# Deligne's theorem on the characterisation of Tannakian categories

I am trying to understand the proof of the Theorem 7.1 from Catégories Tannakiennes by Pierre Deligne https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf.

Essentially, it is proved that if $T$ is a symmetric monoidal $k$-linear category such that every object in it has a finite integer dimension, then there exists a functor $T \rightarrow R\text{-}Mod$, called a fiber functor, to the category of modules over a $k$-ring $R$ which is strong monoidal and has some other properties, like being exact. I will only concentrate on the property of strong monoidality.

The proof establishes an existence of a monoid $A$ in $Ind(T)$ with the property that for any object $X$ in $T$ the tensor product $A\otimes X$ is isomorphic as an $A$-module to $A^{\oplus dim(X)}$. Then, the fiber functor is $T(1, A\otimes -)$ which takes values in $R$-modules with $R = T(1, A)$. The monoidality of this functor should go through the monoidality of $A\otimes -$. My question is why is the latter strong monoidal? Sure, one can formally write

$$A\otimes (X\otimes Y) \cong A^{dim(X)dim(Y)} \cong A^{dim(X)}\otimes A^{dim(Y)} \cong (A\otimes X)\otimes (A\otimes Y).$$

but, I do not see why is this a natural family a priori? The original paper, as well as a few other places where I've seen an exposition, do not seem to discuss anything which would account for this.

• $A\otimes -$ is not strong monoidal as an endofunctor of $Ind(T)$ (your second isomorphism is incorrect!). It is strong monoidal as a functor to $A$-modules. Then $\Gamma$ is lax monoidal from $A$-modules to $\Gamma(A)$-modules, but it is strong when restricted to the full subcategory of dualizable $A$-modules, where $A\otimes -$ lands. – Marc Hoyois Apr 28 '16 at 20:57
• Sorry, by "dualizable" I meant "finite free" in the last sentence. – Marc Hoyois Apr 28 '16 at 21:08
• @MarcHoyois That is correct. Thank you. – Dimitri Chikhladze Apr 29 '16 at 10:46
• Btw, does anyone know how to get hold on A. Rosenberg "The existence of fiber functors" in the series Gelfand Mathematical Seminars? – Dimitri Chikhladze May 3 '16 at 13:55
• sent you a copy by email. – Marc Hoyois May 5 '16 at 22:08