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Stream: theory: mathematics

Topic: Cospans of finite sets

John Baez (Aug 11 2025 at 16:59):

In work on double categorical systems theory, the symmetric monoidal double category $\mathbb{C}\mathbf{sp}$ plays an important role. This has

• finite sets as objects
• functions between finite sets as tight morphisms
• cospans of finite sets as loose morphisms
• maps of cospans as 2-morphisms

Disjoint union (that is, coproduct) provides the symmetric monoidal structure.

I'm wondering if someone has worked out the universal property of this symmetric monoidal double category.

I conjecture that $\mathbb{C}\mathbf{sp}$ is the free symmetric monoidal double category on a 'special commutative Frobeniua monoid' - or probably better, the categorified version of that, which I'd call a 'special symmetric Frobenius pseudomonoid'.

The reason is this. If we water down $\mathbb{C}\mathbf{sp}$ to get a symmetric monoidal category, its universal property is known! To do this, take $\mathbb{C}\mathbf{sp}$ and form its 'loose bicategory': the bicategory of objects, loose morphisms and 2-morphisms for which the source and target tight morphisms are identities. Then decategorify this bicategory and get a category $\mathsf{Csp}$.

Steve Lack showed $\mathsf{Csp}$ is the free symmetric monoidal category on a special commutative Frobenius monoid. This is the reason for my conjecture.

It may be less stressful to characterize $\mathbb{C}\mathbf{sp}$ as a double category internal to the category of cocartesian categories - @Evan Patterson likes that viewpoint.

Simon Burton (Aug 11 2025 at 17:20):

Would any of this work if you forgot the tight morphisms and asked about the monoidal bicategory of spans on finite sets?

Kevin Carlson (Aug 11 2025 at 17:21):

I can't find that result in Lack's paper. Is it in there under some different terminology?

Nathanael Arkor (Aug 11 2025 at 17:55):

I think it's §5.4.

John Baez (Aug 11 2025 at 19:50):

Simon Burton said:

Would any of this work if you forgot the tight morphisms and asked about the monoidal bicategory of spans on finite sets?

It would probably work more easily, at least if we treat that as a symmetric monoidal bicategory. (We need the macrocosm to be symmetric, or at least close, to talk about a symmetric pseudomonoid in it.)

The part I'm really worried about is the tight morphisms. The whole question of how they interact with the other stuff may be under-explored, and it may be tricky.

John Baez (Aug 11 2025 at 19:53):

Nathanael Arkor said:

I think it's §5.4.

Yes. That says $\mathsf{Csp}$ is the prop for special commutative Frobenius monoids - or as Lack puts it, commutative separable algebras. That's another way of saying $\mathsf{Csp}$ is the free prop on a special commutative Frobenius monoid.

John Baez (Aug 11 2025 at 20:05):

Hmm, there's a paper on Frobenius pseudomonoids in the bicategory of spans of finite sets:

• Ivan Contreras, Rajan Amit Mehta and Walker H. Stern, Frobenius and commutative pseudomonoids in the bicategory of spans.

However, this may not help me with my current question.

Nathanael Arkor (Aug 11 2025 at 20:07):

It is worth noting that $\mathbb C\mathbf{sp}$ has a very nice existing universal property: it is the free (virtual) double category on $\mathbf{FinSet}$ with companions and conjoints. On the face of it, this seems quite different from Lack's characterisation, but I think it is not so different after all. Lack constructs the category by "composing" $\mathbf{FinSet}$ with $\mathbf{FinSet}^{\text{op}}$. These are respectively the underlying category of the double categories of squares on $\mathbf{FinSet}$, and its loose opposite, which are respectively the free companion completion and free conjoint completion of $\mathbf{FinSet}$. Lack's distributive law corresponds exactly to a pseudodistributive law between these two double categories, viewed as pseudomonads in the pseudo double category $\mathbb S\mathbf{pan}(\mathbf{Cat})$ of spans of categories.

Nathanael Arkor (Aug 11 2025 at 20:09):

In fact, this is precisely how @Bryce Clarke and I construct the double category of spans in our work on polynomial functors. The paper isn't out yet, but I gave a talk on this at the Topos Institute seminar a few months ago: https://www.youtube.com/watch?v=tCbRfjv6JQ4

John Baez (Aug 11 2025 at 20:10):

Nice! What's a 'covirtual' double category (a rough indication will suffice), and can I think of that adjective as a technical nuance and say that at least morally $\mathbb{C}\mathbf{sp}$ is the free equipment (= double category with companions and conjoints) on the category $\mathsf{FinSet}$?

Nathanael Arkor (Aug 11 2025 at 20:13):

A virtual double category is like a double category, except that we have multiary cells like so, and do not impose the existence of composites of loose morphisms.
image.png

Since $\mathbf{FinSet}$ has pushouts, you can indeed ignore the prefix "virtual" here: the important thing to note is that the universal property is with respect to normal lax functors of double categories, rather than pseudo functors or strict functors.

Nathanael Arkor (Aug 11 2025 at 20:14):

So there is an equivalence of categories between (1) the category of normal lax functors from $\mathbb C\mathbf{sp}$ to a double category $\mathbb D$ with companions and conjoints; and (2) the category of functors from $\mathbf{FinSet}$ to the underlying category of objects and tight morphisms in $\mathbb D$.

Nathanael Arkor (Aug 11 2025 at 20:16):

This universal property is in the literature: it appears as (the dual of) Theorem 3.15 of Dawson–Paré–Pronk's The span construction.

Nathanael Arkor (Aug 11 2025 at 20:16):

(What doesn't yet appear in the literature is the construction of $\mathbb C\mathbf{sp}$ as a pseudodistributive law, which is what is needed to make the connection to Lack's characterisation.)

Nathanael Arkor (Aug 11 2025 at 20:18):

(I fixed a mix-up with virtual/covirtual, because we are talking about cospans rather than spans.)

John Baez (Aug 11 2025 at 21:45):

Oh, good - I do have some understanding of virtual categories, but I'd never heard of 'covirtual' ones.

Since $\mathbf{FinSet}$ has pushouts, you can indeed ignore the prefix "virtual" here.

Even better!

John Baez (Aug 11 2025 at 21:58):

Nathanael Arkor said:

So there is an equivalence of categories between (1) the category of normal lax functors from $\mathbb C\mathbf{sp}$ to a double category $\mathbb D$ with companions and conjoints; and (2) the category of functors from $\mathbf{FinSet}$ to the underlying category of objects and tight morphisms in $\mathbb D$.

Wow, this is very interesting. I think I have a decent intuition for why this is true (though don't ask me to prove it).

John Baez (Aug 11 2025 at 22:23):

Nathanael Arkor said:

(What doesn't yet appear in the literature is the construction of $\mathbb C\mathbf{sp}$ as a pseudodistributive law, which is what is needed to make the connection to Lack's characterisation.)

Okay, so it seems there's still room for me to pose an interesting problem in the paper I'm writing.

Kevin Carlson (Aug 12 2025 at 19:47):

John Baez said:

Oh, good - I do have some understanding of virtual categories, but I'd never heard of 'covirtual' ones.

Covirtual double guys appear pretty naturally for spans in a category that might not have pullbacks. You can define a cell from $x_1\leftarrow x_2\to x_3$ to $y_1\leftarrow y_2\to y_3\leftarrow y_4\to y_5$ without needing pullbacks: you have maps $x_1\to y_1$ and $x_3\to y_5$ as well as maps $x_2\to y_2,x_2\to y_4$ which all together make three squares commute. You can't even define a virtual double category of spans if pullbacks don't exist.