Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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 (4)

June 16, 2019

Applied Category Theory Meeting at UCR

Posted by John Baez

The American Mathematical Society is having their Fall Western meeting here at U. C. Riverside during the weekend of November 9th and 10th, 2019. Joe Moeller and I are organizing a session on Applied Category Theory!

Posted at 10:16 PM UTC | Permalink | Followups (1)

June 13, 2019

What’s a One-Object Sesquicategory?

Posted by John Baez

A sesquicategory, or 1121\frac{1}{2}-category, is like a 2-category, but without the interchange law relating vertical and horizontal composition of 2-morphisms:

(αβ)(γδ)=(αγ)(βδ) (\alpha \cdot \beta)(\gamma \cdot \delta) = (\alpha \gamma) \cdot (\beta \delta)

Better, sesquicategories are categories enriched over (Cat,)(Cat,\square): the category of categories with its “white” tensor product. In the cartesian product of categories CC and DD, namely C×DC \times D, we have the law

(1)(f×1)(1×g)=(1×g)(f×1) (f \times 1)(1 \times g) = (1 \times g)(f \times 1)

and we can define f×gf \times g to be either of these. In the white tensor product CDC \square D we do not have this law, and f×gf \times g makes no sense.

What’s a one-object sesquicategory?

Posted at 6:29 PM UTC | Permalink | Followups (13)

June 5, 2019

Nonstandard Models of Arithmetic

Posted by John Baez

A nice quote:

There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time.

This is from Matthew Katz and Jan Reimann’s nice little book An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics. I’ve been been talking to my old friend Michael Weiss about nonstandard models of Peano arithmetic on his blog. We just got into a bit of Ramsey theory. But you might like the whole series of conversations.

Posted at 6:41 PM UTC | Permalink | Followups (29)

June 3, 2019

Why Category Theory Matters

Posted by John Baez

No, I’m not going to tell you why category theory matters. To learn that, you must go here:

Posted at 12:40 AM UTC | Permalink | Followups (8)