## Definitions and the main question

Recall that a category $\mathcal C$ is **monoidal** if it is equipped with the following data (two functors, three natural transformations, and some properties):

- a functor $\otimes: \mathcal C \times \mathcal C \to \mathcal C$,
- a functor $1: \{\text{pt}\} \to \mathcal C$,
- a natural transformation $\alpha: (X\otimes Y)\otimes Z \overset\sim\to X\otimes(Y\otimes Z)$ between functors $\mathcal C^{\times 3} \to \mathcal C$ (natural in $X,Y,Z$),
- natural transformations $\lambda: 1 \otimes X \overset\sim\to X$, $\rho: X\otimes 1 \overset\sim\to X$ between functors $\mathcal C \to \mathcal C$ (natural in $X$; this uses the canonical isomorphisms of categories $\{\text{pt}\} \times \mathcal C \cong \mathcal C \cong \mathcal C \times \{\text{pt}\}$),
- such that $\alpha$ satisfies a pentagon,
- and $\alpha,\lambda,\rho$ satisfy some other equations.

I tend to be less interested in the unit laws $\lambda,\rho$, which is my excuse for knowing less about their technicalities. In my experience, it's the associativity law $\alpha$ that can have interesting behavior.

Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F: \mathcal C \to \mathcal D$ is **(strong) monoidal** if it comes with the following data (two natural transformations, and three properties):

- a natural isomorphism $\phi: \otimes_{\mathcal D} \circ (F\times F) \overset\sim\to F\circ \otimes_{\mathcal C}$ of functors $\mathcal C \times \mathcal C \to \mathcal D$,
- a natural isomorphism $\varphi: 1\_{\mathcal D} \overset\sim\to F\circ 1\_{\mathcal C}$ of functors $\{\text{pt}\} \to \mathcal D$,
- satisfying some properties, the main one being that the two natural transformations $(FX \otimes_{\mathcal D} FY) \otimes_{\mathcal D} FZ \overset\sim\to F(X\otimes_{\mathcal C} (Y \otimes_{\mathcal C}Z))$ that are built from $\phi, \alpha_{\mathcal C}, \alpha_{\mathcal D}$ agree. This property expresses that the associators in $\mathcal C$, $\mathcal D$ are "the same" under the functor $F$.

My question is whether there is a (useful) weakening of the axioms for a monoidal functor that expresses the possibility that the associators might disagree.

## An example: quasiHopf algebras

Here is my motivating example. Let $A$ be a (unital, associative) algebra (over a field $\mathbb K$), and let $A\text{-rep}$ be its category of representations. I.e. objects are pairs $V \in \text{Vect}\_{\mathbb K}$ and an algebra homomorphism $\pi_V: A \to \text{End}\_{\mathbb K}(V)$, and morphisms are $A$-linear maps. Then $A\text{-rep}$ has a faithful functor $A\text{-rep} \to \text{Vect}\_{\mathbb K}$ that "forgets" the map $\pi$.

Suppose now that $A$ comes equipped with an algebra homomorphism $\Delta: A \to A \otimes_{\mathbb K} A$. Then $A\text{-rep}$ has a functor $\otimes: A\text{-rep} \times A\text{-rep} \to A\text{-rep}$, given by $\pi_{(V\otimes W)} = (\pi_V \otimes \pi_W) \circ \Delta: A \to \text{End}(V\otimes_{\mathbb K}W)$. Just this much data is not enough for $A\text{-rep}$ to be monoidal. (Well, we also need a map $\epsilon: A \to \mathbb K$, but I'm going to drop mention of the unit laws.) Indeed: there might not be an associator.

A situation in which there *is* an associator on $(A\text{-rep},\otimes)$ is as follows. Suppose that there is an invertible element $p \in A^{\otimes 3}$, such that for each $a\in A$, we have
$$ p\cdot (\Delta \otimes \text{id})(\Delta(a)) = (\text{id} \otimes \Delta)(\Delta(a))\cdot p $$
and $\cdot$ is the multiplication in $A^{\otimes 3}$. Then for objects $(X,\pi_X), (Y,\pi_Y), (Z,\pi_Z) \in A\text{-rep}$, define:
$$ \alpha_{X,Y,Z} = (\pi_X \otimes \pi_Y \otimes \pi_Z)(p) :
((X\otimes\_{A\text{-}{\rm rep}} Y) \otimes\_{A\text{-}{\rm rep}} Z) \to (X \otimes\_{A\text{-}{\rm rep}} (Y \otimes\_{A\text{-}{\rm rep}} Z)) $$
You can check that it is in fact a isomorphism in $A\text{-rep}$. Moreover, supposing that $p$ satisfies:
$$ (\text{id} \otimes \text{id} \otimes \Delta)(p) \cdot (\Delta \otimes \text{id} \otimes \text{id})(p) = (1 \otimes p) \cdot (\text{id} \otimes \Delta \otimes \text{id})(p) \cdot (p \otimes 1) $$
where now $\cdot$ is the multiplication in $A^{\otimes 4}$, then $\alpha$ is an honest associator on $A\text{-rep}$.

Then (provided also that $A$ have some sort of "antipode"), the data $(A,\Delta,p)$ is a **quasiHopf algebra**.

Anyway, it's clear from the construction that the forgetful map $\text{Forget}: A\text{-rep} \to \text{Vect}\_{\mathbb K}$ is a faithful exact functor which is weakly monoidal in the sense that $\text{Forget}(X \otimes\_{A\text{-}{\rm rep}} Y) = \text{Forget}(X) \otimes\_{\mathbb K} \text{Forget}(Y)$ — indeed, this is equality of objects, so perhaps it is "strictly" monoidal — but it is not "monoidal" since it messes with the associators.

## Actual motivation

My actual motivation for asking the question above is the understand the Tannaka duality for quasiHopf algebras. In general, we have the following theorem:

**Theorem:** Let $\mathcal C$ be an abelian category and $F: \mathcal C \to \text{FinVect}\_{\mathbb K}$ a faithful exact functor, where $\text{FinVect}\_{\mathbb K}$ is the category of finite-dimensional vector spaces of $\mathbb K$. Then there is a canonical coalgebra $\text{End}^{\vee}(F)$, and $\mathcal C$ is equivalent as an abelian category to the category of finite-dimensional corepresentations of $\text{End}^{\vee}(F)$.

For details, see A Joyal, R Street, An introduction to Tannaka duality and quantum groups, Category Theory, Lecture Notes in Math, 1991 vol. 1488 pp. 412–492.

The Tannaka philosophy goes on to say that if in addition to the conditions in the theorem, $\mathcal C$ is a monoidal category and $F$ is a monoidal functor, then $\text{End}^{\vee}(F)$ is a bialgebra, and $\mathcal C$ is monoidally equivalent to $\text{End}^{\vee}(F)\text{-corep}$. If $\mathcal C$ has duals, $\text{End}^{\vee}(F)$ is a Hopf algebra. If $\mathcal C$ has a braiding, then $\text{End}^{\vee}(F)$ is coquasitriangular. Etc.

My real question, then, is:

What is the statement for Tannaka duality for (co)quasiHopf algebras?

It seems that the standard paper to answer the real question is: S. Majid, Tannaka-Krein theorems for quasi-Hopf algebras and other results. Contemp. Math. 134 (1992), pp. 219–232. But I have not been able to find a copy of this paper yet.