Blog

Can we build a tradition of math talks which invites the audience to engage not only with the speaker but also with each other? That is, a seminar that starts as a lecture and ends as a conversation. In this post, Sophie and Priyaa discuss how such a seminar might look based on previous…

Sophie Libkind, Priyaa Varshinee Srinivasan

2024-08-02

Our Summer Research Associates in 2024

Another year, another cohort of wonderful Summer Research Associates (RAs) at Topos working on exciting projects. As is now the custom, we’ve asked each of them to introduce themselves and write a little bit about what they’ll be working on over the summer. You can expect to see some blog posts…

2024-07-29

How to write a fantastic book in four months, Part III

Priyaa Varshinee Srinivasan

In the third post of this series about Relational Thinking: from abstractions to applications, we look at the story-telling approach that we took in writing the book.

2024-07-09

How to write a fantastic book in four months, Part II

Priyaa Varshinee Srinivasan

In the second post of this series about Relational Thinking: from abstractions to applications, we look at the technologies used to build the book.

2024-07-04

How to write a fantastic book in four months, Part I

Priyaa Varshinee Srinivasan

In the first post of this series, we introduce the freely available online book Relational Thinking: from abstractions to applications, starting with the story of how it came into being and giving a brief overview of its contents.

2024-06-28

Toward compact double categories: Part 2

Last time we framed the puzzle of axiomatizing categorical duality and introduced new double-categorical tools, culminating with the twisted Hom functor. We’ll now get straight to the point by proposing a definition of a compact double category. After that we’ll fill in a few remaining technical…

2024-06-24

Toward compact double categories: Part 1

Lately I’ve been pursuing formal category theory using double categories. A first step in this program is to abstract from categories to category objects in a double category; for instance, internal categories are category objects in a double category of spans and enriched categories are category…

2024-06-20

Applied Category Theory for Engineering Design: A Teaser Video

Prof Gioele Zardini will be teaching a new course on Applied Category Theory for Engineering Design (ACT4ED) this Fall. Here’s a teaser video outline the key challenges in engineering systems design we aim to address with applied category theory.

2024-06-03

Launching new collaboration with Chapman University

We’ve launched a collaboration with Chapman University to explore joint research projects and expand our role mentoring and teaching the next generation of technologists working in the public interest. Each year, many people reach out to ask how they might pursue doctoral studies with our…

2024-05-28

Ontological Commitments for Boundaries

When we do math about things we care about, committing to which mathematics represents which intuitions moves us forward.

Sophie Libkind

2024-04-25

Understanding UMAP

data science

UMAP is a dimension-reduction method that has become very popular, especially in many areas of biology. Recently, a certain figure produced by UMAP has been circulated online to support spurious conclusions about “genetic racial groups”, or, to put it more bluntly, to support scientific racism. In…

2024-04-05

Poly @ Work 2024

Polynomial functors are a prime example of mathematics’ startling capacity to yield deep formal insights and real-world applications from the simplest ingredients. The category of polynomial functors (Poly) offers an abundance of elegant theory linked with computational design patterns that span…

Sophie Libkind, David Spivak

2024-03-27

A Retrospective on the Oxford–Topos Meeting

Sam Staton, Paolo Perrone and I organized a small meeting between Topos Institute and the Oxford CS department. It was kind of like a workshop, but because of the short time frame that we planned it on and limited funding, we didn’t invite all the people we’d like to invite or have it open to…

2024-03-13

SCAI — AI for the Global Good

In December 2023, Topos partnered with the Government of Singapore to host the inaugural Singapore Conference on AI for the Global Good (SCAI). An international, cross-sectoral meeting of leaders, SCAI focussed efforts on AI for the global good through the joint writing of 12 foundational yet…

2024-02-02

Algebras are promonads

An interesting analogy between algebras over a ring and promonads on a category is formalized using the apparatus of double theories.

2024-01-29

Composing springs

Physical systems are often composed of many interacting subsystems. In this post, we take a peek at the math and the software implementation for composing systems of springs using decorated cospans.

