September 19, 2018

p-Local Group Theory

Posted by John Baez

I’ve been trying to learn a bit of the theory of finite groups. As you may know, Sylow’s theorems say that if you have a finite group $G$, and $p^k$ is the largest power of a prime $p$ that divides the order of $G$, then $G$ has a subgroup of order $p^k$, which is unique up to conjugation. This is called a Sylow $p$-subgroup of $G$.

Sylow’s theorems also say a lot about how many Sylow $p$-subgroups $G$ has. They also say that any subgroup of $G$ whose order is a power of $p$ is contained in a Sylow $p$-subgroup.

I didn’t like these theorems as an undergrad. The course I took whizzed through them in a desultory way. And I didn’t go after them myself: I was into group theory for its applications to physics, and the detailed structure of finite groups doesn’t look important when you’re first learning physics: what stands out are continuous symmetries, so I was busy studying Lie groups.

Since I didn’t really master Sylow’s theorems, and had no strong motive to do so, I didn’t like them — the usual sad story of youthful mathematical distastes.

But now I’m thinking about Sylow’s theorems again, especially pleased by Robert A. Wilson’s one-paragraph proof of all three of these theorems in his book The Finite Simple Groups. And I started wondering if the importance of groups of prime power order — which we see highlighted in Sylow’s theorems and many other results — is all related to localization in algebraic topology, which is a technique to focus attention on a particular prime.

Posted at 3:52 PM UTC | Permalink | Followups (1)

September 18, 2018

What is Applied Category Theory?

Posted by John Baez

Tai-Danae Bradley has a new free “booklet” on applied category theory. It grew out of the workshop Applied Category Theory 2018, and I think it makes a great complement to Fong and Spivak’s book Seven Sketches and my online course based on that book:

Abstract. This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field.

Check it out!

Posted at 7:44 PM UTC | Permalink | Followups (1)

September 5, 2018

A Categorical Look at Random Variables

Posted by Tom Leinster

guest post by Mark Meckes

For the past several years I’ve been thinking on and off about whether there’s a fruitful category-theoretic perspective on probability theory, or at least a perspective with a category-theoretic flavor.

(You can see this MathOverflow question by Pete Clark for some background, though I started thinking about this question somewhat earlier. The fact that I’m writing this post should tell you something about my attitude toward my own answer there. On the other hand, that answer indicates something of the perspective I’m coming from.)

I’m a long way from finding such a perspective I’m happy with, but I have some observations I’d like to share with other n-Category Café patrons on the subject, in hopes of stirring up some interesting discussion. The main idea here was pointed out to me by Tom, who I pester about this subject on an approximately annual basis.

Posted at 10:24 PM UTC | Permalink | Followups (35)