How do you count rooted planar n -ary trees with some number of leaves? For n = 2 this puzzle leads to the Catalan numbers. These are so fascinating that the combinatorist Richard Stanley wrote a ...

The monoid of n × n n \times n matrices has an obvious n n -dimensional representation, and you can get all its representations from this one by operations that you can apply to any representation. So ...

Despite the “2” in the title, you can follow this post without having read part 1. The whole point is to sneak up on the metricky, analysisy stuff about potential functions from a categorical angle, ...

We’re brought up to say that the dual concept of injection is surjection, and of course there’s a perfectly good reason for this. The monics in the category of sets are the injections, the epics are ...

In Part 1, I explained my hopes that classical statistical mechanics reduces to thermodynamics in the limit where Boltzmann’s constant k k approaches zero. In Part 2, I explained exactly what I mean ...

In this year’s edition of the Adjoint School we covered the paper Triangulations, orientals, and skew monoidal categories by Stephen Lack and Ross Street, in which the authors construct a concrete ...

is always an isomorphism. The above definition is justified by the following: Theorem: A multicategory 𝒞 is isomorphic to M (𝒟) for some monoidal category 𝒟 if and only if it is representable. (we ...

Outline of this blog Throughout this blog post, we will present many of the ideas in the paper “String Diagrams for lambda calculi and Functional Computation” by Dan R. Ghica and Fabio Zanasi from ...

Bijection statement Here is the statement as I understand it to be, framed as a bijection of sets. My chief reference is the wonderful book Elliptic Curves, Modular Forms and their L-Functions by ...

(Jointly written by Astra Kolomatskaia and Mike Shulman) This is part two of a three part series of expository posts on our paper Displayed Type Theory and Semi-Simplicial Types. In this part, we ...

Why Mathematics is Boring I don’t really think mathematics is boring. I hope you don’t either. But I can’t count the number of times I’ve launched into reading a math paper, dewy-eyed and eager to ...

I’ve long been fascinated by the relation between ‘classical’ and ‘quantum’. One way this manifests is the relation between cartesian monoidal categories (like the category of sets with its cartesian ...