The n-Category Café

Fantastic! I’m looking forward to it. And thank you for timing it so that it’s accessible to lots of us around the world.

The link to the seminar web page (https://sites.google.com/ucr.edu/actucr/) doesn’t work for me. When I click it, I get taken to a Google login page.

The Zulip page requires an invitation; is there a way to make the page open to all visitors?

In case you haven’t already heard of it, Zoombombing is now a phenomenon that everyone should be aware of (and prepared to deal with) if hosting online events with public links.

The first two talks will be:

• Wednesday April 1st, John Baez: Structured cospans and double categories.

Abstract. One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce “structured cospans” as a way to study networks with inputs and outputs. Given a functor $L \colon A \to X$, a structured cospan is a diagram in $X$ of the form $$L(a) \to x \leftarrow L(b).$$ If $A$ and $X$ have finite colimits and $L$ is a left adjoint, we obtain a symmetric monoidal category whose objects are those of $A$ and whose morphisms are certain equivalence classes of structured cospans. However, this arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans, not equivalence classes thereof. We explain the mathematics and illustrate it with an example from chemistry.

• Wednesday April 8th, Prakash Panangaden: A categorical view of conditional expectation.

Abstract. This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones — which are abstract versions of the familiar cones in vector spaces — of measures and related categories cones of $L_p$ functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

Will recordings of the talks be available?