Riehl, Bradley, Cheng, Dancstep, and Lugg: Category theory outreach panel Topos Institute 2026-09-20 | Riehl, Bradley, Cheng, Dancstep, and Lugg: Category theory outreach panel
Adrian Miranda: Kleisli constructions for pseudomonads Topos Institute 2025-01-27 | Topos Institute Colloquium, 30th of January 2025.———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 to 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.
[Berkeley Seminar] Harrison Grodin | Amortized Analysis via Coalgebra Topos Institute 2024-12-18 | Title: Amortized Analysis via CoalgebraAbstract: Amortized analysis is a technique for analyzing the efficiency of operations on a data structure in which cost is studied in aggregate: rather than considering the cost of a single operation in isolation, one bounds the total cost encountered throughout multiple operations in sequence. Traditionally, amortized analysis is phrased inductively, quantifying over finite sequences of operations. Connecting to prior work on coalgebraic semantics for data structures, we develop the alternative perspective that amortized analysis is naturally viewed coalgebraically in a category of cost algebras. In addition to simplifying the precise definition of amortized analysis, this perspective also generalizes the technique to other settings and incorporates type- and category-theoretic intuition.https://topos.institute/events/berkeley-seminar/
E.Subrahmanian & Y.Keraron: Engineering practice and the potential role of CT in systems engineering Topos Institute 2024-12-13 | Topos Institute Colloquium, 12th of December 2024.———In this talk, we will illustrate the role of information structures in engineering practices. In a particular case, we will instantiate an example of the required structures and their interaction in the systems engineering process. We will then identify Categorical representations that could formalize the models embedded in systems engineering standards and support their composition. The talk is not mathematical but an attempt to map the needs of systems engineering modeling to potential categorical representations that serve the purpose of engineering practices.
Matthew Akamatsu: Collective grassroots knowledge generation with lab discourse graphs Topos Institute 2024-12-06 | Topos Institute Colloquium, 5th of December 2024.———Scientific research teams face persistent challenges in coordinating their work and synthesizing collective knowledge into models across projects and labs. Discourse graphs offer a solution by breaking scientific research into its atomic components - questions, claims, and evidence - and connecting them in a semantic graph. In our cell biology lab, we have implemented discourse graphs as a coordination layer for collective research and model building. Lab members use the graphs to identify open research questions, maintain focus on their chosen hypotheses while documenting new results, and share discrete findings with clear provenance and context. Researchers report streamlined thinking, better orientation toward their target question, and a sense of accomplishment from contributing tangible, reusable results. This collaborative approach has allowed the lab to organically evolve its scientific models through bottom-up knowledge synthesis. Our new user pilot aims to extend these benefits through accessible tooling and interoperable graphs, thereby enabling collective grassroots knowledge generation and sensemaking across research communities.
[Berkeley Seminar] Kevin Carlson: What is it like to be a lax double functor? Topos Institute 2024-11-26 | Title: What is it like to be a lax double functor?Abstract: I will attempt to impart the vibe that the models of double theories that are the basis of CatColab are certain kinds of families of categories, even though the definition looks like a generalization of the families of sets we know and love as presheaves. Furthermore, CatColab does (on a sufficiently bleeding-edge branch) contain families of sets that live over these families of categories in an appropriate way, which we call instances of models of double theories. (Deep breath.) Probably only two people know what those are, yet, so let me try to tell you, because they're going to be important; it's not all that bad, just new.