Sophie Libkind

2024-01-18

Why double categories? Part 1

databases

The beginning of a series of posts answering the question “why double categories?”. Our first answer is that double categories give the algebra of relations from universal properties.

2024-01-15

Building dynamic structures

In our day-to-day lives, we all interact with many systems where the structure of the system changes based on external or internal factors. How do we make sense of structures that can change in this way? In mathematics, we often assume the structure we’re working with is “fixed” to some extent…

Samantha Jarvis

2024-01-09

Poly-morphic effect handlers

In computer science, programmers often perform effects to interact with the surrounding environment. For example, a program may print strings or interact with mutable state. Then, effects may be handled, implemented in terms of other effects. In this post, we reconstruct a categorical semantics…

Harrison Grodin, David Spivak

2024-01-03

Responsible mathematics and metaphors of semantics

philosophy

An exposition of a philosophical argument about how words connect with their meanings, and a tentative connection to work done at Topos.

Kris Brown

2023-12-14

Trustworthy Reification

Not all meals created from the same recipe are equally delicious. Not all programs that accurately sort a list are equally useful. High-level specifications, such as a recipe or a prompt, are only practical because I trust the chef or engineer to execute them in a way that aligns with my values…

Sophie Libkind

2023-12-05

Categorical Statistics in Julia

statistics

This summer I created the StatisticalTheories.jl package so we can do categorical probability and synthesize probabilistic programs in the AlgebraicJulia ecosystem. Read all about it!

Harper Hults

2023-11-28

Mathematics for climate change

John Baez is now helping lead a new Fields Institute program on the mathematics of climate change.

John Baez

2023-11-23

Topos researchers win Best Paper Award at AAAI Fall Symposium 2023

Congratulations to Angeline Aguinaldo, Evan Patterson, and their collaborators James Fairbanks, William Regli, and Jaime Ruiz at the University of Florida and the University of Maryland, College Park for winning Best Paper Award at the 2023 AAAI Fall Symposium on Unifying Representations for Robot…

2023-11-17

Introducing InterTypes

Acsets

Announcing the first version of InterTypes: a package for cross-language serialization for ADTs and ACSets

2023-11-14

Solving problem-solving

Our physical world is incredibly complex. Surviving life is a lot about problem-solving, be it waking up in morning and finding your way to the bathroom or solving a math problem for research. We constantly abstract information back and forth from the physical plane to our mental plane for solving…

Priyaa Varshinee Srinivasan, David Spivak

2023-11-13

Lie groups induce Hopf monoids in Poly

A Lie group G is a group object in the category of manifolds; that is, it’s a smooth space equipped with a special point e:G and a multiplication operation G\times G\to G. Every manifold M has a cotangent bundle T^*M\to M, and this can be made into a polynomial functor t^*_M. In this…

2023-10-26

Retrotransformations

Retrotransformations between lax double functors are introduced as the “multi-object” analogue of a cofunctor between categories. Notions of “monoidal cofunctor” between monoidal categories and of “multicofunctor” between multicategories are then derived as special cases.

2023-10-20

Cartesian double theories

Cartesian double theories are a new framework for doctrines based on double-categorical functorial semantics.

2023-10-13

Our Summer Research Associates in 2023

Topos Institute is proud to have ushered in another cohort of research associates (RAs) this summer! Each early career researcher made a positive impact on both the science and culture at Topos, through which their legacies live on. We hope that the collaboration was reciprocal, empowering them to…

Juliet Szatko

2023-10-09

Powers of polynomial monads

The category \mathbf{Poly} is cartesian closed, and in particular, we can raise one polynomial q to the “power” of…

Many of our favorite monads on \mathbf{Set}, such as the Maybe monad and the List monad, are polynomial. It turns out that monads have special “powers” in \mathbf{Poly}.

2023-09-21

Unbiased monoidal categories are pseudo-elements

operads

Categorifying the observation that monoids are generalized elements of multicategories, we show that unbiased pseudomonoids, such as unbiased monoidal categories, are “pseudo-elements” of 2-multicategories.

