are isomorphisms. Definition. A symmetric 2-rig is a 2-rig whose underlying monoidal category is a symmetric monoidal category. One can work through the details of these definitions and show the ...

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 ...

Thanks for a really interesting post. I hadn’t heard of Martianus Capella. Whilst Aristotle got the quantified relationship between force and velocity wrong, I happen to like his notion of force in ...

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 ...

Galois Theory Posted by Tom Leinster I’ve just arXived my notes for Edinburgh’s undergraduate Galois theory course, which I taught from 2021 to 2023. I first shared the notes on my website some time ...

This may be a tempting question when reading about categorical probability, but we might argue that this isn’t completely reinventing traditional probability from the ground up. Instead, we’re ...

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 ...

In ordinary category theory, many results can be extended to double categories. For instance, in an ordinary category, we can determine if it has all limits (resp. finite limits) by checking if it has ...

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 been talking about Grothendieck’s approach to Galois theory, but I haven’t yet touched on Brauer theory. Both of these involve separable algebras, but of different kinds. For Galois theory we ...