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] Dennis Chen: Cartesian polynomial monads in HoTT Topos Institute 2024-11-15 | Title: Cartesian polynomial monads in HoTTDate: November 13, 2024Abstract: Modern math has shown the necessity of higher categories and higher structures. Infinite coherence data arises quite naturally as one considers various types of topological spaces and homotopy theory, as well as modern treatments of algebraic geometry. One can see simple versions of this problem already when looking at path composition in a topological space. Path composition is not strictly associative, but only associative up to a higher cell. One promising method of presenting higher structures is the development of homotopy type theory, which formulates spaces (up to homtopy equivalence) as its basic objects. However one major obstacle is notating the infinite coherences of infinity categories in homotopy type theory. This is due to the autophagy problem: how can one talk about algebraic structures on types, if the algebraic structures themselves are encoded as types? Here we discuss a solution to the problem by Finster, Allioux, and Sozeau by axiomatizing the nature of polynomial monads, hence allowing themselves certain computational equalities. They then go on to use these polynomial monads to discuss higher structures including higher categories. Essential to their method is Baez and Dolan's slice construction, which is able to capture infinite coherences of polynomial monads even if one starts from very strict, classical polynomial monads. This integration of HoTT with polynomial monads I believe is incredibly interesting and could prove to be a very useful foundation/computational system.https://topos.site/events/berkeley-seminar/
Arezoo Islami: Unravelling the Unreasonable Effectiveness of Mathematics Topos Institute 2024-11-15 | Topos Institute Colloquium, 14th of November 2024.———The Applicability Problem is the problem of explaining why mathematics is applicable to the empirical sciences. This problem is revived and reformulated by the physicist Eugene Wigner under the striking title "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". In this influential work, Wigner argues that the applicability of mathematics is a miracle, "a wonderful gift which we neither understand nor deserve". Responses to this problem range from metaphysical claims about the mathematical structure of the universe to epistemic claims about the nature of human cognition, as well as formalist views that characterize mathematics as a type of game.In my view, to find an explanation for this relationship, we must first understand the explanandum itself. More fundamental than the why-question (why is mathematics applicable in the natural sciences) is the how-question (how is mathematics applicable in the natural sciences). By examining how mathematics has been used across different eras and fields within the natural sciences, we can begin to understand the relationship between mathematics and the sciences. More importantly, this exploration allows us to address questions regarding the nature of mathematics as it is used and practiced. By distinguishing pseudo-problems from the genuine problems of applicability, we open new paths for philosophical reflections on the nature of mathematics and the sciences.
Carl Miller: Graphical Methods in Quantum Cryptography Topos Institute 2024-11-08 | Topos Institute Colloquium, 7th of November 2024.———Mathematical security proofs are crucial in the field of cryptography: they allow us to base the theoretical security of a particular protocol on a set of simply-stated assumptions. However, a major challenge (in both education and research) is managing the complexity of these proofs. Fully rigorous security proofs can occupy a lot of space even when the underlying ideas are relatively simple. In the subfield of quantum cryptography, which incorporates quantum states and processes, this challenge has additional complications.Visual reasoning helps illuminate proofs in cryptography, and in some cases, visual reasoning can actually serve as a replacement for symbolic reasoning. In this talk I will discuss how picture-proofs, based on categorical quantum mechanics, can be incorporated into quantum cryptography.References:- Breiner, Spencer, Carl A. Miller, and Neil J. Ross. "Graphical methods in device-independent quantum cryptography." Quantum 3 (2019): 146. arXiv:1705.09213- Breiner, Spencer, Amir Kalev, and Carl Miller. "Parallel Self-Testing of the GHZ State with a Proof by Diagrams." 15th International Conference on Quantum Physics and Logic. Electronic Proceedings in Theoretical Computer Science (EPTCS), 2018. arXiv:1806.04744
Mario Román: Partial Markov Categories Topos Institute 2024-11-01 | Topos Institute Colloquium, 31st of October 2024.———Partial Markov categories are an algebra and syntax for Bayesian inference. They use a string diagrammatic syntax—with a formal correspondence to programs—to reason about continuous and discrete probability, decision problems (Monty Hall, Newcomb's), the compositional properties of normalization, and an abstract Bayes' theorem.Partial Markov categories are a careful blend of Markov categories (from categorical probability theory) and cartesian restriction categories (from the algebraic theory of partial computations). We will discuss the construction, theory, and applications of partial Markov categories.This is joint work with Elena Di Lavore. It is based on "Evidential Decision Theory via Partial Markov Categories", presented at LiCS'23 (arxiv.org/abs/2301.12989).