2023-08-15

A nuclear adjunction between Poly and Dir

There are at least two interesting kinds of maps between bundles: “forward-forward” maps and “forward-backward” maps. That is, both kinds go forward on the base, but the first kind also goes forward on the fibers, whereas the second kind goes backwards on the fibers. In a paper with David Jaz…

2023-07-21

Dialogue on a mathematical approach to the good

This post is a dialogue and consists of two parts. In the first part, Bartłomiej Skowron and David Spivak consider what mathematics and ethics have in common. Though some will think these notions have nothing in common, these authors share the intuition that the opposite is true. In particular…

philosophy

Bartłomiej Skowron, David Spivak, David Corfield

2023-07-11

Agent-based modeling via graph rewriting

Rewrite rules are organized via a graphical syntax into discrete-time simulations which can be understood as agent-based models. This representation is transparent, compositional, and serializable.

Kris Brown

2023-07-07

Welcoming our new Director of Research, Shaowei Lin

We’re delighted to announce that Shaowei Lin has joined Topos as our new Director of Research.

2023-06-22

Acsets with variables

Acsets

rewriting

Acsets are great, but what if attributes could be variables?

Kris Brown, Kevin Arlin

2023-06-20

Singular Learning Theory and Alignment Summit 2023

statistics

There are surprisingly deep connections between algebraic geometry and statistical learning theory, which have been explored over the past few decades by Sumio Watanabe in Singular Learning Theory. This body of work is highly relevant to the problem of understanding the behaviour of large…

Shaowei Lin

2023-06-17

Towards a Research Program on Compositional World-Modeling

In this post, we lay out a vision and challenge for using category theory to build tools for understanding systems at a global scope.

davidad , Owen Lynch

2023-06-15

Natural transformations between cofunctors

Polynomial comonads can be identified with categories, but the morphisms between them are not functors; they’re called cofunctors. There is a reasonable notion of natural transformation between cofunctors, but I always found remembering how it goes to be a slog. Recently I realized that they have…

2023-05-26

Categories for Epidemiology

Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood, Eric Redekopp and I have been creating software for modeling the spread of disease… with the help of category theory! Lots of epidemiologists use “stock-flow diagrams” to describe ordinary differential equation (ODE) models of disease dynamics.…

John Baez

2023-05-12

Symbolic presentations of dynamical systems

A follow-up to Algebraic Geometry for the Working Programmer, this post explains a category-theoretic approach to symbolic open dynamical systems.

2023-05-08

Spooling out syntax from behavior

The behavior of a dynamical system is its entire fate or character: the tree consisting of precisely what it will do when exposed to any particular sequence of inputs. In the world of polynomial functors, behavior for a dynamical system with interface p is formalized using the cofree comonad…

2023-04-24

Conegation rewriting

Rewriting theory studies rules for turning things into other things. Typically, one considers various types of graph, and looks at rewriting rules that have a form like this: “whenever you see a subgraph shaped like I, replace it with one shaped like O, using the common subgraph K to attach it”.…

rewriting

Eigel Rischel

2023-04-11

Imperative Programming with Poly

Category theory has been applied fruitfully to functional programming for decades, leading some to think that functional programming represents the one true way of doing math on a computer. However, the stubborn fact that the internals of computers and the world itself are both stateful means that…

2023-04-05

Seventy Years Using Fixed Points

lambda calculus

Having first heard about theorems on fixed points as an undergraduate, uses for them came into my research on many subsequent occasions. The talk will review some personal history and give some suggestions for possible further applications.

Dana Scott

2023-03-29

Lotteries: a constructive version of the distributions monad

Many category theorists have heard of the distributions monad \mathsf{dist}\colon\mathbf{Set}\to\mathbf{Set}, which sends a set X to the set of finitely supported probability distributions on X. Many have also heard of the operad \Delta of simplices, for which an n-ary operation is a…

2023-03-23

Algebraic geometry for the working programmer

In this series of posts, we investigate the duality between algebra and geometry in order to develop new types of lenses. In this first post, we review some basic ideas about algebraic geometry that will be needed in the coming posts.

