Centers, 2-categories, and the Eckmann-Hilton argument

The center of a group is an interesting construction: it associates to every group an abelian group in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor (unlike the abelianization ). We might wonder, then, exactly what kind of [...]

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Euler characteristic as homotopy cardinality

Let be a finite CW complex with vertices, edges, and in general different -cells. The Euler characteristic is a fundamental invariant of , and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula for a convex polyhedron. But where exactly does this expression come from? The modern story involves the [...]

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SU(2) and the quaternions

The simplest compact Lie group is the circle . Part of the reason it is so simple to understand is that Euler’s formula gives an extremely nice parameterization of its elements, showing that it can be understood either in terms of the group of elements of norm in (that is, the unitary group ) or [...]

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SO(3) and SU(2)

In order to study the hydrogen atom, we’ll need to know something about the representation theory of the special orthogonal group . This post consists of a few preliminaries along the way to doing this. I’ll be somewhat vague about a few things that 1) I don’t have much experience with, and 2) that would [...]

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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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Fractional linear transformations and elliptic curves

The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even). Lemma 1: Fix a field . The group of fractional linear transformations acts triple transitively on and the stabilizer of any triplet of distinct points is trivial. Lemma 2: The group [...]

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Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when [...]

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Irreducible components

If it wasn’t clear, in this discussion all rings are assumed commutative. Given a variety like we’d like to know if there’s a natural way to decompose it into its “components” . These aren’t its connected components in the topological sense, but in any reasonable sense the two parts are unrelated except possibly where they [...]

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The Noetherian condition as compactness

Let’s think more about what an abstract theory of unique factorization of primes has to look like. One fundamental property it has to satisfy is that factorizations should be finite. Another way of saying this is that the process of writing elements as products of other elements (up to units) should end in a finite [...]

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