### Theorems Into Coffee II

#### Posted by John Baez

Nobody instantly solved my first coffee challenge, but I hope that interest is brewing. Maybe some of you will perk up if I throw another $15 in the pot?

It’s a slight variation on the same theme: taking a nice category where the morphisms are $m \times n$ matrices, interpreting it as a PROP, and asking what sort of algebraic gadget is defined by this PROP.

As before, I would love to have access to a proof of this result. So, under the same terms as before, I’ll send a $15 Starbucks gift certificate (or check) to anybody who gives me, in LaTeX, a well-written rigorous proof of the following theorem:

Let $Mat(\mathbb{Z})$ be the category whose objects are natural numbers and whose morphisms $f : n \to m$ are $m \times n$ matrices of integers, with composition being given by matrix multiplication. Think of $Mat(\mathbb{Z})$ as a symmetric monoidal category in the obvious way where the tensor product of objects $n$ and $m$ is $n + m$, and the tensor product of morphisms is direct sum of matrices.

Theorem:$Mat(\mathbb{Z})$ is the PROP for bicommutative Hopf algebras.In a bit more detail: we can talk about the models — I guess most people say ‘algebras’ — of a PROP in any symmetric monoidal category $C$. I’m claiming that the category of models of $Math(\mathbb{Z})$ in $C$ is equivalent to the usual category of ‘bicommutative Hopf algebras’ in $C$. You might prefer to call them ‘bicommutative Hopf objects’, since no linear algebra is involved. They’re just bicommutative monoid objects as defined last time, but now also equipped with an antipode satisfying the usual conditions in the definition of a Hopf algebra — written out in diagrammatic form, of course.

The point is that the antipode $S: H \to H$ of our Hopf object is an operation with one input and one output. It should thus correspond to a $1 \times 1$ matrix of integers. We’ll take this matrix to be $(-1)$. The following equation then encodes the fact that if we comultiply something, apply the antipode to one of the outputs, and then multiply them, we get the unit:

$1 + (-1) = 0$

Anyone who can prove the last theorem should be well on the way to proving this one too. I would love for people to collaborate, though perhaps this ‘prize’ format doesn’t encourage that. Be nice! — coffee tastes better shared with a friend.

If you need some basic information on PROPs, this paper could be a good place to start, along with the Wikipedia article I cited before:

- Lucian Ionescu, From operads and PROPs to Feynman Processes.

This book has a lot more detail:

- Tom Leinster, Higher Operads, Higher Categories.

## Re: Theorems Into Coffee II

Hi,

The proof of the fact that Mat(N) is the PROP for bicommutative bialgebras can be found in the draft version of my paper where I construct the “PRO” corresponding to a category of games and strategies interpreting a fragment of first-order propositional logic (comments are of course very welcome!). I think that the credits and coffees for this result should go to Albert Burroni and Yves Lafont who introduced the general methodology for constructing such PROs. As far as I know, the notions of

polygraphandcanonical formused to prove the result were introduced by Albert in Higher dimensional word problem with application to equational logic and deeply studied and refined by Yves in Towards an Algebraic Theory of Boolean Circuits. I haven’t checked the details, but this method should extend to the case of Mat(Z) without major difficulties (I can give a notion of canonical form for this PROP or more details if anyone is interested).