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August 16, 2019

Graphical Regular Logic

Posted by John Baez

guest post by Sophie Libkind and David Jaz Myers

This post continues the series from the Adjoint School of Applied Category Theory 2019.

Posted at 8:03 AM UTC | Permalink | Followups (6)

Evil Questions About Equalizers

Posted by John Baez

I have a few questions about equalizers. I have my own reasons for wanting to know the answers, but I’ll admit right away that these questions are evil in the technical sense. So, investigating them requires a certain morbid curiosity… and have a feeling that some of you will be better at this than I am.

Here are the categories:

RexRex = [categories with finite colimits, functors preserving finite colimits]

SMCSMC = [symmetric monoidal categories, strong symmetric monoidal functors]

Both are brutally truncated stumps of very nice 2-categories!

Posted at 7:31 AM UTC | Permalink | Followups (3)

August 11, 2019

Even-Dimensional Balls

Posted by John Baez

Some of the oddballs on the nn-Café are interested in odd-dimensional balls, but here’s a nice thing about even-dimensional balls: the volume of the 2n2n-dimensional ball of radius rr is

(πr 2) nn! \frac{(\pi r^2)^n}{n!}

Dillon Berger pointed out that summing up over all nn we get

n=0 (πr 2) nn!=e πr 2 \sum_{n=0}^\infty \frac{(\pi r^2)^n}{n!} = e^{\pi r^2}

It looks nice. But what does it mean?

Posted at 3:13 AM UTC | Permalink | Followups (40)

August 9, 2019

The Conway 2-Groups

Posted by John Baez

I recently bumped into this nice paper:

• Theo Johnson-Freyd and David Treumann, H 4(Co 0,)=/24\mathrm{H}^4(\mathrm{Co}_0,\mathbb{Z}) = \mathbb{Z}/24.

which proves just what it says: the 4th integral cohomology of the Conway group Co 0\mathrm{Co}_0, in the sense of group cohomology, is /24\mathbb{Z}/24. I want to point out a few immediate consequences.

Posted at 8:58 AM UTC | Permalink | Followups (13)

2020 Category Theory Conferences

Posted by John Baez

Here are some dates to help you plan your carbon emissions.

Posted at 7:19 AM UTC | Permalink | Post a Comment

July 23, 2019

Summer Meanderings About Enriched Logic

Posted by David Corfield

Reading the recently appeared article

which treats Gabriel-Ulmer and related dualities in an enriched setting, I was wondering what sense we should make of “enriched logic”.

If, for instance, we may think of ordinary Gabriel-Ulmer duality as operating between essentially algebraic theories and their categories of models, then how to think of a finitely complete 𝒱\mathcal{V}-category as a kind of enriched essentially algebraic theory?

That got me wondering about the case where 𝒱\mathcal{V} is the reals or the real interval, i.e., something along the lines of a Lawvere metric space, which led me to some recent work on continuous logic. This logic is associated with a longstanding program on continuous model theory, but it seems that the time is ripe now for category theoretic recasting, as in:

  • Simon Cho, Categorical semantics of metric spaces and continuous logic, (arXiv:1901.09077).

In this article Cho argues that the object of truth values of continuous logic is to be seen as a “continuous subobject classifier” in the sense of topos theory.

Posted at 6:47 AM UTC | Permalink | Followups (27)

July 22, 2019

Applied Category Theory 2019 Talks

Posted by John Baez

Applied Category Theory 2019 happened last week! It was very exciting: about 120 people attended, and they’re pushing forward to apply category theory in many different directions. The topics ranged from ultra-abstract to ultra-concrete, sometimes in the same talk.

Now the Applied Category Theory 2019 school is about to start. But we shouldn’t let the momentum built up at the conference dissipate.

Posted at 8:02 AM UTC | Permalink | Followups (3)

July 17, 2019

What is the Laplace Transform?

Posted by Mike Shulman

One of the best ways to understand something difficult is to reinvent it. Two weeks ago I did that with the Laplace transform, and now I feel like it finally makes sense to me. (In fact this was while I was at the Magnitude Workshop, trying to make sense of magnitude for infinite metric spaces. Thanks to Richard Hepworth for pointing out that what I was reinventing was the Laplace transform — in fact I was stumbling towards some of the same ideas that he had already formulated, which are described in his excellent talk.)

The short answer is that the Laplace transform is really just a generalization of the familiar Laurent series representation of complex analytic functions, but where the exponents are allowed to be non-integers and to “vary continuously” rather than discretely. Wikipedia sketches this briefly, but I would never have discovered it there because I was looking for that generalization rather than looking for a way to understand the Laplace transform. Moreover, this explanation is obscured by the fact that people generally choose obfuscating coordinates.

In this post I’ll try to explain the Laplace transform as I understand it now — which is probably still quite rudimentary compared to the people who really understand it, but maybe it’ll be helpful for other folks in the audience who think more like me than like an analyst. (And maybe some analysts will come along and offer further insight!) Along the way we’ll also learn what the “Z-transform” is and obtain some insight into the Fourier transform.

