Uploaded May 2025 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 22nd of May 2025.
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Random variables are a central notion of probability theory, but they are defined with reference to an elusive sample space existing the background. Alex Simpson pioneered the use sheaves over sample spaces to make this dependence explicit. The resulting sheaf topos comes with a notion of random variable and thus probability built-in. It turns out that sample spaces can be generalized to a wide variety of probability-like phenomena, such as nondeterminism or fresh name generation. The probability sheaf construction hinges on a notion of independence structure where certain squares of sample spaces are distinguished as independent.
I will give a tour of these abstract samples and then discuss the rich interplay between sample spaces and couplings. These categories determine each other in a precise sense, where couplings behave like relations over sample spaces and compose via independent pullback.
Joint work with Chris Heunen, Matthew Di Meglio and Paolo Perrone
Topos Institute Colloquium, 22nd of May 2025.
———
Random variables are a central notion of probability theory, but they are defined with reference to an elusive sample space existing the background. Alex Simpson pioneered the use sheaves over sample spaces to make this dependence explicit. The resulting sheaf topos comes with a notion of random variable and thus probability built-in. It turns out that sample spaces can be generalized to a wide variety of probability-like phenomena, such as nondeterminism or fresh name generation. The probability sheaf construction hinges on a notion of independence structure where certain squares of sample spaces are distinguished as independent.
I will give a tour of these abstract samples and then discuss the rich interplay between sample spaces and couplings. These categories determine each other in a precise sense, where couplings behave like relations over sample spaces and compose via independent pullback.
Joint work with Chris Heunen, Matthew Di Meglio and Paolo Perrone
![[Oxford Seminar] Jason Brown | Wreaths in Span(Set)](https://i.ytimg.com/vi/tbnRgqf81_A/mqdefault.jpg "[Oxford Seminar] Jason Brown | Wreaths in Span(Set)
Oxford Seminar, 2nd of October 2025
Abstract:
Steve Lack and Ross Street introduced wreaths as a generalisation of distributive laws between monads in their paper The formal theory of Monads II. This paper presents a nice story about how wreaths arise from considering the free completion of a 2-category under Kleisli objects and provides some interesting examples of wreaths. In particular, it is shown that orthogonal factorisation systems on a category can be expressed as wreaths in Span(Set).
In this talk well try to understand general wreaths in Span(Set) as expressing a weaker notion of factorisation system on a category. Well relate these factorisation structures to both familial functors and crossed double categories, and also present examples of where they naturally occur.")
![[Berkeley Seminar] Michael Arntzenius | UC Berkeley](https://i.ytimg.com/vi/tj9n1m9dNuY/mqdefault.jpg "[Berkeley Seminar] Michael Arntzenius | UC Berkeley
Title: A type system for finitely supported functions via pointed sets, Part 1
ABSTRACT:
Finite maps, in the form of dictionaries, associative arrays, or tables, are a key data type in most language’s standard libraries, but constructing and manipulating them is generally very explicit and loopful. In this talk I’ll demonstrate work in progress on a type system that can guarantee that functions written directly as lambda-expressions are finitely supported, and therefore can be represented as tables. This yields a higher-level, more declarative syntax, similar to logic programming languages or database query languages.
The semantics of my language live in Set, the category of pointed sets and point-preserving maps. In order to define the support of a function f: A → B, one needs a point nil ∈ B; then support f = {x ∈ A : f x ≠ nil}. In fact, finitely supported maps form a graded monad on Set; they are graded by their domain A. To fully flesh out the semantics and type system, I also need the free/forgetful adjunction between Set and Set*.
Unfortunately the typing rules get quite complex. I’d like to know if there’s a simpler way to accomplish the same goals, or a simpler or more general framework for presenting the type system itself.
Paper preprint for the interested: https://www.rntz.net/files/finite-functional-programming.pdf
Date: March 31, 2026")
![[2-torial] Owen tells Tim about elaborators for type theories [1/2]](https://i.ytimg.com/vi/uBjuFDs-shw/mqdefault.jpg "[2-torial] Owen tells Tim about elaborators for type theories [1/2]
Accompanying code can be found on github: https://github.com/ToposInstitute/elaboratorial")
![[2-torial] Quantum information theory, Part 1: Some linear algebra](https://i.ytimg.com/vi/uQ7hL9aaPYA/mqdefault.jpg "[2-torial] Quantum information theory, Part 1: Some linear algebra
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] Paul Lang | Categorical Abstractions for Living Systems (CatALiSys)](https://i.ytimg.com/vi/udIhokeYh-I/mqdefault.jpg "[Berkeley Seminar] Paul Lang | Categorical Abstractions for Living Systems (CatALiSys)
Title: Categorical Abstractions for Living Systems (CatALiSys): A Framework for Collaborative, Comprehensive and Customizable Modeling of Biochemical Reaction Systems
Abstract: Modeling is fundamentally the process of choosing useful abstractions. Different researchers studying overlapping biochemical systems often choose different abstractions because they seek answers to different scientific questions. While these choices are individually well motivated, they make it difficult to integrate existing models into larger representations of cellular processes, posing a fundamental obstacle to the long-term vision of scientifically interpretable whole-cell models.
