“Just when I thought I was out, they pull me back in.” — Michael Corleone, The Godfather, Part 3 My interest in category theory goes in cycles. Something will spark my interest in it, and I’ll dig a little further. Then I reach my abstraction tolerance and put it back on the shelf. Then sometime […]
A pullback is a limit over a diagram of the following shape. The pullback is a sort of product of A and B that depends on C, and so it is sometimes written as a product with a subscript C on the product symbol: A ×C B. In all these diagrams, the givens will be […]
I was thumbing through a new book on causal inference, The Effect by Nick Huntington-Klein, and the following diagram caught my eye. Then it made my head hurt. It looks like a category theory diagram. What’s that doing in a book on causal inference? And if it is a category theory diagram, something’s wrong. Either […]
Definitions via universal properties seem strange at first, even evasive. They are non-constructive, but they’re often followed up by a constructive proof that shows that they exist. So you have a pair of definitions: one that says a gadget, if it exists, is something that behaves this way, and another that says that gadgets do […]
A couple days ago, near the end of a post, I mentioned exact sequences. This term does not mean what you might reasonably think it means. It doesn’t mean exact in the sense of not being approximate. It means that the stuff that comes out of one step is exactly the stuff that gets set […]
Wire gauge is a perennial source of confusion: larger numbers denote smaller wires. The reason is that gauge numbers were assigned from the perspective of the manufacturing process. Thinner wires require more steps in production. This is a common error in user interface design and business more generally: describing things from your perspective rather than […]
There are two techniques in software development that have an almost gnostic mystique about them: monads and macros. Pride and pragmatism As with everything people do, monads and macros are used with mixed motives, for pride and for pragmatism. As for pride, monads and macros have just the right barrier to entry: high enough to […]
In my previous post, I looked at the map Δ that takes a column vector to a diagonal matrix. I even drew a commutative diagram, which foreshadows a little category theory. Suppose you have a function f of a real or complex variable. To an R programmer, if x is a vector, it’s obvious that […]
I recently discovered quiver, a tool for drawing commutative diagrams. It looks like a nice tool for drawing diagrams more generally, but it’s designed particularly to include the features you need when drawing the kinds of diagrams that are ubiquitous in category theory. You can draw diagrams using the online app and export the result […]
These have been the most popular math posts this year. Queueing theory: The science of waiting in line US Army applying abstract math The license plate game How category theory is applied Progress on the Collatz conjecture Any number can start a factorial