[Berkeley Seminar] Sophie Libkind: Two takeaways from ACT2Infer Topos Institute 2024-10-29 | Title: Two takeaways from ACT2InferAbstract: In September, David Spivak and I hosted the ACT2Infer workshop which brought together experts in applied category theory and active inference together in order to begin the process of articulating and clarifying the underlying principles of active inference via the formalism of category theory. In this two-part talk I'll (1) share the logistics of hosting this workshop and (2) share my main conceptual takeaway --- a polynomial functor account of active inference.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] David Espinosa: Monad translations compose Topos Institute 2024-10-25 | Title: Monad Translations ComposeAbstract: Many people have observed that monad translations compose. We give this idea a try using the ML module system and see what mileage we can get out of it.https://topos.site/events/berkeley-seminar/
Jason Gross: Compact Proofs: Measuring Quality of Understanding with a Compression-Based Metric Topos Institute 2024-10-25 | Topos Institute Colloquium, 24th of October 2024.———The field of mechanistic interpretability – techniques for reverse engineering model weights into human-interpretable algorithms – seeks to compress explanations model behavior. By studying tiny transformers trained to perform algorithmic tasks, we can make rigorous the extent to which various understandings of a model permit compressing an explanation of its behavior.In this talk, I’ll discuss how we prototyped this approach in our paper where formally proved lower bounds on the accuracy of 151 small transformers trained on a Max-of-K task, creating 102 different computer-assisted proof strategies to assess their length and tightness of bound on each of our models. Using quantitative metrics, we found that shorter proofs seem to require and provide more mechanistic understanding. Moreover, we found that more faithful mechanistic understanding leads to tighter performance bounds.We identified compounding structureless noise as the leading obstacle to generating more compact proofs of tighter performance bounds. I plan to discuss ongoing work to address this challenge by either relaxing the worst-case constraint enforced by using proofs; or by fine-tuning partially-interpreted models to align more closely with our explanations.I’ll conclude by discussing the roadmap I see to scaling the compact proofs approach to rigorous mech interp up to frontier models.
[Berkeley Seminar] Shaowei Lin: Safety by Shared Synthesis Topos Institute 2024-10-24 | Title: Safety by Shared SynthesisAbstract: Today, critical infrastructure is vulnerable to both malicious attacks and unintended failures, and these risks are expected to grow in the foreseeable future. Deploying formal verification (FV) across critical cyber physical systems would dramatically improve safety and security, but has historically been too costly to use outside the simplest or most critical subsystems. AI could allow widespread use of FV in years not decades, shifting cyber risks strongly in favor of defense. In this talk, I will outline our report with Atlas Computing on AI-enabled tools for scaling formal verification (atlascomputing.org/ai-assisted-fv-toolchain.pdf). I will also discuss some lessons that I learnt along the way, especially about shared synthesis - the collaborative construction of formal specifications, implementations and proofs.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] Evan Patterson: Collaborative modeling with domain-specific categorical logics Topos Institute 2024-10-24 | Title: Collaborative modeling with domain-specific categorical logics: the early design of CatColabDate: October 15, 2024Abstract: Category theory is a toolbox to build and interoperate formal languages for a wide array of domains, from logic and programming to data science and statistics to science and engineering. Despite having transformative potential, this viewpoint is not yet widely appreciated outside of specialized research communities. We believe that category-theoretic modeling can become a mainstay if it is embodied in useful technologies that do not require their users to have specialized mathematical knowledge. To this end, we are building CatColab, a new platform for formal, interoperable, conceptual modeling within domain-specific categorical logics. In this talk, we describe our early progress on CatColab, focusing on the interplay between the mathematical foundation and its embodiment as a technology intended for human use.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] Owen Lynch: Stateful Lenses Topos Institute 2024-10-22 | Title: Stateful Lenses - A Recipe for Expressive Systems-Theoretic Cartesian Double CategoriesAbstract: 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/
Thomas Powell: Quantitative results for stochastic processes Topos Institute 2024-10-17 | Topos Institute Colloquium, 16th of October 2024.———It is an elementary fact from real analysis that any monotone bounded sequence of real numbers converges. It turns out that the monotone convergence theorem can be given an equivalent finitary formulation: roughly that any sufficiently long monotone bounded sequence experiences long regions where the sequence is metastable. This so-called "finite convergence principle" is carefully motivated and discussed by Terence Tao in a 2007 blog post ('Soft analysis, hard analysis, and the finite convergence principle'), but was already known to proof theorists, where the use of logical methods to both finitize infinitary statements and provide uniform quantitative information for the finitary versions plays a central role in the so-called proof mining program.
