People / David Spivak

Senior Scientist and Institute Fellow & Treasurer (US)

About

David Spivak leads the Collective Intelligence research group at Topos, after a decade as a researcher at MIT. Since his PhD from UC Berkeley in 2007, he has worked to bring category-theoretic ideas into science, technology, and society, through novel mathematical research, through collaboration with scientists and engineers from disciplines including Materials Science, Chemistry, Robotics, Aeronautics, and Computer Science. His mission at Topos is to help develop the ability for people, organizations, and societies to make sense of—and hence to serve—the systems which sustain them.

Personal website
http://dspivak.net/

Grants

Talks

Here are slides and recordings from talks given since starting at Topos.

Blog posts

 
Discrete vs. filtered accessibility and presentability

Accessible and presentable categories are nice because they make calculating hom sets easy. But not all hom-sets-easily-calculable categories are accessible. In this post we suggest analogous notions, replacing “filtered colimits” with “discrete colimits” (i.e. coproducts), and similarly “compact…

David Spivak
2026-09-24

Who cares about a computation?

Computers were invented to automate cognitive labor for us. But the work we need to get done depends on what we care about. This issue of care is—unlike the mechanism of computation, extracted by Turing—not something we know how to account for. Present-day computers, including AI, do not care…

David Spivak
2026-09-22

Neural wiring diagrams for message passing in multiscale organizations

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

Plausible fiction

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

A polynomial account of Bayesian update

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

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

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

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…

David Spivak
2023-10-26

Powers of polynomial monads

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

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…

David Spivak
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…

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

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…

David Spivak
2023-05-26

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…

David Spivak
2023-04-24
 
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…

David Spivak
2023-03-23
 
Category theorists welcome self-learners in a new outreach panel

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

Promonoidal categories and wiring diagrams

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

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…

David Spivak
2022-12-20

Where matter and pattern meet

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

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

When you light up, I light up

“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

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.

David Spivak
2022-09-29

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…

David Spivak
2022-08-04
 
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.

David Spivak
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…

David Spivak
2022-06-16
 
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…

David Spivak, James Dama
2022-06-03
 
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…

David Spivak
2022-05-18

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…

David Spivak
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…

David Spivak
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…

David Spivak
2022-03-25

Me at the chopped live machine, 1980

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…

David Spivak
2022-01-19

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…

David Spivak
2021-12-30

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

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…

David Spivak
2021-09-22

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

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

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…

David Spivak
2021-07-01
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