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 ...

Here Spec(R) is the set of prime ideals p of R, and Frac(R / p) is the field of fractions of the integral domain R / p. In particular, NF(b) is a coproduct of representables for each b ∈ B. (In the ...

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 ...

for each object X, Y, Z X, Y, Z in C \mathcal {C}. These are subject to the following conditions. The simplex category Δ \mathbf {\Delta} and its subcategory Δ⊥ \mathbf {\Delta}_ {\bot} A simple ...

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 ...

We then consider the problem of constructing a classifier for semi-simplicial diagrams. Specifically, we are interested in Reedy fibrant semi-simplicial diagrams, which are the homotopical counterpart ...

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 ...

The existence of this tensor product is a special case of a result of Hyland and Power. In fact their work shows this tensor product makes SMC SMC into a monoidal 2-category. I’m sure it must be ...