2023-03-23

Structured cospans as a cocartesian equipment

The theory of structured cospans is dramatically simplified by the use of double-categorical universal properties. Specifically, we show that structured cospans form a cocartesian equipment, a result that is stronger yet easier to prove than the usual result that they form a symmetric monoidal…

We are finding that self learning is becoming more and more popular when it comes to learning category theory. In a collaborative effort to assist people in this pursuit, Topos will host a category theory outreach panel moderated by Emily Riehl, and featuring Tai-Danae Bradley, Eugenia Cheng, Paul…

2023-03-15

Category theorists welcome self-learners in a new outreach panel

David Spivak, Alexis Anaya

2023-03-09

Women in Logic: the history so far

This year we’re having the 7th Women in Logic (WiL) Workshop in Rome, associated with FSCD and CADE. Considering that we never had grants to help us defray costs of the workshop, I think getting to our seventh anniversary is a big milestone. Everyone who works for Women in Logic does it for free…

2023-03-06

What made the FRA workshop structure successful?

Elena Di Lavore, Mario Román

The Topos Institute recently hosted “Finding the Right Abstractions for Healthy Systems”, with 24 researchers, mainly from the applied category theory and AI safety communities. This is a post trying to understand what made it successful and some ideas for similar events.

2023-02-22

Ask Eugenia Cheng about Category Theory!

We are excited to announce a book club for The Joy of Abstraction, a new book by Eugenia Cheng. The Joy of Abstraction is a wonderful introduction to category theory for anyone who wants to get into the formality of the subject but does not necessarily have the mathematical background to read…

2023-02-09

Promonoidal categories and wiring diagrams

wiring diagrams

operads

There are (colored) operads for all sorts of different flavors of wiring diagrams: those governing monoidal categories, symmetric monoidal categories, traced categories, compact closed categories, hypergraph categories, etc. But all of these operads are in fact more than mere operads: there’s a…

2023-01-31

Recursive Types via Domain Theory

lambda calculus

Functional programming languages often include recursive types, allowing programmers to define types satisfying a given isomorphism. Such types cannot all be interpreted as sets, but they can be understood using domain theory. We consolidate and present foundational techniques in domain theory to…

Harrison Grodin

2023-01-10

Preparing for Networked Mathematics

Networked mathematics is one of my main projects at the Topos Institute (the other is Dialectica categories and its extended family of formalisms). We have not had much luck getting the funding agencies to pay attention to it, yet. But the work is progressing, albeit slowly. We have been working…

2023-01-05

Lenses are semi-monads, maybe lenses are monads

Have you ever heard of a semi-monad? I hadn’t, but when I came across a “monad without unit” in the wild, I made an analogy with semigroups (a semigroup is a group without unit). It seems others have also thought to use the term semi-monad for such a gadget. There is a semi-monad structure on the…

wiring diagrams

2022-12-20

Research opportunities at Topos in early 2023

We invite applications to join Topos Institute as a Research Associate (summer 2023) or as a Research Software Engineer (regular position).

2022-12-12

Dialectica Categories in Computing

dialectica

In the first half of the year we had an AMS Mathematical Research Community (MRC) on Applied Category Theory, in Beaver Hollow, NY. An MRC is a bit like a version “on steroids” of the ACT Adjoint School, as there are many more researchers in a research subgroup in the MRC than in a Adjoint School…

2022-12-06

Scott’s model of lambda calculus

lambda calculus

Lambda calculus constructs a set of terms starting from a set of variables using operations and rules. If we treat lambda calculus as a programming language, we can consider a model as a way of giving an implementation of lambda calculus. This essentially gives an implementation of a typed…

Anthony Agwu

2022-11-23

Statement on Topos and the FTX Foundation

Topos is a community and an organisation that cares deeply about the public interest, and works to make a world of better systems for all. As part of this, we believe accountability and transparency are crucial. This post is to create transparency around our now declined funding relationship with…

2022-11-19

Mathematical concepts: how do you recognize them?

