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Archive for September, 2012

Previously we discussed categories with finite biproducts, or semiadditive categories. Today, partially as a further warmup for the axioms defining an abelian category, we’ll discuss monomorphisms and epimorphisms.

Monomorphisms and epimorphisms are supposed to be a categorical generalization of the familiar notion of an injective resp. surjective structure-preserving map (such as an injective resp. surjective group homomorphism or an injective resp. surjective continuous function). This idea more or less works out for monomorphisms, but epimorphisms are somewhat infamous for behaving in unexpected ways, and even monomorphisms can behave unexpectedly sometimes.

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There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group G. The kind which is covariantly functorial is some variation on the group algebra k[G], which is the free k-module on G with multiplication inherited from the multiplication on G. The kind which is contravariantly functorial is some variation on the algebra k^G of functions G \to k with pointwise multiplication.

When k = \mathbb{C} and when G is respectively either a discrete group or a compact (Hausdorff) group, both of these algebras can naturally be endowed with the structure of a random algebra. In the case of \mathbb{C}[G], the corresponding state is a noncommutative refinement of Plancherel measure on the irreducible representations of G, while in the case of \mathbb{C}^G, the corresponding state is by definition integration with respect to normalized Haar measure on G.

In general, some nontrivial analysis is necessary to show that the normalized Haar measure exists, but for compact groups equipped with a faithful finite-dimensional unitary representation V it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on G using representation theory. This construction will in some sense explain why the category \text{Rep}(G) of (finite-dimensional continuous unitary) representations of G behaves like an inner product space (with \text{Hom}(V, W) being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on G.

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Previously we described all finite-dimensional random algebras with faithful states. In this post we will describe states on the infinite-dimensional ^{\dagger}-algebra \mathbb{C}[x]. Along the way we will run into and connect some beautiful and classical mathematical objects.

A special case of part of the following discussion can be found in an old post on the Catalan numbers.

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The goal of today’s post is to introduce and discuss semiadditive categories. Roughly speaking, these are categories in which one can add both objects and morphisms. Prominent examples include the abelian categories appearing in homological algebra, such as categories of sheaves and modules and categories of chain complexes.

Semiadditive categories display some interesting categorical features, such as the prominence of pairs of universal properties and the surprising ways in which commutative monoid structures arise, which seem to be underemphasized in textbook treatments and which I would like to emphasize here. I would also like to emphasize that their most important properties are unrelated to the ability to subtract morphisms which is provided in an additive category.

In this post, for convenience all categories will be locally small (that is, \text{Set}-enriched).

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The previous post on noncommutative probability was too long to leave much room for examples of random algebras. In this post we will describe all finite-dimensional random algebras with faithful states and all states on them. This will lead, in particular, to a derivation of the Born rule from statistical mechanics. We will then give a mathematical description of wave function collapse as taking a conditional expectation.

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Notes

I’ve uploaded current notes for the classes I’m taking. I make no claim that these notes are complete or correct, but they may be useful to somebody. The notes for Dylan Thurston’s class are particularly fun to take; I’ve been drawing the pictures in Paper and I’m generally very happy with the way they’re turning out.

Edit: It would probably be a good idea for me to briefly describe these classes.

  • Homological Algebra (Wodzicki): An introduction to homological algebra aimed towards triangulated categories. Taking these notes is a good exercise in live-TeXing commutative diagrams.
  • Curves on Surfaces (Thurston): An introduction to various interesting structures related to curves on surfaces. There are cluster algebra structures involved related to Teichmüller space, the Jones polynomial, and 3- and 4-manifold invariants, but the actual curves on the actual surfaces remain very visualizable. Taking these notes is a good exercise in drawing pictures like this (a curve on a thrice-punctured disc being acted on by an element of the mapping class group, which in this case is the braid group B_3):

  • Quantum Field Theory (Reshetikhin): An introduction to the mathematics of quantum field theory. The course website has more details. Taking this class is a strong incentive for me to learn differential, Riemannian, and symplectic geometry.

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