# Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$

$$\newcommand\Mod{#1\text{-Mod}}$$Does any one have a reference on a explicit equivalence between $$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$

The proof in "Tensor Categories EGNO" uses the universal property of $$\boxtimes$$, but I would like to see an explicit functor that defines an equivalence.

In BaKi definition 1.1.15 there is a explicit description of $$\mathcal{C}_1\boxtimes \mathcal{C}_2$$ when $$\mathcal{C}_1$$, $$\mathcal{C}_2$$ are additive categories over $$k$$. Namely,

• $$\operatorname{Ob}(\mathcal{C}_1\boxtimes \mathcal{C}_2)=$$ finite sums of the form $$\bigoplus X_i\boxtimes Y_i$$, where $$X_i\in \operatorname{Ob}(\mathcal{C}_1)$$, $$Y_i\in \operatorname{Ob}(\mathcal{C}_2)$$.
• and $$\operatorname{Hom}_{\mathcal{C}_1\boxtimes \mathcal{C}_2}(\bigoplus X_i\boxtimes Y_i,\bigoplus X_j'\boxtimes Y_j')= \bigoplus \operatorname{Hom}(X_i,X'_j)\otimes \operatorname{Hom}(Y_i,Y'_j)$$.

Using this description I wanted to construct the equivalence.

I tried showing that \begin{align*} F:\Mod A\boxtimes \Mod B &\rightarrow \Mod{A\otimes_\mathbb{C} B}\\ X\boxtimes Y&\mapsto X\otimes_\mathbb{C}Y \end{align*} is full, faithul, and essentially surjective.

But I got stuck on trying to show that this functor is essentially surjective. There is what seems to be a counter example, shown on MathStack (Link).

Any help/suggestions would be greatly appreciated.

I currently only need the case when the modules are finite dimensional. But dont want to be restricted to finite dimensional algebras. Namely I am working with finite dimensional modules over $$U(\mathfrak{g}_1\oplus \mathfrak{g}_2)\cong U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)$$.

Edit (More context): I was really trying to use the Relative Tambara Tensor product introduced in (Tam01), and saw a sentence in (DSS18) (First Paragraph), that when using $$\mathcal{C}=\operatorname{Vect}$$ it agrees with Deligne's Tensor Product (though I could have misunderstood).

The Relative Tambara Tensor Product has a similar description of objects and morphisms, but also has much more relations.

• For your particular problem it’s probably better to rewrite that as comodules for the corresponding quantized function algebra which will be better behaved under tensor product. I think this is discussed in one of the Ben Zvi-Brochier-Jordan papers Mar 12 at 15:38