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April 19, 2019

Can 1+1 Have More Than Two Points?

Posted by John Baez

I feel I’ve asked this before… but now I really want to know. Christian Williams and I are working on cartesian closed categories, and this is a gaping hole in my knowledge.

Question 1. Is there a cartesian closed category with finite coproducts such that there exist more than two morphisms from 11 to 1+11 + 1?

Cartesian closed categories with finite coproducts are a nice context for ‘categorified arithmetic’, since they have 00, 11, addition, multiplication and exponentiation. The example we all know is the category of finite sets. But every cartesian closed category with finite coproducts obeys what Tarski called the ‘high school algebra’ axioms:

x+yy+x x + y \cong y + x

(x+y)+zx+(y+z) (x + y) + z \cong x + (y + z)

x×1x x \times 1 \cong x

x×yy×x x \times y \cong y \times x

(x×y)×zx×(y×z) (x \times y) \times z \cong x \times (y \times z)

x×(y+z)x×y+x×z x \times (y + z) \cong x \times y + x \times z

1 x1 1^x \cong 1

x 1x x^1 \cong x

x (y+z)xy×xz x^{(y + z)} \cong x y \times x z

(x×y) zx z×y z (x \times y)^z \cong x^z \times y^z

(x y) zx (y×z) (x^y)^z \cong x^{(y \times z)}

together with some axioms involving 00 which for some reason Tarski omitted: perhaps he was scared to admit that in this game we want 0 0=10^0 = 1.

So, one way to think about my question is: how weird can such a category be?

Posted at 4:43 AM UTC | Permalink | Followups (29)

April 17, 2019

Univalence in (∞,1)-toposes

Posted by Mike Shulman

It’s been believed for a long time that homotopy type theory should be an “internal logic” for (,1)(\infty,1)-toposes, in the same way that higher-order logic (or extensional type theory) is for 1-toposes. Over the past decade a lot of steps have been taken towards realizing this vision, but one important gap that’s remained is the construction of sufficiently strict universe objects in model categories presenting (,1)(\infty,1)-toposes: the object classifiers of an (,1)(\infty,1)-topos correspond directly only to a kind of “weakly Tarski” universe in type theory, which would probably be tedious to use in practice (no one has ever seriously tried).

Yesterday I posted a preprint that closes this gap (in the cases of most interest), by constructing strict univalent universe objects in a class of model categories that suffice to present all Grothendieck (,1)(\infty,1)-toposes. The model categories are, perhaps not very surprisingly, left exact localizations of injective model structures on simplicial presheaves, which were previously known to model all the rest of type theory; the main novelty is a new more explicit “algebraic” characterization of the injective fibrations, enabling the construction of universes.

Posted at 12:35 AM UTC | Permalink | Followups (2)

April 14, 2019

The ZX-Calculus for Stabilizer Quantum Mechanics

Posted by John Baez

guest post by Fatimah Ahmadi and John van de Wetering

This is the second post of Applied Category Theory School 2019. We present Backens’ completeness proof for the ZX-calculus for stabilizer quantum mechanics.

Posted at 1:20 AM UTC | Permalink | Followups (1)

April 9, 2019

Postdoctoral Researcher Position in Lisbon

Posted by John Huerta

Applications are invited for a postdoctoral researcher position in the “Higher Structures and Applications” research team, funded by the Portuguese funding body FCT.

For more, read on!

Posted at 6:54 PM UTC | Permalink | Post a Comment

April 4, 2019

Category Theory 2019

Posted by Tom Leinster

I’ve announced this before, but registration is now open! Here we go…

Third announcement and call for contributions

Category Theory 2019

University of Edinburgh, 7–13 July 2019

Invited speakers:

plus an invited tutorial lecture on graphical linear algebra by

and a public event on Inclusion-exclusion in mathematics and beyond by

Posted at 3:39 PM UTC | Permalink | Post a Comment

April 1, 2019

Motion Group Workshop in Leeds

Posted by Simon Willerton

Braid groups are what you get when you let points move around in the plane: motion groups, or loop braid groups, are what you get when you let circles move around in 3-space. There’s an upcoming workshop in Leeds this summer.

This workshop will revolve around loop braid groups and related braided and knotted structures, throughout different areas of geometric topology, with an eye on applications in physics, namely modeling topological phases of matter.

Posted at 6:59 PM UTC | Permalink | Followups (1)

March 24, 2019

Normalising Quantum Circuits

Posted by John Baez

guest post by Giovanni de Felice and Leo Lobski

This post marks the beginning of the second iteration of the Applied Category Theory School. The mentors this year are Miriam Backens, Tobias Fritz, Pieter Hofstra, Bartosz Milewski, Mehrnoosh Sadrzadeh, and David Spivak. Each mentor has provided two papers associated to their proposed project in the field of applied category theory. Our students read these papers, gaining the requisite background to engage the projects. As part of this process, they will write an article on each paper and post it here.

Posted at 7:51 PM UTC | Permalink | Post a Comment

The Kantorovich Monad

Posted by Simon Willerton

guest post by Paolo Perrone

On this blog there has been quite a number of posts about the interaction of category theory and probability (for example here, here, here, and here), as well as about applying category theory to the study of convex structures (for example here, here, and here).

One of the central constructions in categorical probability theory is that of probability monad. A probability monad can be thought of as a way to extend spaces in order for them to include, besides their elements, also their random elements.


