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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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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