In category theory, a notion of monoidal category in which every sequence $X\_1, \ldots , X\_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 of Tom Leinster's [book](https://arxiv.org/pdf/math/0305049.pdf) for more details).


Going down one category level, one can also formulate the notion of an unbiased algebra (over some field):



> 
> **Definition:**
>  An *unbiased algebra* consists of a vector space $A$ together with, for every ordered sequence of elements $a\_1,\ldots,a\_n\in A$ ($n\ge 0$), a new element $m\_n(a\_1,\ldots,a\_n)\in A$, called the product of $a\_1,\ldots,a\_n$. The product depends on the $a\_i$'s in a multilinear way, and is subject to the following axiom: $$m\_{p+1+q}\circ (id\_{A^p}\times m\_n\times id\_{A^q})=m\_{p+n+q}.$$
>  Note that this axiom encodes both the associativity and the unitality of the algebra.
> 
> 
> 


Of course, an unbiased algebra is just the same thing as an algebra. But that's a theorem (a fairly easy theorem).


Now, based on the material contained in the last 5 minutes [this video](https://www.youtube.com/watch?v=CJR-QXHP5GQ&t=0s&index=2&list=PLdex6rFe0dfQRKLmcGC0rUrsswjHd01FS) (the interval 55:30—1:00:00 to be precise), I am tempted to believe that there exists an unbiased definition of Hopf algebra.



> 
> **Question:**
>  Can anyone describe the notion of a Hopf algebra in an unbiased way?
> 
> 
> 



Unbiased Hopf algebras
I'll just consider a finite-dimensional bicommutative Hopf algebra $H$ over a field $k$. 


* For every map $A\xrightarrow{p} B$ of finite sets, we have maps $\mu\_p\colon H^{\otimes A}\to H^{\otimes B}$ and $\psi\_p\colon H^{\otimes B}\to H^{\otimes A}$
* These have the obvious kind of functoriality for composites $A\xrightarrow{p}B\xrightarrow{q}C$
* For any pullback square
$\require{AMScd}$
\begin{CD}
A @>{m}>> B \\
 @VnVV @VVpV\\ 
C @>>q> D
\end{CD}
we have $\psi\_p\mu\_q=\mu\_m\psi\_n\colon H^{\otimes C}\to H^{\otimes B}$


I'll define a balanced Hopf algebra to be a finite dimensional vector space $H$ over $k$, equipped with maps $\mu\_p$ and $\psi\_p$ as above. I think that balanced Hopf algebras are the same as finite dimensional bicommutative Hopf algebras, but I cannot claim to have checked that carefully.


The above formulation does not give you an antipode, but we can force one to exist as follows. Define $p,q\colon 3\to 2$ by $p(0)=p(1)=q(0)=0$ and $p(2)=q(1)=q(2)=1$. Then $\mu\_q\psi\_p$ is a kind of shearing map, and I think it works out that $H$ has an antipode iff $\mu\_q\psi\_p$ is invertible. More generally, given $n\xleftarrow{p}m\xrightarrow{q}n$, we can define a matrix $$M(p)\_{ij}=|\{k\in m:(p(k),q(k))=(i,j)\}|.$$
If $H$ has an antipode and $M(p)$ is invertible over $\mathbb{Z}$ then $\mu\_q\psi\_p\colon H^{\otimes n}\to H^{\otimes m}$ should be invertible. 


