### Spans in 2-Categories: A Monoidal Tricategory

#### Posted by Alexander Hoffnung

A while back I posted a draft of a paper on the construction of a monoidal tricategory of spans. Since then the paper has changed quite a bit, and a new version appeared on the arXiv on Monday.

The following theorem is the main result of the paper:

Given a strict $2$-category $\mathcal{C}$ with finite pseudolimits, there is a monoidal tricategory $Span(\mathcal{C})$, consisting of:

- the same objects as $\mathcal{C}$,
- spans in $\mathcal{C}$ as $1$-morphisms,
- maps of spans in $\mathcal{C}$ as $2$-morphisms,
- maps of maps of spans in $\mathcal{C}$ as $3$-morphisms.

I usually think of spans as categorified linear operators, for example, the spans of groupoids in groupoidification. Also, spans, or more specifically, correspondences, are central in convolution operations in geometric representation theory. Here on the blog we have seen this in Ben-Zvi’s notes on geometric function theories. Of course, spans generalize relations between sets, so they show up in many other contexts as well.

This theorem has not changed from the previous version, although some changes to the structure of $Span(\mathcal{C})$ have been made. I want to comment on some of the changes here. In particular, I want to highlight the inclusion of Todd Trimble’s definition of a tetracategory.

The main changes are to the monoidal structure on the tricategory $Span(\mathcal{C})$. The structure in the previous version was too strict, so the new version corrects for this by specifying additional structure. Of course, this additional structure should satisfy coherence axioms as part of the monoidal structure. The result is a big change from the previous version in which the axioms were trivially satisfied.

In the paper, I define a monoidal tricategory as a one-object tetracategory. The coherence axioms to be satisfied are then Todd’s tetracategory axioms. It took me some time to understand the perturbations and other top-dimensional cells in the axioms well enough to write down the components of the span construction.

Todd was nice enough and patient enough to explain his method for writing down the tetracategory axioms to me. I have tried to pass along some of what Todd explained to me in this paper. Along with these “CliffsNotes” (apparently no longer “Cliff’s Notes”) for tetracategories, I have included the tetracategory definition itself, hopefully making it more accessible. If you have forgotten or have not seen it before, the handwritten definition, which currently lives on John Baez’s website, is 56 pages long!

Comments are very welcome.

## Re: Spans in 2-Categories: A monoidal tricategory

I’d like to say publicly how grateful I am to Alex for all the hard work he poured into this. Over the course of a few weeks, he asked me a bunch of really good questions about that document on JB’s website, and the result is, I think, a much more readable definition now that it’s passed through Alex’s hands. It’s the most in-depth conversation I’ve had about tetracategories with anyone.

I’m sure we’ll be having more discussions. I sure hope so!