Blog

Research articles, statements of vision, and more

An informal representation of the model

Extending mass-action semantics, Part 1
CatColab
modeling

One particular use of formal categorical models is for exposition and learning. Indeed, the act of creating one can be a useful exercise for exposing implicit assumptions. In this post, we walk through the process of creating a diagrammatic representation of a system of ODEs and highlight the…

Tim Hosgood
2026-03-16

Figure 1. Phase space for the system \dot{x}=x-x^3. The systems has attracting equilibria at x=-1 and x=1, and a repelling equilibrium at x=0. The vectors indicate the direction of the flow.

Composition of attractor lattices

This summer at the Topos Institute, under the supervision of Dr. Sophie Libkind, I studied the composition of attractors. The project itself started earlier with my advisor, Dr. William Kalies, who asked me the following question: how do attractor lattices behave when we combine dynamical systems?…

Tony Wehbe
2026-01-30

CatColab v0.4: Robin
CatColab

Since our last blog post about CatColab, we’ve had two releases, meaning we’re now at v0.4: Robin. This brings a few major new features, including compositional notebooks and novel analyses for Petri nets.

Tim Hosgood
2026-01-08

Call for 2026 Summer Research Associates
hiring
personnel

Come spend the summer at the Topos Institute! For early-career researchers, we’re excited to open up applications for our 2026 Summer Research Associate program.

Kris Brown
2025-11-24

Set-sets

An introduction to endofunctors on Set. Algebras and coalgebras of endofunctors model basic objects of study in mathematics, computer science, and logic. This post explains how to interpret algebras and coalgebras of endofunctors on Set as sets equipped with operations or “co-operations”. We also…

Aaron Fairbanks
2025-11-21

DOTS from double theories
double categories
category theory
systems theory

Working as a mathematician at Topos means that I spend a lot of time hearing about double categories these days, which are not something that I have much previous familiarity with. In fact, they come up in two seemingly different ways in Topos’ work at the moment: double theories, and double…

Tim Hosgood
2025-11-07

Open position: Director of UK Operations
Topos
hiring

Topos UK is hiring for a Director of Operations. What does this mean? Read more to find out.

Tim Hosgood
2025-10-07

Linear time frame of w = \langle w_0, w_1, w_2, w_3, \dots\rangle.

Free PLTL algebras and a coalgebraic extension of hyperdoctrines
categorical logic
logic

This blog post provides an overview of the work I had done with José Siqueira this summer. Inspired by the free Boolean/Heyting algebra of a given set, we develop a free-forgetful adjunction between posets and PLTL temporal algebras, where PLTL denotes propositional linear temporal logic. We…

Q Le
2025-09-26

Incremental query updating in adhesive categories
acsets
applied category theory
databases
rewriting
AlgebraicJulia

Working in the general setting of adhesive categories, we derive a practical algorithm for incrementally updating a query’s results with respect to small changes in the object being queried.

Kris Brown
2025-08-15

Substitution is also pushout
rewriting

How we can think about pushouts as applying rules via substitution, featuring examples in categorical databases and Datalog.

Kris Brown
2025-08-06

CatColab for Model Building
CatColab
modeling
crosspost

Nate Osgood, together with 4 students from the Computational Epidemiology and Public Health Informatics Lab in Saskatoon, Canada, recently ran a community group model building event focusing on the drivers for homelessness in their city. This post describes the event, and the next steps that they…

Nathaniel Osgood
2025-07-18
 
Liberating synthetic quasi-coherence from forcing
topos theory
type theory

In this blog post, I’ll reformulate Ingo Blechschmidt’s “synthetic quasi-coherence” axiom — or more precisely his “general nullstellensatz” — as a lifting property inspired by Ivan Di Liberti’s work on coherent toposes and ultrastructures. This lifting property shows that synthetic…

David Jaz Myers
2025-07-13

The Summer RAs, from left to right: Ea, Lucy, Matt, Corinthia, Aaron, Tony

Our Summer Research Associates in 2025
Topos
personnel

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, with more detailed blog posts from each…

Molly White
2025-07-01

How to prove equations using diagrams, part 2
category theory
rewriting
This is the second part in a series about diagrammatic reasoning, inspired by e-graphs. Last time, we reviewed the concept of initial functor and showed by example how to calculate with diagrams and initial functors. This time, we make that calculus more systematic and we reconceive e-graphs in…
Evan Patterson
2025-06-10
 