Humans are very good at recognizing the words they do not know, the concepts they haven’t met yet. To help with these they invented dictionaries, glossaries, encyclopedias, wikipedias, crib sheets, etc. Initially, they did it via laborious manual work, more recently through automated…

Valeria de Paiva, Jacob Collard

2022-11-16

Where matter and pattern meet

operads

Etymologically, the word matter comes from mother and the word pattern comes from father. Like two parents, matter and pattern represent a fundamental dichotomy: matter is the pure material, unconcerned with our ideas about it; pattern is pure structure, unconcerned with what substantiates it.…

2022-11-07

Who is category theory for?

YouTube is our most powerful outreach tool. Yet how does it shape who we reach out to? And is this in line with the values of our community?

2022-11-04

When you light up, I light up

Sophie Libkind, David Spivak

“Cells that fire together, wire together.” This slogan for Hebbian learning evokes a strategy for reorganization in which an individual strengthens their connection with another if they have similar behavior. Here we give a mathematical account of Hebbian learning as a dynamic monoidal category.

2022-10-28

Project cards: a tool for research transparency

This blogpost presents some early-stage, exploratory work on a kind of tool that may be useful for Topos (and other similar institutes!) as it matures as a company. In a nutshell, it is meant to serve as a framework to facilitate accessibility and discussion—internally and externally—of the…

Alejandra Arciniegas

2022-10-24

JMM Workshop on ACT for teaching and learning

My name is Ted Theodosopoulos and I am a mathematician working at the Nueva School, an independent school for gifted students in San Mateo, California. It was largely as a result of the encouragement that folks at Topos offered me that I decided to propose a workshop on “ACT as a transformative…

Ted Theodosopoulos

2022-10-14

In celebration of Dana Scott

The greatest privilege of working at Topos is being part of a community of incredible people. One of these people is Dana Scott. Dana’s research achievements need little introduction: his ideas have changed our understanding of computing and logic, and have been recognised with numerous awards…

2022-10-09

Ethics, Technology, and Embracing the 21st Century

Last week our chair Ilyas Khan hosted Lord Rees for a discussion on Ethics, Technology, and Embracing the 21st Century at the Royal Society. The conversation ranged broadly, from the need for public intellectuals; to the limits of science and what we might never know, and the importance of sincere…

2022-10-07

So, what is category theory anyway?

Ask most category theory experts what it is, and they’ll launch into a session at a white board that can leave your head spinning. That is, if you are not a card carrying mathematician, as many of us are not. Even at Topos Institute, we have staff, donors, and friends who are not mathematicians.…

Tish Tanski, Paul Dancstep

2022-10-04

Nate & Jesse’s adjoint 5-tuple

In August, Nate Soares visited the Topos Institute. We told him a little about Poly and Proly, and he told us about what he wanted from a type theory. Probably the high point in the discussion for me is when he drew the following picture of what he wanted from a type theory.

It’s breakfast time! You wake up and walk to your kitchen and notice a loaf of bread, a knife, a raw egg (in its shell), a skillet, and a stove burner sitting on the counter. You’re hungry and your preferred state of existence is to, instead, have an egg sandwich sitting on your counter. You are…

2022-09-29

Using categorical logic for AI planning

rewriting

Acsets

Angeline Aguinaldo

2022-09-20

A dynamic monoidal category for strategic games

After last month’s wonderful ACT conference at University of Strathclyde in Glasgow, David Spivak and I spent some time with Matteo Capucci and Riu Rodriguez Sakamoto talking about how our theory of dynamic monoidal categories can model the non-cooperative strategic games that they study at…

Brandon Shapiro

2022-09-12

Data Operations are Functorial Semantics

Double categories illustrate that data munging is functorial semantics. This post reviews the functional and relational approaches to ologs and database schemas. Double-categorical ologs are meant to incorporate aspects of each. These should also be viewed as double-categorical database schemas…

databases

Michael Lambert

2022-09-06

What does it take to run a research nonprofit?

We’re exploring a new role at Topos: Director of Research. We’d love to hear from or about anyone who might be interested!

