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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.
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.
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!
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 via the functor category where is the one object category with as its set of morphisms, on the other hand, you cannot in ordinary category theory encapsulate the category of representations of , 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 .
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!