[Berkeley Seminar] Kevin Carlson: Free cocompletions made friendly? Topos Institute 2024-10-17 | Title: Free cocompletions made friendly?Abstract: Lots of us are familiar with computations in Poly, some of which are made really nice by the fact that Poly is the category of families in Set^op, more formally, the free cocompletion of Set^op under coproducts. Free cocompletions under all colimits are also really important; they're presheaf categories! However, presheaf categories are kind of annoying to actually calculate colimits in (coequalizers of sets are scary), in contrast to how trivial it is to calculate coproducts of polynomials. Free cocompletions are supposed to be closing a category up under "formal" colimits in some sense, so shouldn't taking the colimits also be "formal"? Actually, it can be! There is a category structure on the class of all small diagrams in C that models the free cocompletion of C in an extremely elementary way with lots of nice properties. This has been known to a few people, such as Andrée Ehresmann, for decades, and in the case of free cocompletions under filtered colimits to a lot more people, such as Grothendieck, but has only been known to me for a couple of weeks. I'd like to make it known to you too. I'm hoping this will be helpful both for generalizations of Poly beyond mere sets of positions and for better expressing data migrations, which is all about writing down functors from a category into a free cocompletion.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] Keri DAngelo: Composing Instantaneous Machines Topos Institute 2024-10-15 | Title: Composing Instantaneous MachinesAbstract: In this talk, I’ll discuss recent progress Sophie and I have made on composing instantaneous machines. Instantaneous machines means that at any point in time, input can be given to the machine and the machine will give output based on this input and its current state. In this talk, I’ll show how we can compose such machines. We first create an extended category of directed wiring diagrams accounting for the dependency between input and output, and then define an operad algebra giving us the semantics that defines composition. Depending on where the interest lies, we can delve into some details including that since the output now depends on the input, we come into the “problem” that every time the output changes, the input may also change. This can be accounted for by giving a fixed point that shows after finite time, our input and output will stabilize.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] Kris Brown: Categorical approaches to inferentialist semantics Topos Institute 2024-10-15 | Title: Categorical approaches to inferentialist semanticsAbstract: There are remarkable similarities between applied category theory and inferentialist semantics in the philosophy of language: a focus on making pre-existing structure explicit, sensemaking without rigid foundations, characterizing content in terms of external structure rather than internal structure, emphasis on open systems, and putting syntax and semantics in the same playing field. I will present some formalizations of inferentialism (a generalization of Girard's phase semantics) due to Hlobil and Brandom and show some progress towards understanding what is taking place, categorically.The accompanying slides are available at krisb.org/role/role-0034.xml, and a related blog post is available at https://topos.site/blog/2024-10-11-nonlogical-concepts/.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] Sam Vandervelde: Where is the middle of a Fibonacci sequence? Topos Institute 2024-10-10 | Title: Where is the middle of a Fibonacci sequence?Abstract: As one might expect, a generalized Fibonacci sequence is a doubly-infinite sequence of integers in which every term is the sum of the two previous, such as ..., -11, 8, -3, 5, 2, 7, 9, 16, 25, 41, ... We will see that it is possible to give an unambiguous mathematical definition of the middle term of such a sequence, and that doing so leads to a means of organizing all Fibonacci sequences that is natural, compelling, and beautiful.https://topos.site/events/berkeley-seminar/
Amélia Liao: Cubical types for the working formalizer Topos Institute 2024-10-04 | Topos Institute Colloquium, 3rd of October 2024.———Cubical type theory is an extension of dependent type theory designed to make the univalence principle *provable*, rather than an axiom. Then, by what is essentially a happy coincidence, it also provides a design for working with higher inductive types and with coinductive types. However, the tradeoff for these features is a hit to usability: In practice, the user of cubical type theory is directly exposed to the complicated primitive operations that let us implement these higher features, even if they're working on set-level mathematics. Worse, any implementation of HoTT imposes additional proof obligations to stay in the realm of sets rather than escaping off into coherence purgatory. This talk discusses the experience of using cubical type theory to build the 1Lab— in particular, the automation we've been building so the end-user of the library does not have to memorise the cubical type theory papers if all they want is to formalise traditional, low-homotopy level mathematics.