Tish Tanski

2022-09-01

Strategies for ethically oriented pursuit of research and technology

At Topos we’re motivated to build technologies that help us understand the complexity of the systems around us, and to cooperate to improve them, for everyone’s sake. But sometimes good intentions are not quite enough. For example, cooperation requires trust, and we want our technologies to help…

Alejandra Arciniegas, Brendan Fong

2022-08-24

Imagining Bicomodules with Type Theory

Some say that the objects of mathematics really exist in some Platonic realm, and some say they are fictions in the minds of mathematicians. Either way, it is evident that they cannot be directly grasped with the ordinary five senses. Nobody has ever seen a perfect circle or a natural number.…

Joshua Meyers

2022-08-19

Computation and Category Theory

In a recent talk, David Spivak, my advisor at Topos Institute, described Poly as “the language of computation”, due to its facility in describing concepts in computer science such as data migration, dependent types, and Turing machines. But is Poly really the language of computation? To address…

Joshua Meyers

2022-08-10

It’s Proly like Poly but better

Every mathematician I talk to agrees that Poly is an incredibly rich category. How could they not: it’s complete, cocomplete, three orthogonal factorization systems, two monoidal closed structures, its comonoids are categories, etc. And yet there is a common complaint about it, when it comes to…

2022-08-04

Our Summer Research Associates in 2022

The Topos Institute is pleased to introduce the 2022 cohort of Summer Research Associates! Each of them brings a unique and meaningful perspective to the Topos team, and we consider ourselves lucky to have the opportunity to engage in this important learning and teaching experience. Education is a…

Juliet Szatko

2022-07-14

Dialectica categories and polynomial functors (Part 1)

Like many stories in math, this one is about building bridges: uniting the dialectica categories from Valeria’s 1991 doctoral thesis with David’s work on polynomial functors. I’ve been working with David on polynomial functors for two years now, so when I was placed in Valeria’s team at the AMS’s…

dialectica

Nelson Niu

2022-07-12

A 15-year history of my research program, in 10 minutes

This post presents a video presentation, originally shown to the Topos Board of Directors, about David’s research program over the past 15 years, from 2007 to present.

2022-07-07

Graphs in Poly

It’s really amazing to me that comonoids in Poly are categories, and I think it would be cool if more people understood that on a gut level. But explaining it takes more time than people are sometimes prepared to give. So today, I want to explain a much simpler case—one which gives a lot of the…

2022-06-16

All models are wrong, but…

What makes a good mathematical model? For recent work at Topos, it’s when the model helps people cooperate to achieve collective goals. This has a few implications for how we design modelling software.

2022-06-07

An account of sense-making

When you say “oh, that makes sense” or “no, that doesn’t make sense”, you’re talking about whether or not the story fits together or adds up right. But you may also say “I have a sense of when people are feeling awkward” or “I seem to use my sense of smell more than other people do”. Could it be…

philosophy

David Spivak, James Dama

2022-06-03

Decorated cospans via the Grothendieck construction

Building on the double Grothendieck construction introduced last time, we explain how decorated cospans are instance of the Grothendieck construction. This perspective suggests a natural generalization of decorated cospans, which we illustrate through several examples.

2022-05-30

Grothendieck construction for double categories

What is the Grothendieck construction for double categories? We explore one possible answer to this question, based on the perspective that double categories are categories internal to \mathsf{Cat}. In fact, we suggest a general procedure for doing the Grothendieck construction on any structure…

2022-05-23

Join Topos as a senior leader in finance and operations!

Topos is more than just a collection of researchers: we are a community pioneering a new, international, and interdisciplinary tech organisation. We do this in order to help create a future where the systems around us, large and small, allow every member of society to flourish. While our…

2022-05-20

The “artificial” distinction

I’m like, “oh man, people always talk about ‘artificial’ this, ‘artificial’ that, but what’s up with that? I mean aren’t we humans part of nature too? Are fish and monkeys natural in some way that we humans are not? Are our products somehow not natural, whereas termite mounds are natural? This…

2022-05-18

The many facets of Networked Mathematics

Last year we started a discussion on the production of mathematics, the pre-industrial kind of process that we all still follow of solving problems and taking the solutions to the market of ideas (conferences, seminars, blog posts, twitter, coffee breaks in the department, etc.) as…

