The Yoneda lemma I

For two categories let denote the functor category, whose objects are functors and whose morphisms are natural transformations. For a locally small category, the Yoneda embedding is the functor sending an object to the contravariant functor and sending a morphism to the natural transformation given by composition. The goal of the next few posts is [...]

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Ultrafilters in topology

Remark: To forestall empty set difficulties, whenever I talk about arbitrary sets in this post they will be non-empty. We continue our exploration of ultrafilters from the previous post. Recall that a (proper) filter on a poset is a non-empty subset such that For every , there is some such that . For every , [...]

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Boolean rings, ultrafilters, and Stone’s representation theorem

Recently, I have begun to appreciate the use of ultrafilters to clean up proofs in certain areas of mathematics. I’d like to talk a little about how this works, but first I’d like to give a hefty amount of motivation for the definition of an ultrafilter. Terence Tao has already written a great introduction to [...]

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The adjoint functor theorem for posets

Recently in Measure Theory we needed the following lemma. Lemma: Let be non-constant, right-continuous and non-decreasing, and let . Define by . Then is left-continuous and non-decreasing. Moreover, for and , . If you’re categorically minded, this last condition should remind you of the definition of a pair of adjoint functors. In fact it is [...]

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Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials

SPUR is finally over! Instead of continuing my series of blog posts, I thought I’d just link to my paper, Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials, and the first few blog posts should more or less provide enough background to read it. My project ended up changing direction. The formula I was [...]

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Chevalley-Bruhat order

Before we define Bruhat order, I’d like to say a few things by way of motivation. Warning: I know nothing about algebraic groups, so take everything I say with a grain of salt. A (maximal) flag in a vector space of dimension is a chain of subspaces such that . The flag variety of is, [...]

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Kraft’s inequality for prefix codes

Kraft’s inequality: Let be a prefix code on an alphabet of size , and let denote the number of codewords of length . Then . In lieu of reposting my old blog posts I thought I’d revisit some of their topics instead. In an old post of mine I set this inequality as a practice [...]

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The “correct” definition of a homomorphism

(Warning: I’m trying to talk about things I don’t really understand in this post, so feel free to correct me if you see a statement that’s obviously wrong.) Why are continuous functions the “correct” notion of homomorphism between topological spaces? The “obvious” way to define homomorphisms for a large class of objects involves thinking of [...]

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