Uploaded January 2025 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 30th of January 2025.
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The passage from a monad (A,S) to its category of algebras (resp. category of free algebras) can be seen as a V = Cat weighted limit (resp. colimit) construction [1]. The colimit case also has a description involving maps of the form X to SY and the so-called Kleisli composition.
When we move to the two-dimensional setting, the 2-category of pseudoalgebras can be seen as a V= Gray enriched weighted limit [2], but neither of the familiar descriptions of the Kleisli category categorify to give a weighted colimit [3]. We give a third, less well-known description of the Kleisli category which does categorify to the pseudomonad setting to give a weighted colimit. We show that comparisons induced by pseudoadjunctions splitting the pseudomonad are biequivalences if and only if their left pseudoadjoints are biessentially surjective on objects. This allows the more familiar Kleisli constructions for pseudomonads to be seen as tricategorical colimits, and for the development of the formal theory of pseudomonads pt. 2.
Topos Institute Colloquium, 30th of January 2025.
———
The passage from a monad (A,S) to its category of algebras (resp. category of free algebras) can be seen as a V = Cat weighted limit (resp. colimit) construction [1]. The colimit case also has a description involving maps of the form X to SY and the so-called Kleisli composition.
When we move to the two-dimensional setting, the 2-category of pseudoalgebras can be seen as a V= Gray enriched weighted limit [2], but neither of the familiar descriptions of the Kleisli category categorify to give a weighted colimit [3]. We give a third, less well-known description of the Kleisli category which does categorify to the pseudomonad setting to give a weighted colimit. We show that comparisons induced by pseudoadjunctions splitting the pseudomonad are biequivalences if and only if their left pseudoadjoints are biessentially surjective on objects. This allows the more familiar Kleisli constructions for pseudomonads to be seen as tricategorical colimits, and for the development of the formal theory of pseudomonads pt. 2.
![[Berkeley Seminar] Ea Thompson | A Characterization of Pro-representable Virtual Double Categories](https://i.ytimg.com/vi/CLwKSyVtjLc/mqdefault.jpg "[Berkeley Seminar] Ea Thompson | A Characterization of Pro-representable Virtual Double Categories
Title: A Characterization of Pro-representable Virtual Double Categories
Abstract: Virtual double categories have been shown to provide an effective framework for formal category theory, from characterizations of adjoints and liftings to descriptions of pointwise Kan extensions and Yoneda structures. In studying virtual double categories themselves, an interesting question comes from asking what kind of structure virtual double categories are enriched over. In this talk, we take a first step in answering this question by characterizing the exponentiable, or pro-representable, virtual double categories. We will provide examples of pro-representable virtual double categories, including pseudo double categories and cospan virtual double categories, while also providing a simple counter-example to the claim that all virtual double categories are exponentiable. If there is time, we will discuss another internal-hom candidate coming from Libkind, Myers, Carlson, and Brown’s theory of loose bimodules.")
![[2-torial] Quantum information theory, Part 2: Quantum theory isnt real](https://i.ytimg.com/vi/EacdwqbOzlY/mqdefault.jpg "[2-torial] Quantum information theory, Part 2: Quantum theory isnt real
2-torial, May 20 2026
You can find accompanying notes and other information on this video at https://topos.institute/work/lectures/2-torials/2026-05-20_hosgood_quantum/
Tutor: Tim Hosgood
Tutee: Jason Brown
The elements of quantum information theory are not so complicated: some linear algebra over the complex numbers and a bit of classical probability theory. But it doesnt take long before you run into some intriguing problems that showcase how established experimental results contradict our classical intuition. In this 2-torial we will learn the very basics of quantum theory, see how a Mach–Zehnder interferometer gives a real-world implementation of a classically impossible construction and suggests that we should go all in on complex numbers, and finally how this all applies to the world of theoretical bird watching via the Quantum Zeno effect.")
![[Berkeley Seminar] Sophie Libkind | Two takeaways from ACT2Infer](https://i.ytimg.com/vi/EcEJQtUAYMs/mqdefault.jpg "[Berkeley Seminar] Sophie Libkind | Two takeaways from ACT2Infer
Title: Two takeaways from ACT2Infer
Abstract: In September, David Spivak and I hosted the ACT2Infer workshop which brought together experts in applied category theory and active inference together in order to begin the process of articulating and clarifying the underlying principles of active inference via the formalism of category theory. In this two-part talk Ill (1) share the logistics of hosting this workshop and (2) share my main conceptual takeaway a polynomial functor account of active inference.
https://topos.site/events/berkeley-seminar/")
![[Berkeley Seminar] Kevin Carlson | Free cocompletions made friendly?](https://i.ytimg.com/vi/EwtMivSKIYE/mqdefault.jpg "[Berkeley Seminar] Kevin Carlson | Free cocompletions made friendly?
Title: Free cocompletions made friendly?
Abstract: Lots of us are familiar with computations in Poly, some of which are made really nice by the fact that Poly is the category of families in Set^op, more formally, the free cocompletion of Set^op under coproducts. Free cocompletions under all colimits are also really important; theyre presheaf categories! However, presheaf categories are kind of annoying to actually calculate colimits in (coequalizers of sets are scary), in contrast to how trivial it is to calculate coproducts of polynomials. Free cocompletions are supposed to be closing a category up under formal colimits in some sense, so shouldnt taking the colimits also be formal? Actually, it can be! There is a category structure on the class of all small diagrams in C that models the free cocompletion of C in an extremely elementary way with lots of nice properties. This has been known to a few people, such as Andrée Ehresmann, for decades, and in the case of free cocompletions under filtered colimits to a lot more people, such as Grothendieck, but has only been known to me for a couple of weeks. Id like to make it known to you too. Im hoping this will be helpful both for generalizations of Poly beyond mere sets of positions and for better expressing data migrations, which is all about writing down functors from a category into a free cocompletion.
https://topos.site/events/berkeley-seminar/")
![[Berkeley Seminar] Keri DAngelo | Composing Instantaneous Machines](https://i.ytimg.com/vi/FevtvBgF1RU/mqdefault.jpg "[Berkeley Seminar] Keri DAngelo | Composing Instantaneous Machines
Title: Composing Instantaneous Machines
Abstract: In this talk, I’ll discuss recent progress Sophie and I have made on composing instantaneous machines. Instantaneous machines means that at any point in time, input can be given to the machine and the machine will give output based on this input and its current state. In this talk, I’ll show how we can compose such machines. We first create an extended category of directed wiring diagrams accounting for the dependency between input and output, and then define an operad algebra giving us the semantics that defines composition. Depending on where the interest lies, we can delve into some details including that since the output now depends on the input, we come into the “problem” that every time the output changes, the input may also change. This can be accounted for by giving a fixed point that shows after finite time, our input and output will stabilize.
https://topos.site/events/berkeley-seminar/")
