Uploaded September 2025 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 25th of September 2025.
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We define a bicategory 𝟚TDX whose 1-cells provide a categorification of transducers, computational devices extending finite-state automata with output capabilities. This bicategory is a mathematically interesting object: among many equivalent characterizations, given an input category 𝒜 and an output category ℬ, its 1-cells (𝒬, t) : 𝒜 ⇝ ℬ can be seen as “families of profunctors over 𝒬, indexed over 𝒜, and enriched over (a universal monoidal category obtained from) ℬ”; more precisely, 2-transducers are functors of type
t : 𝒜 × 𝒬ᵒᵖ × 𝒬 × (ℬ* )ᵒᵖ ⟶ Set
where ℬ* denotes the free monoidal category over ℬ. Extending t to 𝒜*, in the obvious way, for each “word” 𝑎̲ in 𝒜* we obtain an endoprofunctor over a category 𝒬 of “states” and enriched in presheaves over the free monoidal category ℬ*.
We discuss a number of other characterizations of 𝟚TDX(𝒜, ℬ) in detail; we establish a Kleisli-like universal property for 𝟚TDX(𝒜, ℬ) and explore the connection of 𝟚TDX to other bicategories of computational models, such as Walters’ “bicategory of circuits”. It is convenient to regard 𝟚TDX as the loose bicategory of a double category 𝔻TDX; the bicategory (resp. double category) of profunctors is naturally contained in the bicategory (resp. double category) 𝟚TDX (resp. 𝔻TDX); we study the completeness and cocompleteness properties of 𝔻TDX, as well as monads, adjunctions, and other structures/properties that naturally arise from the definition.
Topos Institute Colloquium, 25th of September 2025.
———
We define a bicategory 𝟚TDX whose 1-cells provide a categorification of transducers, computational devices extending finite-state automata with output capabilities. This bicategory is a mathematically interesting object: among many equivalent characterizations, given an input category 𝒜 and an output category ℬ, its 1-cells (𝒬, t) : 𝒜 ⇝ ℬ can be seen as “families of profunctors over 𝒬, indexed over 𝒜, and enriched over (a universal monoidal category obtained from) ℬ”; more precisely, 2-transducers are functors of type
t : 𝒜 × 𝒬ᵒᵖ × 𝒬 × (ℬ* )ᵒᵖ ⟶ Set
where ℬ* denotes the free monoidal category over ℬ. Extending t to 𝒜*, in the obvious way, for each “word” 𝑎̲ in 𝒜* we obtain an endoprofunctor over a category 𝒬 of “states” and enriched in presheaves over the free monoidal category ℬ*.
We discuss a number of other characterizations of 𝟚TDX(𝒜, ℬ) in detail; we establish a Kleisli-like universal property for 𝟚TDX(𝒜, ℬ) and explore the connection of 𝟚TDX to other bicategories of computational models, such as Walters’ “bicategory of circuits”. It is convenient to regard 𝟚TDX as the loose bicategory of a double category 𝔻TDX; the bicategory (resp. double category) of profunctors is naturally contained in the bicategory (resp. double category) 𝟚TDX (resp. 𝔻TDX); we study the completeness and cocompleteness properties of 𝔻TDX, as well as monads, adjunctions, and other structures/properties that naturally arise from the definition.
![[Oxford Seminar] Owen Lynch | An introduction to the geolog project](https://i.ytimg.com/vi/d6xMeIxsCM4/mqdefault.jpg "[Oxford Seminar] Owen Lynch | An introduction to the geolog project
Oxford Seminar, April 2 2026
Speaker: Owen Lynch
Full Title: An introduction to the geolog project
Abstract: Within the ARIA program, a number of researchers are working on a project called geolog. Geolog has some ambitious goals and some controversial theses; in this talk we will give an overview of these goals and theses alongside an intro to some of the math and computer science that we are using to attempt to achieve the goals and confirm the theses.")
