### Describing PROPs Using Generators and Relations

#### Posted by John Baez

Here’s another post asking for a reference to stuff that *should* be standard. (The last ones succeeded wonderfully, so thanks!)

I should be able to say

$C$ is the symmetric monoidal category with the following presentation: it’s generated by objects $x$ and $y$ and morphisms $L: x \otimes y \to y$ and $R: y \otimes x \to y$, with the relation

$(L \otimes 1)(1 \otimes R)\alpha_{x,y,x} = (1 \otimes R)(L \otimes 1)$

Here $\alpha$ is the associator. Don’t worry about the specific example: I’m just talking about *a presentation of a symmetric monoidal category* using generators and relations.

Right now Jason Erbele and I have proved that a certain symmetric monoidal category has a certain presentation. I defined what this meant myself. But this has got to be standard, right?

So whom do we cite?

You are likely to mention PROPs, and that’s okay if they get the job done. But I don’t actually know a reference on describing PROPs by generators and relations. Furthermore, our actual example is not a *strict* symmetric monoidal category. It’s equivalent to one, of course, but it would be nice to have a concept of `presentation’ that specified the symmetric monoidal category only up to equivalence, not isomorphism. In other words, this is a ultimately a 2-categorical concept, not a 1-categorical one.

If it weren’t for this, we could use the fact that PROPs are models of an algebraic theory. But our paper is actually about control theory—a branch of engineering—so I’d rather avoid showing off, if possible.

## Re: Describing PROPs Using Generators and Relations

Well, (non-strict) symmetric monoidal categories are the (strict) algebras for a 2-monad. So you could use the standard notion of “generators and relations” for algebras over a monad, i.e. a coequalizer of maps between free algebras. That would specify it up to isomorphism, though.

Your example suggests that maybe you’re thinking instead of a codescent object of maps between free algebras, where you would give generating objects, generating morphisms, and relations on the morphisms. Codescent objects (unlike coequalizers) are flexible colimits, so there the strict colimit is equivalent to the bicategorical one, and the latter is specified only up to isomorphism.