Uploaded September 2025 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 4th of September 2025.
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What does it mean for a system to have "beliefs" and take "actions" in service of "goals"? Philosophy and cognitive science have offered a variety of answers to this question, including Dennett's intentional stance, the Bayesian brain hypothesis, and various forms of 'enactivism', among many others. I will talk about several closely related approaches to telling these stories, or simplified versions of them, in a mathematical way. Surprisingly (to me at least), doing so makes them seem much more compatible and closely related than they might at first appear.
Dennett's core idea is that the key question to ask about a system is not whether it is an agent or not, but whether its behaviour can be well described by treating it as one. This entails ascribing beliefs and goals to it, and predicting its behaviour by assuming it will take actions that would bring the goals about if the beliefs were true. For some systems this yields excellent predictions of their behaviour, and the behavioural patterns that enable this can be said to be a real property of the system.
Mathematically this suggests a paradigm where instead of starting with a problem and asking what systems can solve it, we start with a system and ask what problems it can solve. In doing this we are forced to grapple with questions such as what exactly constitutes a belief; how should beliefs change over time in response to new information; what happens if two different belief attributions give rise to the same behaviour; and where should we draw the boundary of a system, if we want to treat it as an agent? Ideas from categorical systems theory and category-theoretic probability are used throughout the work.
Topos Institute Colloquium, 4th of September 2025.
———
What does it mean for a system to have "beliefs" and take "actions" in service of "goals"? Philosophy and cognitive science have offered a variety of answers to this question, including Dennett's intentional stance, the Bayesian brain hypothesis, and various forms of 'enactivism', among many others. I will talk about several closely related approaches to telling these stories, or simplified versions of them, in a mathematical way. Surprisingly (to me at least), doing so makes them seem much more compatible and closely related than they might at first appear.
Dennett's core idea is that the key question to ask about a system is not whether it is an agent or not, but whether its behaviour can be well described by treating it as one. This entails ascribing beliefs and goals to it, and predicting its behaviour by assuming it will take actions that would bring the goals about if the beliefs were true. For some systems this yields excellent predictions of their behaviour, and the behavioural patterns that enable this can be said to be a real property of the system.
Mathematically this suggests a paradigm where instead of starting with a problem and asking what systems can solve it, we start with a system and ask what problems it can solve. In doing this we are forced to grapple with questions such as what exactly constitutes a belief; how should beliefs change over time in response to new information; what happens if two different belief attributions give rise to the same behaviour; and where should we draw the boundary of a system, if we want to treat it as an agent? Ideas from categorical systems theory and category-theoretic probability are used throughout the work.
![[Berkeley Seminar] Evan Patterson | How to Prove Equations Using Diagrams](https://i.ytimg.com/vi/3wLDkjocE2c/mqdefault.jpg "[Berkeley Seminar] Evan Patterson | How to Prove Equations Using Diagrams
Title: How to Prove Equations Using Diagrams: A Category Theorist Looks at E-Graphs
Abstract: First invented in 1980, e-graphs are receiving new attention as a powerful and adaptable data structure to keep the books for equational reasoning. In this talk on work in progress, we explore an operational semantics for e-graphs based on category-theory. We argue that classic concepts from category theory, particularly diagrams and initial functors, provide a conceptual foundation for the diagrammatic reasoning performed in e-graphs.
https://topos.institute/blog/2025-05-27-e-graphs-1/
Date: 2025-05-20
https://topos.institute/events/berkeley-seminar/")
![[Berkeley Seminar] David Jaz Myers | Categorical Algebra with Segal Conditions](https://i.ytimg.com/vi/52AsVTxHXU0/mqdefault.jpg "[Berkeley Seminar] David Jaz Myers | Categorical Algebra with Segal Conditions
Title: Categorical Algebra with Segal Conditions
Abstract: There are many ways to present algebraic structures categorically: monads, Lawvere theories, limit sketches, and more. In this talk, well learn about Yet Another Way to Present Algebra (YAWPA): Segal conditions and Chu and Haugsengs algebraic patterns. The basic idea behind algebraic patterns is that in many cases of categorical algebra we have a notion of *pasting diagram* which tells us how we are going to compose things. These pasting diagrams are made out of elementary features things like boxes and wires and can swallow other diagrams whole by putting subdiagrams into boxes. These two kinds of morphisms, diagram inclusions and swallowings, organize into a factorization system on a category of pasting diagrams. Presheaves on this category of diagrams which send any pasting diagram to the limit over all its elementary subdiagrams is an algebra composing according to those pasting diagrams; that single axiom is the so-called Segal condition. Well see how this works for categories and double categories in particular, and, time-permitting, see how to derive the notion of virtual double category from the algebraic pattern for categories and muse about the appropriate virtual triple categories which are similarly derived from the algebraic pattern for double categories.