2022-04-18

Diagrammatic equations and multiphysics (Part 2)

physics

Previously, we surveyed some of the fundamental concepts from our paper on the use of diagrams to present equations from mathematical physics. In this blog post we take a look at how we can enrich our framework, introducing cartesian and symmetric monoidal products in order to be able to express…

Tim Hosgood, Evan Patterson

2022-04-15

Diagrammatic equations and multiphysics (Part 1)

physics

Using diagrams to encode equations between physical quantities is something that has become more and more common over the past few decades. To a category theorist, however, the use of such diagrams is not as formal as one would perhaps like. In this new paper, we fix this problem: systematising…

Tim Hosgood, Evan Patterson

2022-04-08

Co-design of dynamical systems

In 2015, Andrea Censi invented a beautiful category-theoretic way to collaboratively and computationally design new things under complex systems of constraints. For example, suppose that a robot is made of a chassis and a motor. Since the motor powers the chassis and the chassis carries the motor…

2022-04-06

2022 Workshop on Polynomial Functors

This year, Topos sponsored a second workshop on polynomial functors. It consisted of 20 talks, spread out over five days (all of which have been recorded and can be found on our YouTube channel) Like last year, the organizers were myself and my friend and colleague Joachim Kock. Two things made…

2022-03-29

Creating mathematics

People have a wide variety of feelings about math: one considers it horrifically painful whereas another considers it exquisitely beautiful. Some see it as the furthest thing from nature, a human construct of black-and-white thinking; others see it as the most natural thing: the enduring forms…

2022-03-25

Brits in Berkeley

Throughout the past calendar year, Topos Oxford has mostly comprised just two people: Toby St Clere Smithe and myself (Tim Hosgood). Since Berkeley has been the Topos site of initial focus, we’ve been working mostly remotely from the rest of the team. But last week was an exciting one for us all…

2022-03-22

Launch of the Women in Logic website

World Logic Day is an international day proclaimed by UNESCO in association with the International Council for Philosophy and Human Sciences (CIPSH) in 2019. The idea is to celebrate it on the 14th of January every year. This year we decided to celebrate Logic Day by launching the Women in Logic…

2022-03-17

So, what’s the plan?

Over the past six months, we’ve conducted interviews across the Topos community, and drafted and re-drafted our 2021–2024 strategic plan, so that we can coordinate effectively around a shared strategy as well as a shared vision. We’d like to share that plan with you all today.

2022-03-15

Call for 2022 Summer Research Associates

Come spend the summer at the Topos Institute! For early-career researchers, we’re excited to open up applications for our summer research associate program.

2022-03-01

Topos seeks Research Software Engineer

We invite applications to join the Topos Institute as a Research Software Engineer, to work on software for compositional data integration and scientific modeling in the Julia programming language.

2022-02-21

Playing with Shapes at Topos

This month sees a new person joining the team here at Topos: we’re thrilled to welcome Brandon Shapiro! He’s working with David Spivak on the theory and applications of polynomial functors. We asked him to write a short blog post introducing both himself and his mathematics.

Brandon Shapiro

2022-02-15

Poly makes me happy and smart

There are many reasons people like math; Poly checks all the boxes. My colleague Valeria says our Topos blog posts are often too technical and asked me to write a ‘fluffier’ blog post about Poly, one with a few unicorns and sprinkles. So here goes! Caveat: the author retains the right to…

2022-01-19

Topos Institute seminars in 2022

With a new calendar year comes a new seminar! On the 25th of January, our very own David Spivak will be giving the inaugural talk ("Categorical interaction in the polynomial ecosystem") of the Intercats seminar, and the seminars will then be held every other Tuesday at 17:00 UTC. The Topos…

2022-01-07

Creating new categories from old: Selection categories

In this short post, I’ll describe a way of creating new categories from old. It reminds me of a ‘particle filter’ or ‘natural selection’. This method comes from the theory of polynomial functors, but I’ll confine all the technical details to a single section, so you don’t need to know anything…

2021-12-30

Topos in 2021: a retrospective

As this calendar year draws to a close, I asked the members of Topos to each write a little bit about what they’ve worked on this year, how things have gone, and what plans they might have for the future. Here’s what they wrote!

