Combinatorics, or at least part of it, is the art of counting. For example: how many derangements does a set with n n elements have? A derangement is a bijection with no fixed points. We’ll count them ...

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 ...

I’ve been blogging a bit about medieval math, physics and astronomy over on Azimuth. I’ve been writing about medieval attempts to improve Aristotle’s theory that velocity is proportional to force, ...

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 ...

and the identities as x ↦ {x} x \mapsto \{x\} gives us the Markov category FinSetMulti \mathsf{FinSetMulti} of possibilities!. The same data from the example can be used in a possibilistic way as well ...

The study of monoidal categories and their applications is an essential part of the research and applications of category theory. However, on occasion the coherence conditions of these categories ...

Sure! Nominal sets are the objects of the Schanuel topos, which is the category of sheaves for a Grothendieck topology on FinSet mono op FinSet_{mono}^{op}.Whereas, the natural semantics of nullary ...

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 ...

By the way, my proof here that the ring of symmetric functions Λ \Lambda is the free λ \lambda-ring on one generator is a bit ‘tricky’, since I was wanting to deploy things we’d already shown and not ...

There are actually two questions here: who invented the concept of monoidal category, and who introduced the term ‘monoidal category’.. I had always assumed both were done by Mac Lane in this paper: ...

For questions 1 and 2, isn’t that true for any group G, not just the fundamental groups of a manifold? And moreover, I think of this as the definition of the profinite completion of a group: as an ...