In biochemical reaction systems, such incompatibilities typically arise from mismatches in the granularity of chemical species and reactions, as well as differing assumptions about the surrounding environment (the “host cell”). Our preliminary work on CatALiSys aims to overcome these barriers to scalable modeling by enabling collaborative knowledge sharing, while preserving individual research interests through generation of customized models.
Catering to an interdisciplinary audience, this talk will provide a brief introduction to biochemical reaction systems, existing modeling formalisms and the categorical ideas underlying our approach.
Based on this, we demonstrate how symmetric monoidal indexed categories can formalize biochemical systems. In this framework, an indexing category explicitly represents the relationships between chemical species types and environmental contexts. This indexing category can be pruned to the desired scope and granularity. Finally, we showcase RuleTT, our preliminary type-theoretic Rust implementation. Using minimal working examples, we demonstrate how RuleTT handles knowledge updates, enables customization of scope and granularity, and helps to make implicit environmental assumptions explicit. We hope these ideas represent a next step toward the collaborative construction of scientifically interpretable and predictive virtual cells.
Date: July 21, 2026")
![[Oxford Seminar] Jana Nickel | Coend calculus in a compact closed virtual equipment](https://i.ytimg.com/vi/uwey3deuN7E/mqdefault.jpg "[Oxford Seminar] Jana Nickel | Coend calculus in a compact closed virtual equipment
Oxford Seminar, June 29 2026
You can view the listing for this talk online at https://topos.institute/events/oxford-seminar/talks/2026-06-29_nickel_coend.html
Speaker: Jana Nickel
Full Title: Coend calculus in a compact closed virtual equipment
Abstract: The talk is based on a project joint with Nathanael Arkor. We lay the foundations for a coend calculus in the framework of virtual double categories. To this end, we construct the concept of a /compact closed virtual equipment/ (mathbb{X}) and introduce the notion of a coend for each tight arrow (fcolonthinspace Xotimes A^circotimes Aotimes Yto C) in (mathbb{X}). The goal is to generalize the coend calculus for locally internal categories due to Betti and Walters (R. Betti and R.F.C. Walters, 1989) to compact closed virtual equipments and recover some prominent results, such as the Fubini theorem for coends. To cite an example, we will consider the virtual equipment (mathbb{S}textsf{pan}(mathcal{E})) of spans in an ordinary category (mathcal{E}) with pullbacks and describe a compact closure on it.
R. Betti and R.F.C. Walters (1989). /The calculus of ends over a base topos/, Journal of Pure and Applied Algebra.")
![[Oxford Seminar] Amitai Nachmany | Hyperdoctrines in DOTS](https://i.ytimg.com/vi/ve6Xotcuxbs/mqdefault.jpg "[Oxford Seminar] Amitai Nachmany | Hyperdoctrines in DOTS
Oxford Seminar, August 27 2026
You can view the listing for this talk online at https://topos.institute/events/oxford-seminar/talks/2026-08-27_nachmany_hyperdoctrine.html
Speaker: Amitai Nachmany
Full Title: Hyperdoctrines in DOTS
Abstract: In this talk we motivate and define (regular) hyperdoctrines and sketch how to view them as double-operadic systems theories. We relate this to the notion of /systems with certificates/, which allows us to formally verify a systems theorys properties in a way that is reminiscent of the internal language of a category. Finally, we sketch a program to generalise this procedure to other logics (and functorialise it), and express some of the benefits of this program for strengthening the toolbox of categorical systems theory.
*Assumed knowledge:* General category theory, especially monoidal categories, double categories and adjunctions. Familiarity with functorial semantics and/or DOTS would help but isnt necessary.")
![[Berkeley Seminar] Owen Lynch | Stateful Lenses](https://i.ytimg.com/vi/veQbMfPSi9M/mqdefault.jpg "[Berkeley Seminar] Owen Lynch | Stateful Lenses
Title: Stateful Lenses - A Recipe for Expressive Systems-Theoretic Cartesian Double Categories
Abstract: This is a talk on work-in-progress that was started with David Spivak, and this abstract should be seen as well-motivated conjecture rather than math that has been completely worked out. Double categorical systems theory tells us how to unify various types of system under a single heading: operad algebras of symmetric monoidal double categories. Both resource sharers being composed by undirected wiring diagrams and Moore machines being composed by directed wiring diagrams are examples of double categorical systems theories. However, it turns out that both resource sharers and Moore machines can be found in the same (cartesian!) double category: resource sharers make up the horizontal morphisms into the (vertically) terminal object, and Moore machines make up the horizontal morphisms out of the (vertically) terminal object. Even better, variants of this construction produce both discrete and continuous resource sharers/Moore machines. Another special case of these stateful lenses include the energy-driven open systems from Spivak, Capucci, and _s Organizing Physics paper. Finally, the fact that stateful lenses form a cartesian double category dramatically reduces the amount of structure in formulating them compared to the double operad/operad algebra perspective on Moore machines/resource sharers.
https://topos.site/events/berkeley-seminar/")
