These are some lecture notes for a 4 1 2 \frac {1} {2} -hour minicourse I’m teaching at the Summer School on Algebra at the Zografou campus of the National Technical University of Athens. To save time ...

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 ...

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 ...

Then show that the passage n ↦ n n \mapsto \mathbf {n} respects addition, multiplication and exponentiation. For example, if we write p = m + n p = m + n then p \mathbf {p} is the coproduct of m ...

The more I read of your notes, the more I wonder if it is really a course in category theory or in set theory. You are certainly teaching a lot of category-theoretic techniques and ideas, even if the ...

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 ...

Physicists often say things like this: “Special relativity reduces to Newtonian mechanics as the speed of light, c c, approaches ∞ \infty.” “Quantum mechanics reduces to classical mechanics as ...

I forget exactly what Connes said, so I imagine it’s something about how the Todd class in algebraic topology, which is usually defined using the function x / (1 − e − x) x/ (1 - e^ {-x}) (whose power ...

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 ...

A note from my QFT class. Finally, I understand what Batalin-Vilkovisky anti-fields are for. The Ward-Takahashi Identities are central to understanding the renormalization of QED. They are an ...

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 ...