2021-12-02

Left adjoints, lenses, and localisation

In this post I am going to recount three stories that I’ve been thinking about recently: Cauchy completion, internal lenses, and GZ/zigzag-localisation. The first two of these stories are probably more likely to be familiar to the (applied) category theory audience of this blog than the third. For…

2021-11-24

Deep neural networks as nested dynamical systems

This week, we (David Spivak and Tim Hosgood) uploaded a new paper to the arXiv: “Deep neural networks as nested dynamical systems”. In it, we discuss how to think of both deep neural networks and interacting dynamical systems as some encompassing generalisation, which we call deeply interactive…

Tim Hosgood, David Spivak

2021-11-05

Categorical systems theory

Categorical systems theory is the study of the design and analysis of systems using category theory. Category theory is a hopelessly abstract branch of pure mathematics, so abstract that even the other mathematicians refer to it as “abstract nonsense”. Why would anyone want to study such complex…

David Jaz Myers

2021-11-04

We’re looking for a postdoc!

Topos is hiring! We invite applicants for a one-year postdoc position, from November 2021, to work on a project related to dynamical systems and data using the language of polynomial functors.

2021-10-08

Poly inside Poly

In this post, I’ll explain how to use Steve Awodey’s notion of universe to represent the set of polynomials as a certain naturally-derived object in Poly: “Poly inside Poly”. That is, given a universe u there’s an associated polynomial functor w that encodes u-small polynomials, and also…

2021-09-22

Compositional Thermostatics

In this post, we explore the compositionality of thermodynamic systems at equilibrium (thermostatic systems). The main body of this uses no category theory, and reviews the physics of thermostatic systems in a formalism that puts entropy first. Then there is a brief teaser at the end showing how…

physics

2021-09-09

Enriched categories as dynamical systems

Often in applied category theory, we hear about how systems form categories — categories of systems. But what about when the systems are categories? In this lecture, we introduce the idea of enriched categories, taking the point of view that an enriched category is a sort of dynamical system.

David Jaz Myers

2021-09-02

Announcing Em-Cats

The Em-Cats seminar series that was announced a while back is almost about to start! We will be welcoming our first speaker (Jade Master) on Wednesday the 25th of August. Here is some more information about this opening talk.

2021-08-24

Dirichlet polynomials and entropy

Recently, we (David Spivak and Tim Hosgood) put a preprint on the arXiv called “Dirichlet polynomials and entropy”, which explains how to recover Shannon entropy from a rig homomorphism involving the weighted geometric mean. In this blog post, we explain the main story, from a slightly different…

Tim Hosgood, David Spivak

2021-07-25

Get to know our Summer Research Associates

We’re excited to welcome our 2021 summer research associates to our Berkeley campus! This summer we have Sophie Libkind, Owen Lynch, David Jaz Myers, and Nelson Niu joining our work on connected intelligence and model-based scientific computing. They bring new exciting energy and ideas, and we’re…

Brendan Fong, Sophie Libkind, David Jaz Myers, Owen Lynch, Nelson Niu

2021-07-19

Introducing the MathFoldr Project

At Topos we believe knowledge empowers people, and that our community’s expertise should be available to all who seek it out. But simple availability is not enough. True access is more than an open door: it’s clear, legible street signs, elevators, and gently sloping on-ramps. And with modern AI…

Brendan Fong, Valeria de Paiva

2021-07-11

Seminars and workshops so far

Although this blog has only just launched, the Topos Institute has been running quite a few public-facing events over the past five months or so, including a workshop and regular seminar series. With even more on the horizon, let’s take a look at what has been, and will be, going on.

2021-07-08

Jump monads: from conjugation to dependent types

The world of polynomial functors often seems so rich that it should contain the whole universe, and there are some tantalizing suggestions that, in fact, it does. Given a dependent type theory, we can follow Steve Awodey and encode its syntax into a “universe polynomial” indexed by the types of…

2021-07-01

Welcome to the Topos Blog

Welcome to the Topos blog! Here you’ll find news and announcements, statements of our visions and projects at Topos, expository articles on topics of interest to the Topos community, and reports of research in progress.