Let G be a directed graph whose nodes are the positive integers and whose edges represent relations between two integers. In our first example we’ll draw an edge from x to y if x is a multiple of y. In our second example we’ll draw an edge from x to y if x ≥ y. […]
I was bewildered by my first exposure to category theory. My first semester in graduate school I had a textbook with definitions like “A gadget is an object G such that whenever you have this unfamiliar constellation of dots and arrows, you’re allowed to draw another arrow from here to there.” What? Why?! I revisited […]
I imagine most programmers who develop an interest in category theory do so after hearing about monads. They ask someone what a monad is, and they’re told that if they really want to know, they need to learn category theory. Unfortunately, there are couple unnecessary difficulties anyone wanting to understand monads etc. is likely to […]
I’m cautiously optimistic about applications of category theory. I believe category theory has the potential to be useful, but I’m skeptical of most claims to have found useful applications. Category theory has found useful application, especially behind the scenes, but a lot of supposed applications remind me of a line from Colin McLarty: [Jean-Pierre] Serre […]
Instead of asking whether an area of mathematics can be applied, it’s more useful to as how it can be applied. Differential equations are directly and commonly applied. Ask yourself what laws govern the motion of some system, write down these laws as differential equations, then solve them. Statistical models are similarly direct: propose a […]
From Paul Phillips: I see adjoint functors. How often do you see them? All the time. They’re everywhere. pic.twitter.com/6PkGJ9wP4A — Paul Phillips (@contrarivariant) May 27, 2017 Mashup of Saunders Mac Lane’s quip “Adjoint functors arise everywhere” and Haley Joel Osment’s famous line from Sixth Sense. Related: Applied category theory
From Drew Armstrong’s notes on adjoint functors: Once upon a time, my opinion of category theory was the same as my opinion of Facebook: if I ignore it for long enough, hopefully it will go away. It is now my educated opinion that category theory will not go away, and in fact the language of […]
When I was in college, I sat in on a communication workshop for Latin American preachers. This was unusual since I’m neither Latin American nor a preacher, but I’m glad I was there. I learned several things in that workshop that I’ve used ever since. For example, when you’re gesturing about something moving forward in time, move your […]
Category theory can be very useful, but you don’t use it the same way you use other kinds of math. You can apply optimization theory, for example, by noticing that a problem has a certain form, and therefore a certain algorithm will converge to a solution. Applications of category theory are usually more subtle. You’re […]
The diagram below shows how category theory definitions build on each other. Based on definitions in The Joy of Cats. Related: More mathematical diagrams Applied category theory
