### Theorems Into Coffee

#### Posted by John Baez

As the famous quote goes, “a mathematician is a machine for turning coffee into theorems”. But every chemical reaction is reversible, at least under the right conditions. So, there’s *got to be some way turn theorems into coffee!*

And now you can do it here.

I haven’t quite worked out the details. I’d like to harness your competitive spirit and yearning for coffee, but also the marvelous *cooperative* spirit of this blog. I’m not sure how yet, but let’s just dive in and see how it goes.

I’ll send a $15 Starbucks gift certificate to anybody who gives me, in LaTeX, a well-written rigorous proof of the following theorem:

Let $Mat(\mathbb{N})$ be the category whose objects are natural numbers and whose morphisms $f : n \to m$ are $m \times n$ matrices of natural numbers, with composition being given by matrix multiplication. Think of $Mat(\mathbb{N})$ 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:$Mat(\mathbb{N})$ is the PROP for bicommutative bialgebras.More explanations: a symmetric monoidal category whose objects are just natural numbers, with addition as the tensor product of objects, is called a PROP. We can talk about the models (also known as ‘algebras’ ) of a PROP in any symmetric monoidal category $C$. I’m claiming that the category of models of $Math(\mathbb{N})$ in $C$ is equivalent to the usual category of ‘bicommutative bialgebras’ in $C$. You might prefer to call these ‘bicommutative bimonoids’ in $C$: they’re commutative monoid objects that are also cocommutative comonoid objects, where the monoid operations are required to be comonoid homomorphisms. (Or, equivalently: the comonoid operations are required to be a monoid homomorphisms!)

I haven’t worked out what I’ll do if I get multiple responses, or responses that don’t quite satisfy me. For now, let me say I’ll give the coffee to the first person who gives me a satisfactory proof in LaTeX. Posting a link on this blog is best; if you can’t manage that, send it via email.

But in fact, I have a bunch of theorems I’d like proved, and I’d like to evolve a system where groups of people could collaborate to prove them and *jointly* receive a reward. That seems a lot more fun than a kind of race to be first. But I don’t know to organize it — suggestions?

I haven’t fully worked out the issue of credit and publishing rights, but let’s say this. If you collect the gift certificate, that means you give me the right to use your LaTeX source code in future work — for example, an expository paper, wiki, or book on higher-dimensional algebra. But I will certainly credit you, and I’ll certainly let you publish this work elsewhere.

Of course, the last part only matters much if the “Theorems into Coffee” idea catches on — the above theorem is just one of a collection of similar results, and they’re probably more publishable *en masse* than one at a time.

I haven’t worked out what I’ll do if someone posts a reference to an existing proof of the theorem. Let’s say for now that I’ll give you the same reward regardless of whether the theorem has been proved, *if* you write up a self-contained clear proof.

I also haven’t worked out what to do if someone outside the US collects the award — for example, Robin Houston or Tom Leinster or Eugenia Cheng. The Starbucks card system I know seems designed to work best inside the US… but I haven’t even tried it yet. Let’s say this: if it turns out to be a nuisance to send you a Starbucks card, *or you don’t want a Starbucks card*, I’ll send you a check for $15. Or, if I bump into you during my travels this summer, I’ll be glad to pay you the approximate equivalent in local currency.

(Frankly the check option looks simpler than the Starbucks card; you just need to promise to spend it on coffee. Or tea, if you’re a wimp. Something that helps you do math, anyway!)

There are probably lots of other things I haven’t worked out.

But for starters, let’s see if *anybody* tackles this theorem.

## Re: Theorems Into Coffee

If a Mathematician is a machine for turning coffee into theorems, then isn’t a co-mathematician a co-machine for turning theorems into ffee?