The n-Category Café

It looks like Brendan Fong has been volunteered to make videos of my talks — so with luck, people who want to see the talks can cut their carbon footprint and do it that way. I’ll definitely make my slides available. And I plan to blog about my paper with Tobias. In fact I already have, on Azimuth:

• Relative entropy: Part 1, Part 2 and Part 3.

It actually makes sense to read these backwards, starting from Part 3, which states our main result.

People interested in control theory and categories can already see slides for the talk on that topic I gave here in Erlangen:

• Categories in control.

People interested in chemical reaction networks, Petri nets and Feynman diagrams can read this book:

• John Baez and Jacob Biamonte, A Course on Quantum Techniques for Stochastic Mechanics.

And people curious about the phylogenetic operad can read this:

• Operads and the tree of life.

So, there’s already a lot of gunk out there, but in my talks I plan to synthesize it, and I’ll make a webpage with the talks and links to all these references.

By the way, in this talk I mistakenly said that relative entropy gave a continuous functor; in fact it’s just lower semicontinuous. I’ve fixed this now in the slides.

The last part of my talk was my own interpretation of Brendan Fong’s master’s thesis:

• Brendan Fong, Causal Theories: a Categorical Perspective on Bayesian Networks.

I took a slightly different approach, by saying that a causal theory $\mathcal{C}_G$ is the free category with products on certain objects and morphisms coming from a directed acyclic graph $G$. In his thesis he said $\mathcal{C}_G$ was the free symmetric monoidal category where each generating object is equipped with a cocommutative comonoid structure. This is close to a category with finite products, though perhaps not quite the same: a symmetric monoidal category where every object is equipped with a cocommutative comonoid structure in a natural way (i.e., making a bunch of squares commute) is a category with finite products. However, I believe this tiny difference makes no difference for what he does, because he is mainly concerned with symmetric monoidal functors from $\mathcal{C}_G$ into a category $\mathcal{D}$ with finite products, and here the cocommutative comonoid structure on any object of $\mathcal{C}_G$ is inevitably mapped to the unique cocommutative comonoid structure on the corresponding object of $\mathcal{D}$, which is automatically natural.

By making this slight change, I am claiming that causal theories can be seen as algebraic theories in the sense of Lawvere. This would be a very good thing, since we know a lot about those.

the last talk is on YouTube: Network Theory III.

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https://www.youtube.com/watch?v=qX8fSYu7ors