https://topos.institute/events/berkeley-seminar/")
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Oxford Seminar, April 30 2026
Speaker: Olga Paris-Romanskevich
Full Title: Which results on gender and science every mathematician should
know
Abstract: Based on a selection of works in social psychology, sociology and cultural studies, I will discuss several epistemological approaches of how one can think the exclusion of women and gender minorities from sciences and mathematics in particular. In the discussion following the talk, we will collectively approach what can be done, at the individual and collective level, in order to prevent violence and exclusion from happening.")
![[Oxford Seminar] Owen Lynch | Markov Semigroups](https://i.ytimg.com/vi/5bBPz4GpBIw/mqdefault.jpg "[Oxford Seminar] Owen Lynch | Markov Semigroups
Oxford Seminar, 14th of February 2025")
![[DOTS Lectures] 15. Representability of reachability and double operad algebras](https://i.ytimg.com/vi/6UGVkOVvM5U/mqdefault.jpg "[DOTS Lectures] 15. Representability of reachability and double operad algebras
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")
![[2-torial] Jason tells Tim about Doctrinal Adjunctions](https://i.ytimg.com/vi/6oDQcA6dU8w/mqdefault.jpg "[2-torial] Jason tells Tim about Doctrinal Adjunctions
[2-torial] Jason tells Tim about Doctrinal Adjunctions")
![[Berkeley Seminar] B. Rousse | Heidegger, Skill, and the Conversational Structure of Human Work](https://i.ytimg.com/vi/6xwODsGadn4/mqdefault.jpg "[Berkeley Seminar] B. Rousse | Heidegger, Skill, and the Conversational Structure of Human Work
Title: Heidegger, Skill, and the Conversational Structure of Human Work
Abstract: AI is transforming how we think about work and intelligence. Many are claiming that AI systems will soon be able to do all the work that human beings do. This claim has a long and somewhat dubious history in the field of AI. To assess its plausibility, it helps to develop a deeper account of the nature of human work and agency. In this talk, I share work in progress aimed at this goal. I trace a tradition that runs from Heidegger’s ontology of human agency, to the theory of skilled activity articulated by Stuart and Hubert Dreyfus (the Dreyfus Skill Model), to the account of the conversational structure of human work developed by Terry Winograd and Fernando Flores in Understanding Computers and Cognition.
Date: 2025-04-29
https://topos.institute/events/berkeley-seminar/")
![[Oxford Seminar] Khyathi Komalan | There is No Band (AQFT and Double Categories)](https://i.ytimg.com/vi/7kMZ8P0hNJo/mqdefault.jpg "[Oxford Seminar] Khyathi Komalan | There is No Band (AQFT and Double Categories)
Oxford Seminar, 21st of August 2025
Title: There is No Band: Double Categories, Fragmented Spacetime, and an AQFT
Abstract: David Lynchs neo-noir classic Mulholland Drive challenges our sense of continuity, identity, and causality. In this talk, I use its fragmented narrative structure as a metaphorical guide to construct an algebraic quantum field theory using a double functor between free globularly generated double categories.
We define a double category of spacetime regions where vertical morphisms are inclusions, horizontal morphisms are causal maps, and squares ensure compatibility between the two. On the algebra side, we use Juan Orendains double category of von Neumann algebras and bimodules using a general construction from decorated bicategories. The result is a functor that captures how observables evolve and embed across spacetime in a way that traditional approaches don’t cleanly separate. We discuss how gluing and locality can be modelled in this construction.
Along the way, we use the storyline of Mulholland Drive to guide our intuition and structure the talk — from fragments of time and overlapping realities to a new way of understanding gluing and locality in field theory. No background in physics is assumed — just curiosity, and a willingness to follow a story where not everything is what it seems.")
