Modular Tensor Categories

In this section we describe the category-theoretic structure that has been developed to support a mathematical theory of anyons. Much of what we describe here is in [9], though we have modified some aspects and rearranged others.

Throughout this section, we let A be a category, intended to describe the quantum- mechanical behavior of a system of anyons. A will carry several sorts of additional structure, roughly classified as “additive” and “multiplicative” structure, all subject to various axioms. We describe the structures and the axioms a little at a time. We begin with the additive structure, because this is where Hilbert spaces enter the picture, so it is the basis for the connection with the usual formalism of quantum theory.

The vectors in our Hilbert spaces will be the morphisms of A. Specifically, for each pair of objects X, Y of A, the set Hom(X, Y) of morphisms from X to Y will have the structure of a Hilbert space. So we have many Hilbert spaces, one for each pair X, Y of objects. Some of these Hilbert spaces will be mere combinations of others, but there will still be several different “basic” Hilbert spaces. This means physically that we regard the system as being subject to superselection rules, which keep these Hilbert spaces separate.

We assume familiarity with some basic notions of category theory, specifically, the notions of product (including terminal object, which is the product of the empty family), coproduct (including initial object), equalizer, coequalizer, monomorphism, epimorphism, isomorphism, functor, and natural transformation. Definitions and examples can be found in [7] or [3, Chap. 1].

 
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