Growing the Topos tech team
CatColab
technology
personnel
Mathematics and science achieve their greatest impact when innovative basic research is coupled with translation into robust and usable technology. That is easier said than done. Today, research and development, despite appearing conjoined in the acronym “R&D,” are rarely done together. Research…
Evan Patterson
2025-05-29

How to prove equations using diagrams, part 1
category theory
rewriting
An e-graph, short for “equality graph,” is a data structure that maintains a congruence relation on expression trees: an equivalence relation stable under forming new expressions. First devised by Nelson and Oppen in 1980 (Nelson 1980; Nelson and Oppen 1980), e-graphs received a surge of new…
Evan Patterson
2025-05-27

The Sun is Good
informal
vision
CatColab
AI
philosophy

This is a shortened version of an essay I wrote last month for the Cosmos Ventures fellowship. Cosmos is about grappling with how more sophisticated philosophical thinking can help us adapt to advancing AI technologies. I hope you like it!

Kevin Carlson
2025-05-09

TopOx seminar
events
micro

Next week we’re co-hosting an in-person seminar talk in Oxford. This post contains the mailing announcement.

Tim Hosgood
2025-05-08
 
The three functions of a name: reference, identity and display
programming
double categories
This is a brief little post about naming. Naming is well-known to be one of the hardest problems in computer science. But why is this the case? I claim that one of the reasons is because often naming systems are being asked to do triple-duty; a single name is used to perform the three functions of …
Owen Lynch
2025-05-07

Graded categories as double functors
category theory
double categories
At last week’s Topos Colloquium, Rory Lucyshyn-Wright told us about categories graded by a monoidal category, following his recent preprint (Lucyshyn-Wright 2025). Graded categories, short for locally graded categories, were first introduced by Richard Wood under a different name (Wood 1976, 1978).…
Evan Patterson
2025-05-01

Wiring Euclid for manufacturing
wiring diagrams

In this blog post we consider the practical tradition of Euclid’s elements, in contrast to its more familiar role as a model of axioms and proof. Inspired by that tradition and combining it with the notion of wiring diagrams we propose an abstract model to think about and problem solve issues of…

Edmund Harriss
2025-04-30

Care at the Edge of Automation
technology
AI
vision

Technologies don’t just solve problems, they change us. We invent technologies, and they invent us in turn, shaping our lives and worlds. This is the phenomenon that Terry Winograd and Fernando Flores, in Understanding Computers and Cognition (1986), called “ontological design.” It matters now…

B. Scot Rousse
2025-04-29

Distilling Research at Scale
technology
vision

New concepts must be explained to be understood, yet we rarely invest sufficient effort in clarifying them. While explanations demand time and care, this investment is essential, because without it, ideas remain opaque, inaccessible, and dormant. Today we see that the pace of explanation…

Hamidah Oderinwale
2025-04-04

A Category Theory-Inspired BBN
crosspost
micro

Brendan recently spoke with Eric Gilliam and the interview resulted in a blog post on Eric’s blog.

Brendan Fong
2025-03-11

Towards codified context, durable documentation, and process preservation
technology
vision

Here, my goal is to introduce my research interests and offer some exposition on how they came to be. Libraries are troves of information, but they also include the systems that help people make use of the information inside them. We should consider rethinking the library and ensure it includes…

Hamidah Oderinwale
2025-02-27

CatColab 0.2: Wren
CatColab

Today we’re excited to announce the first alpha release of CatColab 0.2: Wren. CatColab is software for making models of the world together.

Kevin Carlson
2025-02-05

Polynomial universes and natural models
polynomial functors
type theory

In this post, I summarize the work I conducted over the summer of 2024 at the Topos Institute, together with David Spivak, wherein we have aimed to extend, refine, and generalize the treatment of models of type theory in terms of polynomial functors via the concept of polynomial universes. We…

C.B. Aberlé
2024-12-10

Structure-aware version control via observational bridge types
type theory
version control

In this post, we suggest a way to use observational bridge types (as in the Narya proof assistant) for structure aware version control.

David Jaz Myers
2024-11-13

Call for 2025 Summer Research Associates
hiring
personnel

Come spend the summer at the Topos Institute! For early-career researchers, we’re excited to open up applications for our 2025 Summer Research Associate program.