Seth Frey: Online communities as model systems for commons governance Topos Institute 2024-09-27 | Topos Institute Colloquium, 26th of September 2024.———The best citizens of a large-scale democracy are those who have built and broken several small ones to see how they work. By empowering people to build any kind of community together, the Internet has become a laboratory for self-governance experimentation. Groups who start online communities must overcome the challenges of recruiting finite resources around difficult common goals. Fortunately, they can draw on a growing range of support technologies, peer networks, and scholarship. With their transparency, the Internet's millions of online communities can be surveyed for insights into their design and functioning. Looking at three large platforms for small self-governing online communities, we will pose several questions of institutional processes at the population level, as drawn from the literatures on common-pool resource management and institutional analysis and design.
Chris Fields: What is the Identity operator? Topos Institute 2024-09-20 | Topos Institute Colloquium, 19th of September 2024.———The idea that objects have associated Identity morphisms, or operators, is fundamental to mathematics, physics, and the theory of Active Inference. All of these, moreover, support the idea that identity can be maintained via multiple paths. We will explore the relationship between the idea of identity and the notions of space, time, and memory, with an emphasis on how these latter three relate to the idea of an object in both physical theory and active inference.
Christine Tasson: Semantics for Reactive Probabilistic Programming Topos Institute 2024-09-13 | Topos Institute Colloquium, 12th of September 2024.———Synchronous languages are now a standard industry tool for critical embedded systems. Designers write high-level specifications by composing streams of values. These languages have been extended with Bayesian reasoning to program state-space models which compute a stream of distributions given a stream of observations [1].This talk aims at describing semantics for probabilistic synchronous languages, based on a joint work with Guillaume Baudart and Louis Mandel [2]. The key idea is to interpret probabilistic expressions as a stream of un-normalized density functions which maps random variable values to a result and positive score. Two equivalent semantics are presented: the co-iterative semantics is executable while the relational semantics is easy to use for proving program equivalence. The semantical framework is then applied to prove the correctness of a program transformation required to run an optimized inference algorithm.[1] Reactive Probabilistic Programming, Guillaume Baudart et al, PLDI 2020[2] Density-Based Semantics for Reactive Probabilistic Programming, Guillaume Baudart, Louis Mandel, Christine Tasson, arxiv:2308.01676
Spencer Breiner: Polynomial Interfaces Topos Institute 2024-09-06 | Topos Institute Colloquium, 5th of September 2024.———I will argue that applied category theory would benefit, both practically and strategically, from greater attention to user interface. Starting from some very basic examples, I will discuss some of the conceptual challenges in this area and explain how they map onto categorical structures like monads, comonads and polynomial functors. I will close with a discussion of the Semagrams user interface library, its goals and future prospects.
Cyrus Omar: Totally Live Programming and Proving in Hazel Topos Institute 2024-08-30 | Topos Institute Colloquium, 29th of August 2024.———This talk will review progress on the Hazel programming environment and its underlying theoretical developments. Hazel is the first totally live typed general-purpose programming environment, meaning that it deploys error localization and recovery mechanisms, rooted in language-theoretic developments, that ensure that every editor state is syntactically well-structured and statically and dynamically meaningful. The talk will review the underlying theory and include a live demonstration of various Hazel features, including its editor, training mode, and its stepper, which is forming the basis for ongoing work on a Hazel-based theorem prover. The talk will also discuss various other ongoing and future directions of interest to the community, including our vision for a "computational commons" that operates like a planetary-scale live program.
Will Crichton: How to Make Mathematicians Into Programmers (And Vice Versa) Topos Institute 2024-08-23 | Topos Institute Colloquium, 22nd of August 2024.———Tools for formalized mathematics (FM), such as proof assistants and model checkers, are increasingly capable of handling the real-world problems of both mathematicians and software developers. Yet, these tools are only as effective as the people who use them. The FM community clearly needs to invest in better education and better tooling. But... which curricula are actually effective for learners? What tooling will actually make users more productive? In this talk, I will lay out some preliminary ideas for how to systematically investigate these questions, i.e., develop a science of human factors for FM. My core proposal is to combine experimental psychological methods (e.g., lab studies, IDE telemetry) and cognitive theories (e.g., working memory, mental models) to study how people use FM tools. Then that understanding can be applied to make principled predictions about the efficacy of curricula, tooling, and language design.