Here I would like to talk about a probability monad on the category of metric spaces: the Kantorovich monad. As some of you may suspect, this is related to Kantorovich duality, which appeared on this blog in the context of enriched profunctors. The Kantorovich monad was introduced by Franck van Breugel for compact metric spaces in his note The Metric Monad for Probabilistic Nondeterminism (pdf). In the work that resulted in my PhD thesis (pdf), Tobias Fritz and I extended the construction, and discovered some very particular properties of this monad. In particular, this monad can be described purely in terms of combinatorics of finite sequences of elements! Most of the work explained in this post can be found in the paper by me and Tobias:

Posted at 10:36 AM UTC | Permalink | Followups (6)

Network Models from Petri Nets with Catalysts

Posted by John Baez

Here’s another paper from the network theory gang:

• John Baez, John Foley and Joe Moeller, Network models from Petri nets with catalysts.

Check it out! And please report typos, mistakes, or anything you have trouble understanding! I’m happy to answer questions here.

Posted at 7:21 AM UTC | Permalink | Post a Comment

March 21, 2019

Entropy mod p

Posted by Tom Leinster

I’ve just arXived a new paper:

Tom Leinster, Entropy modulo a prime, arXiv:1903.06961

clock with 13 hours

In it, you’ll find answers to these questions and more — and I’ll also answer them in this post:

  • What are logarithms and derivations mod pp?

  • What is the right definition of entropy for probability distributions whose “probabilities” are integers modulo a prime pp?

  • What is the right mod pp analogue of information loss?

  • What does it mean to say that log83(mod7)\log \sqrt{8} \equiv 3 \pmod{7}?

  • Why is entropy mod pp a polynomial?

  • What does all this have to do with polylogarithms and homology?

Posted at 8:05 PM UTC | Permalink | Followups (15)

March 14, 2019

The Myths of Presentability and the Sharply Large Filter

Posted by Mike Shulman

The theory of locally presentable and accessible categories and functors is an elegant and powerful tool for dealing with size questions in category theory. It contains a lot of powerful results, many found in the standard reference books Locally presentable and accessible categories (by Adamek and Rosicky) and Accessible categories (by Makkai and Pare), that can be quoted without needing to understand their (sometimes quite technical) proofs.

Unfortunately, there are a couple of such “results” that are occasionally quoted, but are actually nowhere to be found in AR or MP, and are in fact false. These are the following claims:

Myth A: “If 𝒞\mathcal{C} is a locally λ\lambda-presentable category and μ\mu is a regular cardinal with μλ\mu\ge\lambda, then every μ\mu-presentable object of 𝒞\mathcal{C} can be written as a μ\mu-small λ\lambda-filtered colimit of λ\lambda-presentable objects.”

Myth B: “If F:𝒞𝒟F:\mathcal{C}\to \mathcal{D} is an accessible functor between locally presentable categories, then FF preserves μ\mu-presentable objects for all sufficiently large regular cardinals μ\mu.”

Below the fold I will recall the definitions of all these words, and then discuss how one might fall into believing these myths, why they are is not true, and what we can do about it.

Posted at 9:29 PM UTC | Permalink | Followups (12)

Sporadic SICs and Exceptional Lie Algebras III

Posted by John Baez

guest post by Blake C. Stacey

On the fourteenth of March, 2016 — so, exactly three years ago — Maryna Viazovska published a proof that the E 8\mathrm{E}_8 lattice is the best way to pack hyperspheres in eight dimensions. Today, we’ll see how this relates to another packing problem that seems quite different: how to fit as many equiangular lines as possible into the complex space 8\mathbb{C}^8. The answer to this puzzle is an example of a SIC: a Symmetric Informationally Complete quantum measurement.

Posted at 1:33 AM UTC | Permalink | Followups (3)

March 11, 2019

Postdoc at Macquarie

Posted by Tom Leinster

Posted on behalf of my friends in Sydney

The Centre of Australian Category Theory at Macquarie University is advertising a 2-year postdoc in relation to a project entitled “Working synthetically in higher categorical structures”. It would be particularly suitable for people who work in higher categories, 2-categories, or homotopy type theory.

Further information is available here, or feel free to get in touch with Richard Garner ( or Steve Lack (

Posted at 2:12 AM UTC | Permalink | Followups (6)

March 10, 2019

How Much Work Can It Be To Add Some Things Up?

Posted by Tom Leinster

Here’s a puzzle for you. Calculating anything takes work. How much work? Well, obviously it depends how you’re doing the calculating. But let’s make some assumptions.

Imagine we have a machine that can add up any finite sequence x 1,,x nx_1, \ldots, x_n of nonnegative reals. Doing so costs W(x 1,,x n)W(x_1, \ldots, x_n) \in \mathbb{R} units of work — maybe measured in electricity, or potatoes, or whatever our machine uses as fuel. What could the function WW reasonably be?

Posted at 2:14 PM UTC | Permalink | Followups (14)

March 6, 2019

Left Adjoints Between Categories of Posets

Posted by John Baez

I have a bunch of related questions about things like “the free lattice on a poset”, and so on. I would hope that at least some of these questions have nice answers, at least for finite posets. But I’m not having much luck finding them!

Posted at 8:06 AM UTC | Permalink | Followups (17)