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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.

May 30, 2023

Galois’ Fatal Duel

Posted by John Baez

On this day in 1832, Evariste Galois died in a duel. The night before, he summarized his ideas in a letter to his friend Auguste Chevalier. Hermann Weyl later wrote “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”

That seems exaggerated, but within mathematics it might be true. On top of that, the backstory is really dramatic! I’d never really looked into it, until today. Let me summarize a bit from Wikipedia.

Posted at 5:16 AM UTC | Permalink | Followups (10)

May 9, 2023

Symmetric Spaces and the Tenfold Way

Posted by John Baez

I’ve finally figured out the really nice connection between Clifford algebra and symmetric spaces! I gave a talk about it, and you can watch a video.

Posted at 8:55 PM UTC | Permalink | Followups (23)

May 5, 2023

Categories for Epidemiology

Posted by John Baez

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!

Posted at 10:55 AM UTC | Permalink | Post a Comment

May 3, 2023

Metric Spaces as Enriched Categories II

Posted by Simon Willerton

In the previous post I set the scene a little for enriched category theory by implying that by working ‘over’ the category of sets is a bit like working ‘over’ the integers in algebra and sometimes it is more appropriate to have a different base category just as it is sometimes more appropriate to have a different base ring. Below, we’ll see with the case of metric spaces that changing the base category can seemingly change the flavour quite a lot.

An example which I was using for illustration in the last post was that whilst, on the one hand, you can encapsulate the group actions of a group GG via the functor category [G,𝒮et][{\mathbf{\mathcal{B}}} G, \mathbf{\mathcal{S}et}] where G{\mathbf{\mathcal{B}}} G is the one object category with GG as its set of morphisms, on the other hand, you cannot in ordinary category theory encapsulate the category of representations of AA, an algebra over the complex numbers, as a category of functors into the category of vector spaces as you might hope. Indeed in ordinary category theory you can’t really see the structure of a vector space lurking in the one-object category A{\mathbf{\mathcal{B}}} A.

In this post I’ll explain what an enriched category is and how enriched category theory can, for example, allow a natural expression for a representatation category as a functor category. I’ll go on to show, following Lawvere’s insight, how metric spaces and much metric space theory can be seen to live within the realm of enriched category theory.

I’ll finish with an afterword on my experiences and thoughts on why enriched categories should be more appreciated but aren’t!

Posted at 2:16 PM UTC | Permalink | Followups (27)