Arthur J Parzygnat: A generalization of inversion using Bayes rule with applications to quantum Topos Institute 2024-08-09 | Topos Institute Colloquium, 8th of August 2024.———Bayes' rule has recently been given a categorical definition in terms of string diagrams due to Cho and Jacobs. This definition of Bayesian inversion, however, is not robust enough for categories that include reasoning about quantum systems due to the no-cloning theorem. In this talk, I will explain how semi-cartesian categories (which have less structure than Markov categories) provide a suitable framework to define Bayesian inversion categorically. In particular, I will provide axioms for such an abstract form of Bayesian inversion. It remains an open question whether these axioms characterize Bayesian inversion for quantum systems.
[Berkeley Seminar] CB Aberle: All Concepts are Essentially Algebraic Topos Institute 2024-08-07 | Title: All Concepts are Essentially AlgebraicAbstract: Lawvere's categorical formulation of algebraic theories enables one to study some of the most common structures found in mathematics – e.g. groups, rings, etc. – at a high level of precision and generality. However, many significant mathematical concepts, including categories, topological spaces, etc., turn out not to be algebraic, in this sense. Notably, the very framework used by Lawvere to describe algebraic theories and their models – categories with finite products and product-preserving functors between them – cannot be described as an algebraic theory, and so it seems that algebra alone cannot encompass the whole of mathematics (nor even itself). There is, however, a deeper sense in which all of mathematics is essentially algebraic. What is needed to reveal this fact is to adapt the classical notion of algebraic theories, which are fundamentally simply typed, to an appropriate notion of dependently typed algebraic theories. At this level of generality, one is capable of defining not only individual mathematical structures, but structures that themselves encompass whole universes of mathematics, including topoi, models of type theory, etc. In particular, the theory of dependently-typed algebraic theories is itself describable as a dependently-typed algebraic theory. This fact has many profound consequences, of which I shall highlight just one: using the framework of dependently-typed algebraic theories, one can construct a type theory whose types themselves correspond to type theories, with functions between these types corresponding to translations between the corresponding type theories.https://topos.site/events/berkeley-seminar/
[Berkeley Seminar] Priyaa Srinivasan: Communicating Relational Thinking Topos Institute 2024-07-30 | Title: Communicating Relational ThinkingAbstract: In this talk I will give an overview of our recently completed experiment of communicating categorical thinking to a STEM-oriented audience outside mathematics. Angeline, Brendan, Paul and myself recently authored a free online textbook titled "Relational thinking: from Abstractions to Applications". In this talk, I will walk through the contents of the book and equally importantly the open source technologies behind the book. This talk is an invitation to the audience not only to read the book but also to create their own inclusive material around category theory / math leveraging new technologies.https://topos.site/events/berkeley-seminar/
Fabio Gadducci: From gs-monoidal to cartesian categories: a structural analysis Topos Institute 2024-07-12 | Topos Institute Colloquium, 11th of July 2024.———It is now folklore that cartesian categories can be viewed as symmetric monoidal ones equipped with two natural transformations, modelling diagonals and projections. In this talk we explore the taxonomy obtained by relaxing the naturality requirement, from gs-monoidal/cd-categories to restriction and Markov ones. We show how these possibly order-enriched categories are related by suitable commutative monads and the shape of the arrows of the free categories generated by an algebraic signature.
Mason Porter: Topological Data Analysis of Spatial Systems Topos Institute 2024-06-28 | Topos Institute Colloquium, 27th of June 2024.———I will discuss topological data analysis (TDA), which uses ideas from topology to quantify the "shape" of data. I will focus in particular on persistent homology (PH), which one can use to find "holes" of different dimensions in data sets. I will briefly introduce these ideas and then discuss a series of examples of TDA of spatial systems. The examples that I'll discuss include voting data, the locations of polling sites, and the webs of spiders under the influence of various drugs.
Bartosz Milewski: Parametric Profunctor Preoptics Topos Institute 2024-06-14 | Topos Institute Colloquium, 13th of June 2024.———Locally graded categories and parametric optics provide a compositional model of neural networks. I will show how to generalize this approach to pre-optics. I'll introduce a parametric profunctor representation of preoptics and use it to implement a perceptron in Haskell.
