## 2 Content

We gathered together things that people wrote at the workshop into Oxford-Topos Meeting 2024 - Outcomes. Some people ended up writing a lot, others none at all. I ended up not writing very much because I was hovering around helping people install forester. Hopefully in future events, everyone will have forester installed and be used to forester syntax before designated writing times.

So my dream of having all of the discussions captured on paper for those who weren’t present didn’t quite materialize. But fortunately I can talk a little bit now about some of the topics of the discussions that I participated in.

### 2.1 Port-Hamiltonian Systems

I ended up in two discussions on port-Hamiltonian systems. Both of these discussions were somewhat one-sided, in that they mostly consisted of me explaining what I did in my masters thesis. I want to emphasize that I did not set out for this meeting to consist of me shilling for my own work, but it seemed like people were interested and enjoyed learning about it.

However, I was especially pleased that after I went through some of the big gaps in my thesis which had to do with my lack of knowledge of differential geometry: Paolo Perrone was kind enough to teach me some intuition about integrable forms. Specifically, the kernel of an integrable 1-form is the tangent bundle of a codimension-1 foliation. He told me to imagine this like sedimentary rocks: the manifold is divided up into layers, and the kernel of consists of directions that travel along a single layer.

Then, as far as I understand it, the idea Paolo was proposing was to replace the relations that I use in my thesis with something like forms which vanish on the relations. The problem that I was running into in my thesis is that, when thought of as relations, linear subbundles of vector bundles don’t necessarily compose because of constant-rank issues. Perhaps moving to forms would allow me to talk about non-constant-rank linear subbundles? Anyways, I’m excited to investigate this direction, and not having a good intuition for integrable forms was something that had bothered me for a while so I was happy to learn about that.

### 2.2 Stochastic Behaviour

Another group I participated in tackled the problem of stochastic behavior of dynamical systems. There is a good story for “representable behaviors” within categorical systems theory, but it was unknown how to generalize this to talk about behaviors of a Markov chain.

We were able to come up with a definition for representable stochastic behavior which mimicked the classical notion of “a stochastic process adapted to a filtration” using some techiniques from quasi-Borel spaces. I wrote up some preliminary notes on this here, but that does not capture where we ended up going on this topic, and hopefully there may end up being a paper on this.

Funnily enough, our group was originally interested in trying to make a categorical systems theory for stochastic differential equations, but we ended up getting sidetracked after we slogged through an hour of half-remembering functional analysis. There were some promising directions here that I hope we circle back around to though.

### 2.3 Double Operads

I was not in the group that discussed double operads, and in fact to a certain extent, I don’t think it was a group, it was a one-man show of Kevin Arlin sitting down and grinding out higher category theory, and the result are here: Double operads.

This was especially cool because in David Jaz’s opening talk he said that he’s wanted a good definition for double operad for years.

I think the lesson from this is that sometimes it’s OK to have a group of 1! Working with other people can spark ideas that it is best to work out individually.

### 2.4 Combinatorial Type Theory

It was a lucky coincidence that Thorsten Altenkirch happened to have been scheduled to give a talk during the meeting, because I learned about the concept of observational type theory and higher observational type theory from this talk.

Or rather, what really happened is that Thorsten gave a talk, and then later on, David Jaz explained to me why it was really cool.

As far as I can understand, the idea of higher observational type theory is that each type constructor in a type theory (i.e. sigma, pi, etc.) should come along with a definitional equality for what the equality type on that type is equivalent to. For instance, equality for the universe type should be definitionally equal to isomorphism, so univalence becomes definitional instead of propositional.

This seems really cool to me, because it is exactly what I want for combinatorial type theory. Namely, if I write down a combinatorial type, I want to automatically compute a definition for identifications between two elements of that combinatorial type: I want to automatically derive from the definition of a graph a definition of graph isomorphism!

I also want to take this one step further: from the definition of a graph, I want to automatically derive a notion of edit of a graph!

Unfortunately, it seems like there hasn’t been much published on Higher Observational Type Theory yet: it’s being kept somewhat under wraps as it develops.

I think I only really need a fragment of the full power of Higher Observational Type Theory to do what I want, so David Jaz and I discussed some ways of doing HOTT “on the cheap”.

But these were not all of the topics discussed at the meeting: these are only the topics that I know about and participated in! I encourage anyone who attended the meeting but didn’t get much of a chance to write during the meeting to write up thoughts while the thoughts are still fresh, and if they feel comfortable, share those thoughts!