Shaowei Lin
2024-11-12

Neural wiring diagrams for message passing in multiscale organizations
polynomial functors
operads
wiring diagrams

In our recent paper, “Dynamic task delegation for hierarchical agents”, Sophie Libkind and I described a task delegation from an agent to a team of subordinates as morphisms in a certain operad called \mathbb{O}\mathbf{rg}_{\mathfrak{m}}. However, such morphisms include a great deal of data, and…

David Spivak
2024-11-08

Declarative Models and Collaborative Modeling
modeling
hiring

Although we at the Topos Institute spend much of our days applying category theory, underlying our activities is a philosophy around the activity of science and engineering that is not explicitly category-theoretic in nature. In this blog post, we lay out some of the core ideas of this philosophy…

Owen Lynch
2024-10-31

Taking nonlogical concepts seriously
vision
language
philosophy

What is the right math to capture our concepts which don’t have formal definitions? Recent work in philosophy of language clarifies the relationship between logic and good reasoning, with consequences for science, applied math, and AI. In this post we will introduce logical expressivism, its…

Kris Brown
2024-10-11

Introducing CatColab
technology
categorical logic
double categories
Topos
CatColab

Today we’re excited to announce the first pre-alpha release of our new software CatColab 0.1: Hummingbird. CatColab is software for making models of the world together.

Kevin Carlson
2024-10-02

Open curricula and assessment tools workshop in Kisumu
community
technology

In August, Brendan and I attended the American Institute of Mathematics workshop “Open source mathematics curriculum and assessment tools”, hosted by Maseno University in Kisumu, Kenya. The workshop was a fantastic opportunity to meet new specialists in a variety of domains and hear about what…

Tim Hosgood
2024-09-06

Postdoc job openings in Topos Oxford, UK
hiring
personnel
Topos

Topos is opening its doors in Oxford, UK, and is inviting applications for two postdoctoral researchers to come work with us on one of our first projects.

Tim Hosgood
2024-08-30

Plausible fiction
informal
vision

Each of us can see problems in our world or our community, cases where something we care about is in need of attention. At the same time, technology is vastly increasing our ability to spread and collectively consider diverse ideas. In this post, I imagine a future world in which people use new…

David Spivak
2024-08-27

Wiring diagrams for Mealy machines
wiring diagrams

Operad algebras are a known tool for the composition of dynamical systems, but what happens when we have a machine where the output of the machine also depends on the input into it? In this post, we give background on how we want to extend the operad of directed wiring diagrams to the operad of…

Keri D’Angelo
2024-08-19

A polynomial account of Bayesian update
polynomial functors

One of the things I do at Topos is make sense of some aspect of the world by articulating it in mathematics. Follow along as I make sense of Bayesian update using the mathematics of polynomial functors.

Sophie Libkind, David Spivak
2024-08-12

Rethink math talks
outreach
informal
events

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

The Summer RAs, from left to right: C.B., Keri, and Owen

Our Summer Research Associates in 2024
Topos
personnel

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…

Tim Hosgood
2024-07-29

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

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.

Priyaa Varshinee Srinivasan
2024-07-09

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

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

Priyaa Varshinee Srinivasan
2024-07-04

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

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.

Priyaa Varshinee Srinivasan
2024-06-28

Toward compact double categories: Part 2
category theory
double categories
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…
Evan Patterson
2024-06-24

Toward compact double categories: Part 1
category theory
double categories
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…
Evan Patterson
2024-06-20

Applied Category Theory for Engineering Design: A Teaser Video
outreach
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.

Brendan Fong
2024-06-03

Launching new collaboration with Chapman University
Topos

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…

Brendan Fong
2024-05-28

Ontological Commitments for Boundaries
dynamical systems

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…

Tim Hosgood
2024-04-05

Poly @ Work 2024
events
polynomial functors

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
events
systems theory
category theory

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…

Owen Lynch
2024-03-13

SCAI — AI for the Global Good
events
AI
ethics
video

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…

Brendan Fong
2024-02-02

Algebras are promonads
category theory
double categories

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

Evan Patterson
2024-01-29

A handwritten solution to solving the acceleration of a spring on earth

Composing springs
AlgebraicJulia
operads
dynamical systems
physics

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
category theory
double categories
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.

Evan Patterson
2024-01-15

Building dynamic structures
polynomial functors

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
polynomial functors
type theory
category theory

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
language

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

A Rosetta Stone for our vignettes

Trustworthy Reification
language

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
programming languages
AlgebraicJulia

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
applied category theory

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
Topos

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…

Brendan Fong
2023-11-17

Introducing InterTypes
acsets
AlgebraicJulia
crosspost

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

Owen Lynch
2023-11-14

Solving problem-solving
polynomial functors
category theory

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
polynomial functors
category theory
geometry

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…

David Spivak
2023-10-26

Retrotransformations
category theory
categorical logic
double categories
lenses

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.