Alex Simpson: Three toposes for probability and randomness Topos Institute 2024-06-07 | Topos Institute Colloquium, 6th of June 2024.———I shall give a brief guided tour of three toposes that have arisen in a research programme to model aspects of probability and randomness from a topos perspective. The first topos of "probability sheaves" supports a synthetic style of probabilistic reasoning about random variables. The second "random topos" makes sense of the notion of "random element" and models a world in which all sets are measurable. The third topos of "random probability sheaves" combines the previous two and provides a home for a more radical style of "synthetic probability theory" expunged of all concerns about sigma-algebras, measurability and the like.
Steve Vickers: The Fundamental Theorem of Calculus: point-free Topos Institute 2024-05-31 | Topos Institute Colloquium, 30th of May 2024.———Point-free topology is known in various guises (locales, formal topology), but can be boiled down to a procedure of defining the points of a space not (“point-set”) as elements of a set, but as models of a logical theory. This is for a constrained “geometric” logic, so that the opens of the space correspond to propositional formulae derived from the theory: thus the theory defines both the points and the topology. Then continuity of maps just means that they are constructed in accordance with the constraints of the logic.Why bother to do topology that way? After all, the logic is even more constrained than constructive reasoning, and we still don’t know how far it reaches.The first reason is that it quite painlessly extends to toposes, viewed as generalized spaces. This uses the machinery of classifying toposes, but only in an unobtrusive way [1]. Many proper classes can then be viewed as point-free spaces - just write down a geometric theory whose models are the elements of the class.The second reason follows from the first but can then be applied back to ungeneralized spaces in a very natural treatment of bundles. Theory presentations can themselves be described as the models of a geometric theory, and this allows us to view bundles as continuously mapping base points to spaces (the fibres). For physics in particular, this invites exploration of how much can be done using geometric methods. See eg [2].Meanwhile, there is the question of how much ordinary mathematics, and in particular real analysis, can be done in this style. I present as a case study the Fundamental Theorem of Calculus [3]. This illustrates some typically geometric features of the reasoning, such as attention paid to one-sided reals and the use of hyperspaces and their analogues, and some exploitation of the geometric fact that everything is continuous, as well as a cute new trick using uniform probability measures.[1] Steven Vickers “Topical categories of domains”, Mathematical Structures in Computer Science 9 (1999)[2] Bas Spitters, Steven Vickers and Sander Wolters “Gelfand spectra in Grothendieck toposes using geometric mathematics", Electronic Proceedings in Theoretical Computer Science 158 (2014)[3] Steven Vickers “The Fundamental Theorem of Calculus point-free, with applications to exponentials and logarithms", arXiv:2312.05228
Berkeley Seminar: David Spivak, A tour of EM(Cat#) (5/20/2024) Topos Institute 2024-05-29 | Title: A tour of EM(Cat#)Abstract: Small categories are graphs with extra structure: namely, every path "composes" to form an arrow. As such, small categories are algebras for a certain monad—the "paths" monad—on the category of graphs. This monad is familial, i.e. its underlying functor can be identified with a loose map in the double category Cat#. But what is an appropriate sort of morphism between monads in Cat#? Shulman's "monoids and modules" construction doesn't apply, because Cat# doesn't have local coequalizers. Instead we consider Lack and Street’s (2002) construction of the free completion of Cat# under Eilenberg-Moore objects, denoted EM(Cat#). Its objects are monads m in Cat# and its morphisms m→n are "quintets" satisfying expectable properties; as the name would suggest, these morphisms induce functors m-Alg → n-Alg between the associated Eilenberg-Moore categories. The 2-cells in this "free completion" are slightly surprising—in particular, they are more flexible than those of (Street 1972)—but still simple to describe. We'll discuss various examples of objects in EM(Cat#), i.e. monads in Cat#, e.g. those arising from cofunctors out of a category, from functors into a category, from Grothendieck topologies on a category, from multicategories, and more. Then we'll discuss morphisms in EM(Cat#), e.g. a free-forgetful adjunction for algebras, a method for turning convex spaces into monoids, one that takes the category of elements of a copresheaf, and perhaps more.
