensor category case? As in, I don't know if A(n) is a product of duals of simple objects. $\endgroup$
    – AT0
    Commented May 27, 2022 at 0:47
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    $\begingroup$ Hi @AT0! In the multi- case this argument is the same, and you are still summing over all simples. In the nonsemisimple case, one needs to use ends. The algebra is expressed as an internal hom which is a right adjoint, and this clues us in to the fact that the construction in this case ought to be an end as opposed to a coend. The formula should be something like $$A(n)\cong\int_{V_{i_1},V_{i_2}\cdots,V_{i_n}}V_{i_n}^*\boxtimes V_{i_{n-1}}^*\boxtimes\cdots\boxtimes V_{i_1}^*\boxtimes V_{i_1}\otimes V_{i_2}\otimes\cdots\otimes V_{i_n}$$ $\endgroup$
    – Sean Sanford
    Commented Jun 11, 2022 at 22:38
  • $\begingroup$ Thanks! This clears it up completely! As an aside, do you know who introduced this algebra? I cannot find anything before EGNO but maybe it is older under another name? $\endgroup$
    – AT0
    Commented Jun 13, 2022 at 16:59

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