Four things can happen when you take an elliptic curve with integer coefficients and look at it over a finite field. There’s good reduction, bad reduction, ugly reduction and weird reduction. Let’s ...

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 ...

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 ...

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 ...

That is correct. There are finite index subgroups of profinite groups that are not open, i.e., there are profinite groups that do not equal their own profinite completion. However, by definition, the ...

My PhD student Ruben Van Belle has just published his first paper! Ruben Van Belle, Probability monads as codensity monads. Theory and Applications of Categories 38 (2022), 811–842. It’s a treasure ...

Elementary Petri nets are the objects of a category N𝒞 whose morphisms can represent simulations or refinements of them. The interpretations of the linear logic connectives of the dialectica category ...

where K K is the separable closure of k k, G = Gal(K | k) G = \mathrm {Gal} (K|k) is the Galois group, and we’re taking the group cohomology of G G with coefficients in the group of units K∗ K^\ast, ...

Back to modal HoTT. If what was considered last time were all, one would wonder what the fuss was about. Now, there’s much that needs to be said about type dependency, types as propositions, sets, ...