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The Topos Institute Colloquium is an expository virtual talk series on topics relevant to the Topos mission and community. We usually meet over Zoom on Thursdays at 17:00 UTC, but this may vary depending on the speaker's time-zone. Talks are recorded, and remain available on our YouTube channel.
If you wish to be subscribed to the mailing list, then simply send an email to seminars+subscribe@topos.institute. For any queries about the colloquium, contact tim+colloquium@topos.institute.
The colloquium centres around following four themes, with many talks fitting multiple themes.
Topos aims to shape technology for public benefit via creating and deploying a new mathematical systems science. What does this mean, and how do we do it? How does technology shape people’s lives today? What are the risks that future technologies may bring? What is the role of mathematicians and computer scientists in shaping how their work has been deployed? What lessons can we learn from the past and present about successes and failures? This theme aims to spark discussion among mathematicians and computer scientists about these questions, and bring them into contact with experts on these subjects.
Thinking clearly about today’s most pressing scientific, technical, and societal questions requires finding the right abstractions. Category theory can help with this by offering design principles for structuring how we account for phenomena in a specific domain, as well as how we translate problems and solutions between different domains. This theme aims to highlight recent developments in applied category theory, in domains such as computation, neuroscience, physics, artificial intelligence, game theory, and robotics.
Like many parts of pure mathematics, results that are initially seen as purely theoretical may not be ripe for application until much later. At Topos we foster the entire pipeline, from the creation of elegant theory to the development of its effective application. The goal of this theme is to foster the pure side of this pipeline, to take in the most beautiful results in category theory, logic, type theory, and related fields, as well as to scout for not-yet-categorical work that appears ripe for "categorification".
Beyond its intrinsic interest, the purpose of applied mathematics is to provide a foundation for new capabilities and technologies that benefit the public. In this theme, we highlight work, at the intersection of research and engineering, that transfers ideas from applied category theory and other parts of mathematics into viable technologies, with an emphasis on software systems and tools. Topics may include scientific modeling, functional programming, differential programming, probabilistic programming, quantum computing, formal verification, and software and systems engineering.
To see the talks from previous years, use the links to the archives at the top of the page.
Adrian Miranda (30th of January)
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 -->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.
Paul-Andre Mellies (6th of February)
Reporting on recent joint work with Nicolas Behr and Noam Zeilberger, I will describe the rabbit calculus, a convolution product over presheaves of double categories motivated by term and graph rewriting. As I will explain, the convolution product generalizes to any double category the usual Day tensor product of presheaves of monoidal categories. An interesting aspect of the construction is that the resulting convolution product is in general only oplaxl associative. Therefore, I will identify several classes of double categories for which the convolution product is not only oplax associative, but fully associative. These include framed bicategories on the one hand, and double categories of term and graph rewriting on the other. For the latter, we establish a formula that justifies the view that the convolution product categorizes the rule algebra product, and captures the basic intuitions of causality in rewriting theory.
Andrew Dudzik (13th of February)
As category theory grows in popularity within the machine learning community, those of us interested in both theory and practice are faced with the question: which abstractions are directly useful when innovating in the design of neural networks? We explain why this is a hard problem, while making use of Joyal’s theory of species to give a down-to-earth description of the data types we see in contemporary neural networks, paying close attention to performance.
Joe Moeller (27th of February)
In his 1892 thesis, Lyapunov developed a method for certifying the stability of an equilibrium point $x^*$ of a dynamical system without actually having to solve the differential equations. He showed that if you can construct a function $V$ (now called a *Lyapunov function*) on the state space which is both positive definite relative to $x^*$ and always decreasing in the direction the system is pointing, then $x^*$ is necessarily (asymptotically) stable. The theory and methodology built on Lyapunov's theorem form the foundations for modern nonlinear control.
In this talk, we present a categorical framework in which we can develop a generalization of Lyapunov theory. This comes in two parts: coalgebras of an endofunctor play the role of dynamical systems in a category, and internal monoid actions play the role of solutions of these systems. We prove a generalization of Lyapunov's theorem in this framework, namely, that an equilibrium point of a coalgebra is stable if there is a *Lyapunov morphism*. This generalization allows us to recover both the classical continuous and discrete time versions of Lyapunov's theorem, as well as for dynamics in Lawvere metric spaces and more generally quantale-enriched categories.
Matteo Capucci (6th of March)
Category theory has a long history of being applied to the study of general systems. Double Categorical Systems Theory (DCST) condenses many lessons learned along the way regarding compositional structures for the representation of systems, their behaviour and the interaction of these two aspects. In this talk I'll revisit old and new wisdom regarding functorial behaviour of systems represented by a category of timepieces, and prove old and new compositionality theorems for them.
Nina Otter (13th of March)
The algebraic analysis of social systems, or algebraic social network analysis, refers to a collection of methods designed to extract information about the structure of social systems modelled as directed graphs. Central among these are methods to determine the roles that exist within a given system, namely compound relations, and the social positions, which one can think of as groups of similarly connected actors. In these methods the focus is on pairwise relationships between social actors, which are modelled through directed graphs. The importance of higher-order relationships in the modelling of social systems has received considerable interest in recent years. In this talk I will explain how we can use the theory of universal coalgebra to formalise these notions for graphs, generalise them to models of social systems that also take into account higher-order interactions between social actors, such as hypergraphs, and finally tie together the analyses of social roles and positions through a functoriality result.
The talk is based on joint work with Nima Motamed and Emily Roff.
B. Scot Rousse (20th of March)
This talk explores "care" as a missing dimension in contemporary discussions on technology, using debates around LLMs and AI alignment as a case study. Our civilization’s technological prowess is largely identified with abstraction and optimization—powers that have yielded immense benefits but which have also reshaped our relationship with reality. Increasingly, our drive to abstract and optimize takes on a life of its own, leading both technologists and non-technologists alike to engage with the world primarily through questions of modeling, measurement, and control. But what we care about cannot be appropriately tended to in such an instrumental framework, for it lacks a defined ultimate goal, a why for which we model, measure, and control. Our challenge is designing how to pursue technological progress within a broader context of care, not mere "values" that are relative to imperatives of efficiency and optimization.
Juergen Jöst (27th of March)
Many machine learning algorithms try to visualize high dimensional metric data in 2D in such a way that the essential geometric and topological features of the data are highlighted. Recent schemes like the popular UMAP depend on sophisticated mathematical tools from Riemannian geometry and category theory. We develop and improve the mathematical theory in a rigorous way, leading to an improved algorithm, IsUMap. The category theoretical foundations include Spivak's fuzzy simplicial sets, and they also offer a link to Topological Data Analysis. They provide a conceptual tool for merging different generalized metric structures, and we will also put them in a wider context.