Posted at 12:45 AM UTC | Permalink | Followups (14)

July 14, 2019

What Happened At The Magnitude Workshop

Posted by Tom Leinster

A week ago we had a short workshop on magnitude at the University of Edinburgh, organized by Heiko Gimperlein, Magnus Goffeng and me. If that sounds familiar to you, it might be because I advertised it here before. The slides from the talks are now on the website. You can also see a list of open problems.

Anyway, it was a great meeting, focused on the magnitude of metric spaces (as opposed to enriched categories more generally), and roughly evenly split between the analytic and homological aspects of magnitude. It included talks from our own Simon Willerton and Mike Shulman, as well as other experts in a wide variety of different fields (as the official name of the workshop suggests: “Magnitude 2019: Analysis, Category Theory, Applications”). And Emily Roff, who’s doing a PhD with me, spoke about our work on the maximum diversity of a compact metric space.

Heiko and Magnus also invited some experts in the theory of capacity to help us out, knowing that this is something highly relevant to magnitude — even though it now seems that the kinds of questions about capacity that we’re asking do not yet have answers. I was particularly happy to see people from the algebraic side taking part in discussions on primarily analytic questions, and vice versa.

The talks were arranged so that each day started with some introductory stuff (schedule here), before going into more depth. So if you’re curious to find out more and read some of the talk slides, that’s where you might want to start.

Posted at 10:24 AM UTC | Permalink | Followups (4)

July 6, 2019

The Riemann Hypothesis Says 5040 is the Last

Posted by John Baez

There are many equivalent ways to phrase the Riemann Hypothesis. I just learned a charming one from this fun-filled paper:

Posted at 12:39 PM UTC | Permalink | Followups (16)

July 5, 2019

Type Theory, Category Theory and Philosophy

Posted by David Corfield

I held a small gathering this week in Kent, Type theory, Category theory and Philosophy. Currently I only have my own slides up there. I’ll see which others I can add.

My talk was on events and temporal type theory. That events are spoken of in ways unlike objects has long been observed by philosophers. Looking for a type-theoretic rendition of natural language, one would expect at the very least that the object/event distinction should be recognised.

My belief is growing the philosophy, computer science and linguistics have much to say to one another, so it was great to hear from Dominic Orchard, a local computer scientist, of his work on graded modalities, and how philosophers such as Kit Fine and Lou Goble had uncovered them in the early 70s. There’s a body of work by linguistics to go through too, such as Lassiter’s Graded Modality.

Posted at 11:08 AM UTC | Permalink | Followups (3)

July 1, 2019

Structured Cospans

Posted by John Baez

My grad student Kenny Courser gave a talk at the 4th Symposium on Compositional Structures. He spoke about his work with Christina Vasilakopolou and me. We’ve come up with a theory that can handle a broad class of open systems, from electrical circuits to chemical reaction networks to Markov processes and Petri nets. The idea is to treat open systems as morphisms in a category of a particular kind: a ‘structured cospan category’.

Posted at 7:39 PM UTC | Permalink | Followups (5)

June 29, 2019

Behavioral Mereology

Posted by John Baez

guest post by Toby Smithe and Bruno Gavranović

What do human beings, large corporations, biological cells, and gliders from Conway’s Game of Life have in common?

This week in the Applied Category Theory School, we attempt some first steps towards answering this question. We turn our attention to autopoiesis, the ill-understood phenomenon of self-perpetuation that characterizes life. Our starting point is Behavioral Mereology, a new take on the ancient question of parthood: how should we understand the relationship between a part and a whole? The authors Fong, Myers, and Spivak suggest that we cleave a whole into parts by observing – and grouping together — regularities in its behavior. Parthood entails behavioral constraints, which are mediated by the whole. We describe the corresponding logic of constraint passing, and the associated modalities of compatibility and ensurance. We propose that autopoiesis entails a special kind of parthood — one that ensures its compatibility with its environment — and end with a list of open questions and potential research directions.

Posted at 3:16 AM UTC | Permalink | Followups (18)

June 25, 2019

Meeting the Dialogue Challenge

Posted by John Baez

guest post by Dan Shiebler and Alexis Toumi

This is the fourth post in a series from the Adjoint School of Applied Category Theory 2019. We discuss Grammars as Parsers: Meeting the Dialogue Challenge (2006) by Matthew Purver, Ronnie Cann and Ruth Kempson as part of a group project on categorical methods for natural language dialogue.

Posted at 12:45 AM UTC | Permalink | Followups (2)

June 20, 2019

Katrina Honigs meets Grothendieck

Posted by John Baez

Here’s a fun story by Katrina Honigs about how she found Grothendieck’s house in the Pyrenees, jumped the fence, knocked on the door, and offered him some pastries.

Posted at 4:06 AM UTC | Permalink | Followups (5)