Nicola Gambino: Monoidal bicategories, differential linear logic, and analytic functors Topos Institute 2024-05-24 | Topos Institute Colloquium, 23rd of May 2024.———The aim of this talk is to present bicategorical counterparts of the notions of a linear explonential comonad, as considered in the study of linear logic, and of a codereliction transformation, introduced in the study of differential linear logic via differential categories. As an application, the differential calculus of Joyal's analytic functors will be extended to analytic functors between presheaf categories, in a way that is analogous to how ordinary calculus extends from a single variable to many variables. This is based on joint work with Marcelo Fiore and Martin Hyland (arxiv.org/abs/2405.05774).
Elena Di Lavore: Effectful trace semantics via effectful streams Topos Institute 2024-05-17 | Topos Institute Colloquium, 16th of May 2024.———Effectful streams are a coinductive semantic universe for effectful dataflow programming and traces. As an example, we formalise the stream cipher cryptographic protocol. In monoidal categories with conditionals and ranges, effectful streams particularize to families of morphisms satisfying a causality condition. Effectful streams allow us to develop notions of trace and bisimulation for effectful Mealy machines; bisimulation implies effectful trace equivalence.This is recent joint work with Filippo Bonchi and Mario Román.
Shaowei Lin: AI-assisted Coding. Correct by Construction, Not by Generation Topos Institute 2024-05-14 | ...
Evan Patterson: Domain specific Logics for Scientific Modeling Theory and Practice Topos Institute 2024-05-14 | ...
Daniele Struppa: An Introduction to Superoscillations and Supershift Topos Institute 2024-05-14 | ...
Alexander Kurz: Canonical Extensions of Quantale Enriched Categories Topos Institute 2024-05-14 | ...
Berkeley Seminar: Sophie Libkind, Probabilistic and Namespace Conditioning in Poly (4/29/2024) Topos Institute 2024-05-14 | In the category of Polynomial functors, the substitution monoidal product is not symmetric. In this talk, we will explore how distributive maps in the substitution monoidal products give notions of "conditioning". These notions in probabilistic conditions (such as "conditioned on it being sunny, what is the probability that I will go outside") and namespace conditioning (such as searching "what directories satisfy the condition of containing 3 files"). Of particular interest will be a derivation of Bayesian inference, whose key insight is a simple braiding map.
Polynomial functors: Lecture 2/9 Topos Institute 2024-05-14 | Lecture 2/9 in a series on polynomial functors given by David Spivak. These lectures were given as part of the Poly at Work workshop held at the Topos Institute in February 2024. Topics0:00 Definition of Poly12:43 Poly(Sy^S,By^A)37:08 Coalgebras58:30 Composing coalgebras1:20:55 Lott
Polynomial functors: Lecture 6/9 Topos Institute 2024-05-14 | Lecture 6/9 in a series on polynomial functors given by David Spivak. These lectures were given as part of the Poly at Work workshop held at the Topos Institute in February 2024.Unfortunately, in this video there is no audio for the last 20 minutes.Topics0:00 ◁ and state machines1:10:26 Exercises1:42:42 Org1:47:12 ◁-monoids
Polynomial functors: Lecture 9/9 Topos Institute 2024-05-14 | Lecture 9/9 in a series on polynomial functors given by David Spivak. These lectures were given as part of the Poly at Work workshop held at the Topos Institute in February 2024.Topic0:00 Bicomodules
Polynomial functors: Lecture 4/9 Topos Institute 2024-05-14 | Lecture 4/9 in a series on polynomial functors given by David Spivak. These lectures were given as part of the Poly at Work workshop held at the Topos Institute in February 2024.Topics0:00 tensor monoids (continued)15:26 tensor comonoids37:42 times (co)monoids1:03:15 internal hom1:04:58 internal hom for times1:57:53 internal hom for tensor
Polynomial functors: Lecture 5/9 Topos Institute 2024-05-14 | Lecture 5/9 in a series on polynomial functors given by David Spivak. These lectures were given as part of the Poly at Work workshop held at the Topos Institute in February 2024.Topics0:00 Composition product (◁)26:01 Facts about ◁33:50 Duoidality
