Uploaded April 2025 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 3rd of April 2025.
———
I shall tell the tale of how I came to understand polynomials. In doing so, I will exhibit a novel universal property of the double category of polynomials in a locally cartesian closed category. In fact, we shall see how polynomials may be liberated from the assumption of exponentiability entirely. This talk is based on joint work with Bryce Clarke.
Topos Institute Colloquium, 3rd of April 2025.
———
I shall tell the tale of how I came to understand polynomials. In doing so, I will exhibit a novel universal property of the double category of polynomials in a locally cartesian closed category. In fact, we shall see how polynomials may be liberated from the assumption of exponentiability entirely. This talk is based on joint work with Bryce Clarke.
![[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.")