Evan Patterson
2023-10-20
 
Cartesian double theories
category theory
categorical logic
double categories
video

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

Evan Patterson
2023-10-13

Our Summer Research Associates in 2023
Topos
personnel

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
polynomial functors
category theory

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}.

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

David Spivak
2023-09-21

Unbiased monoidal categories are pseudo-elements
category theory
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.

Evan Patterson
2023-08-15

A nuclear adjunction between Poly and Dir
polynomial functors
category theory

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…

David Spivak
2023-07-21
 
Dialogue on a mathematical approach to the good
ethics
category theory
philosophy

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…

Bartłomiej Skowron, David Spivak, David Corfield
2023-07-11

Agent-based modeling via graph rewriting
rewriting
dynamical systems
acsets
AlgebraicJulia
crosspost

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

Welcome Shaowei!

Welcoming our new Director of Research, Shaowei Lin
personnel
Topos

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

Brendan Fong
2023-06-22

Acsets with variables
acsets
AlgebraicJulia
rewriting
crosspost

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
events
AI

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
dynamical systems
category theory

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 functors
category theory

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…

David Spivak
2023-05-26

Stock-flow diagram for SIR disease model

Categories for Epidemiology
applied category theory
modeling
AlgebraicJulia

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
geometry
dynamical systems
crosspost

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

Owen Lynch
2023-05-08

Spooling out syntax from behavior
polynomial functors
dynamical systems
programming languages

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…

David Spivak
2023-04-24

Conegation rewriting
category theory
logic
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”.…

Eigel Rischel
2023-04-11

Imperative Programming with Poly
polynomial functors
programming languages

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…

Owen Lynch
2023-04-05
 
Seventy Years Using Fixed Points
lambda calculus
type theory

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

Algebraic geometry for the working programmer
geometry
logic
lenses
crosspost

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.

Owen Lynch
2023-03-23
 
Lotteries: a constructive version of the distributions monad
polynomial functors
category theory

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…

David Spivak
2023-03-23

Structured cospans as a cocartesian equipment
applied category theory
double categories
systems theory

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…

Evan Patterson
2023-03-15
 
Category theorists welcome self-learners in a new outreach panel
events
category theory
community
outreach

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…

David Spivak, Alexis Anaya
2023-03-09

Picture from aninfinityofhypotheses.wordpress.com

Women in Logic: the history so far
logic
community

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…

Valeria de Paiva
2023-03-06
 
What made the FRA workshop structure successful?
events

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.

Elena Di Lavore, Mario Román
2023-02-22

Ask Eugenia Cheng about Category Theory!
category theory
events
video

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…

Brendan Fong
2023-02-09

Promonoidal categories and wiring diagrams
wiring diagrams
operads
category theory

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…

David Spivak
2023-01-31

Recursive Types via Domain Theory
programming languages
type theory
lambda calculus
logic

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

The map of mathematics [Source]

Preparing for Networked Mathematics
MathFoldr
AI
NLP

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…

Valeria de Paiva
2023-01-05

Lenses are semi-monads, maybe lenses are monads
polynomial functors
lenses
wiring diagrams

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…

David Spivak
2022-12-20

Research associates from last summer’s program

Research opportunities at Topos in early 2023
hiring
Topos

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

Evan Patterson
2022-12-12

The AMS poster for MRCs

Dialectica Categories in Computing
dialectica
events

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…

Valeria de Paiva
2022-12-06
 
Scott’s model of lambda calculus
lambda calculus
logic

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

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…

Brendan Fong
2022-11-19

Mathematical concepts: how do you recognize them?
AI
NLP
MathFoldr
language

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
applied category theory
language

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.…

David Spivak
2022-11-07

Who is category theory for?
community
video
outreach

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?

Brendan Fong
2022-11-04

A screenshot of Definition 3.7 from (Shapiro and Spivak 2022)

When you light up, I light up
polynomial functors
dynamical systems
applied category theory

“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.

Sophie Libkind, David Spivak
2022-10-28
 
Project cards: a tool for research transparency
ethics

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
events
applied category theory
community

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
video
events

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…

Brendan Fong
2022-10-09

Ethics, Technology, and Embracing the 21st Century
ethics
events

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…

Brendan Fong
2022-10-07
 
So, what is category theory anyway?
category theory
video

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
polynomial functors
type theory

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.

