Talks
Product-like universal structures in double categories
Kyoto University, 8th October 2026
Limits in double categories were first explored by Grandis and Paré in their seminal paper ‘Limits in double categories’. However, there exist universal properties in double categories not captured by their perspective. For example, companions and conjoins of an arrow in a double category exhibit a universal property but are not 'double limits'. In this talk, we will begin to unfold a more nuanced story. By first generalising products in double categories, we will see more complicated universal structures. These universal structures will allow us to define, for example, a ‘double category of partial maps’ whose underlying bicategory is a bicategory of partial maps as defined by Carboni in 1987.
[Partial functions and other generalised coproduct completions](https://www.youtube.com/watch?v=xa04dVG4fEshttps://www.youtube.com/watch?v=xa04dVG4fEs)
Topos Colloquium, 1st September 2026 — slides
The category of sets and functions is the free coproduct completion of the terminal category. In a double-categorical setting, Evan Patterson defined a new notion of double (co)product and used it to prove that the double category of sets, functions and spans is the free coproduct completion of the terminal double category. In this talk we both generalise and refine this notion. First, we extend Patterson’s original definition to include (co)products in (co)virtual double categories, which can capture examples such as coproducts in Span(C) even when C does not have pullbacks. Second, we isolate finer classes of double (co)products, denoted (L,R)-(co)products, where L and R are certain classes of functions. As an application, we exhibit the double category of sets, functions and partial functions as a free (L,R)-coproduct completion of the terminal double category.
Last updated: November 2025