![[2-torial] Categorical algebraic geometry, Part 2](https://i.ytimg.com/vi/dZKDEhOhz8E/mqdefault.jpg "[2-torial] Categorical algebraic geometry, Part 2
2-torial, July 28 2026
You can find accompanying notes and other information on this video at https://topos.institute/work/lectures/2-torials/2026-07-28_hosgood_geometry/
Speaker: Tim Hosgood
There are many approaches to algebraic geometry, and many different ways to arrive at the subject. Here were going to take an incredibly specific and biased approach: what if you already love 2-categories and want to be able to say the phrase fpqc sheaf as quickly as possible, but not /too/ quickly? In this series of exercises we will build towards an understanding of *relative algebraic geometry*, which allows us to work in arbitrary (nice) symmetric monoidal categories.")
![[DOTS Lectures] 9. Behavioural modules of systems](https://i.ytimg.com/vi/de20OAaAqW0/mqdefault.jpg "[DOTS Lectures] 9. Behavioural modules of systems
Part of a lecture series on the Double Operadic Theory of Systems (DOTS) presented by David Jaz Myers.
Some material from these lectures can be found in Davids book on categorical systems theory:
https://www.davidjaz.com/Papers/DynamicalBook.pdf")
![[Oxford Seminar] Elizabeth Amelia | AI for precision oncology](https://i.ytimg.com/vi/eDZuVRE-XVU/mqdefault.jpg "[Oxford Seminar] Elizabeth Amelia | AI for precision oncology
Oxford Seminar, 29th of May 2025")
![[TopOx] Jakub Opršal: Homotopy theory in the complexity of homomorphism problems](https://i.ytimg.com/vi/fU2pZbrE2aA/mqdefault.jpg "[TopOx] Jakub Opršal: Homotopy theory in the complexity of homomorphism problems
27th of November 2025. Slides available at https://topos.institute/events/topox/
I will talk about an emerging connection between homotopy theory and computational complexity of discrete problems. I will outline a theorem stating that contractibility is necessary for tractability (assumin P ≠ NP) in the realm of finite-template constraint satisfaction problems (CSPs).
There are many ways the CSP can be formulated. One of them is as a homomorphism problem: given two relational structures A and B, decide whether there is a homomorphism from A to B. We usually study a restricted version of this problem where B is fixed, e.g., if B is the k-clique graph K_k, the problem is the same as k-colouring of a given graph A. If B is finite (and of finite signature), such a problem is called finite-template. Famously, finite-template CSPs exhibit a P vs NP-complete dichotomy as proved independently by Bulatov and Zhuk in 2017.
The main theorem of the talk states a sufficient condition for NP-completeness in terms of the topology of ‘solution spaces’ and provides both all necessary hardness for the Bulatov–Zhuk dichotomy and also a new proof of an earlier Hell–Nešetřil dichotomy of graph homomorphism problems.")
![[Oxford Seminar] Khyathi Komalan | Adult Brainrot: Mandela Effect, Misinformation & Conspiracies](https://i.ytimg.com/vi/gk0jJtjNQGI/mqdefault.jpg "[Oxford Seminar] Khyathi Komalan | Adult Brainrot: Mandela Effect, Misinformation & Conspiracies
Oxford Seminar, 25th of September 2025
Abstract: The internet is a strange place — one moment youre
convinced the Starbucks logo never had a star on top, the next youre
knee-deep in a rumor that spreads everywhere at once, and before long
youre staring at a series of pictures that prove the moon landing
was fake. These phenomena may look like pure nonsense, but they also
show how perception, memory, and belief are shaped by our
experiences, the hidden biases behind what we perceive, and the
difficulty of stitching multiple pieces of information, sometimes
conflicting, into a coherent story that fits the most into ones
worldview.
In this talk I introduce adult brainrot as a categorical framework
for these inconsistencies. I will show how categorical models of
quantum processes can capture order-sensitive memory and
indistinguishable sources of misinformation, and how categorical
notions of centers help explain why people’s theories sometimes fail
to form a consistent picture of reality. The aim is to argue that
quantum categorical structures offer a natural mathematical framework
for modeling the inconsistencies and oddities of human perception and
belief.")
![[2-torial] Quantum information theory, Part 4: Quantum bird watching (continued)](https://i.ytimg.com/vi/gl2Ni8ksGsc/mqdefault.jpg "[2-torial] Quantum information theory, Part 4: Quantum bird watching (continued)
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.")
