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Stream: learning: questions

Topic: Commutative Monoids and Functors from Finite Spans

Jade Master (May 27 2025 at 10:39):

In a seminar advertisement, I recently found the following claim:

Commutative monoids in a category with finite products can be
identified with product-preserving functors from spans of finite
sets.

Has anyone heard this or know the detail of how this works? Looks super interesting!

John Baez (May 27 2025 at 11:10):

The idea must be that the symmetric monoidal category of spans of finite sets is generated by spans from 2 to 1 (corresponding to multiplication in your monoid) and from 1 to 2 (corresponding to the diagonal map from your monoid to its square).

John Baez (May 27 2025 at 11:10):

Btw, I sent you an urgent email over the weekend.

Ralph Sarkis (May 27 2025 at 11:18):

You can get all the details in this paper on deconstructing Lawvere theories. It is very concisely stated in Example 6.2a, but to digest the story a bit:

• Generate a category $\mathsf{F}$ of string diagrams with generators $2 \to 1$ and $0 \to 1$ and the evident equations for commutative monoids. This is the prop of commutative monoids in the sense that symmetric monoidal functor $\mathsf{F} \rightarrow \mathbf{C}$ correspond to commutative monoids in $\mathbf{C}$.
• You can also recognize that $\mathsf{F}$ is the opposite of the category of finite sets and functions.
• A main theorem in that paper is that if you add generators $1 \to 2$ and $1 \to 0$ with equations for cocommutative comonoids plus a distributive law between the monoid and comonoid structures (i.e. you add a copy and delete map), you get the Lawvere theory of commutative monoids. (Theorem 6.1)
• Another result in that paper is that this Lawvere theory is obtained with a distributive law between props, and there you can see that the category obtained is the category of spans of finite sets. (Example 3.4b)

You can recognize John's comments in this.

Nathanael Arkor (May 27 2025 at 11:21):

Theorem 1.24 of Gambino and Kock's Polynomial functors and polynomial monads is also worth taking a look at.

John Baez (May 27 2025 at 11:22):

I'm more familiar with the prop for commutative monoids than what we're talking about here, the Lawvere theory for commutative monoids. The prop is just FinSet $^{\text{op}}$, since it lacks the map from 1 to 2.

James Deikun (May 27 2025 at 11:30):

See also https://arxiv.org/abs/1109.1598 ...