David Spivak
2022-09-29

Using categorical logic for AI planning
AI
rewriting
acsets

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…

Angeline Aguinaldo
2022-09-20

A dynamic monoidal category for strategic games
polynomial functors
dynamical systems
applied category theory
lenses

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
databases

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…

Michael Lambert
2022-09-06
 
What does it take to run a research nonprofit?
hiring
Topos

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
ethics

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
type theory
polynomial functors

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
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
polynomial functors

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…

David Spivak
2022-08-04

Our Summer Research Associates in 2022
Topos
personnel

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)
polynomial functors
dialectica

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…

Nelson Niu
2022-07-12
 
A 15-year history of my research program, in 10 minutes
category theory
polynomial functors
dynamical systems
video

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.

David Spivak
2022-07-07

Graphs in Poly
polynomial functors

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…

David Spivak
2022-06-16

All models are wrong, but…
modeling

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.

Brendan Fong
2022-06-07
 
An account of sense-making
informal
philosophy

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…

David Spivak, James Dama
2022-06-03

Decorated cospans via the Grothendieck construction
applied category theory
double categories
systems theory

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.

Evan Patterson
2022-05-30

Grothendieck construction for double categories
category theory
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…

Evan Patterson
2022-05-23

Join Topos as a senior leader in finance and operations!
Topos
hiring

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…

Brendan Fong
2022-05-20
 
The “artificial” distinction
informal

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…

David Spivak
2022-05-18

The many facets of Networked Mathematics
MathFoldr
AI
NLP

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…

Valeria de Paiva
2022-04-18

Diagrammatic equations and multiphysics (Part 2)
applied category theory
physics
modeling

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)
applied category theory
physics
modeling

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
category theory
polynomial functors
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…

David Spivak
2022-04-06

2022 Workshop on Polynomial Functors
events
polynomial functors
category theory

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…

David Spivak
2022-03-29
 
Creating mathematics
category theory
informal

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…

David Spivak
2022-03-25

Brits in Berkeley
personnel
Topos
informal

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…

Tim Hosgood
2022-03-22

Launch of the Women in Logic website
logic
community

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…

Valeria de Paiva
2022-03-17

So, what’s the plan?
Topos

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.

Brendan Fong
2022-03-15

Summer RAs and mentors hard at work at the beginning of summer 2021

Call for 2022 Summer Research Associates
hiring
personnel

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.

Evan Patterson
2022-03-01
 
Topos seeks Research Software Engineer
hiring
personnel
video

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.

Evan Patterson
2022-02-21

Playing with Shapes at Topos
Topos
personnel

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

Me at the chopped live machine, 1980

Poly makes me happy and smart
polynomial functors
informal

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…

David Spivak
2022-01-19

Topos Institute seminars in 2022
events

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…

Tim Hosgood
2022-01-07

Creating new categories from old: Selection categories
polynomial functors

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…

David Spivak
2021-12-30

The seminar room at Topos Berkeley over the summer, with visitors and more permanent members all gathered together for some productive conversations

Topos in 2021: a retrospective
Topos

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!

Tim Hosgood
2021-12-02

Left adjoints, lenses, and localisation
lenses
geometry
category theory

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…

Tim Hosgood
2021-11-24

Deep neural networks as nested dynamical systems
polynomial functors
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

A circuit

Categorical systems theory
category theory
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!
hiring
Topos
personnel

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.

Brendan Fong
2021-10-08

Poly inside Poly
polynomial functors

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…

David Spivak
2021-09-22

Two lumps of metal, one cold and one hot

Compositional Thermostatics
applied category theory
physics

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…

Owen Lynch
2021-09-09

Enriched categories as dynamical systems
category theory
dynamical systems
video

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

The cohort of Em-Cats speakers and their talk titles/topics. [PDF]

Announcing Em-Cats
events

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.

Tim Hosgood
2021-08-24

Our presentation for the ACT poster session (click here for a larger version)

Dirichlet polynomials and entropy
polynomial functors

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

Left to right: Sophie Libkind, David Jaz Myers, Nelson Niu, Owen Lynch

Get to know our Summer Research Associates
Topos
personnel

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

A telegram tape

Introducing the MathFoldr Project
MathFoldr
AI
NLP
language

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

Clockwise, from top left: Johan García Vargas, Jade Master, Gun Pinyo, Nuiok Dicaire, Jennifer Brown, Minani Iragi, Juan F. Meleiro; not pictured, Eli Hazel

Seminars and workshops so far
events
Topos

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.

Tim Hosgood
2021-07-08

Jump monads: from conjugation to dependent types
polynomial functors

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…

David Spivak
2021-07-01

Our online celebration of the official Day 1 of Topos Institute, back in January 2021

Welcome to the Topos Blog
Topos

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.

Brendan Fong
2021-06-30
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