A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an Abelian category enriched over Hilb with a V-structure, conjugate-linear on the homsets, satisfying ( fg, h) =( g, f *h) =( f, hg*). We also define monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call 2-H*-algbebras, braided 2-H*-algebras, and symmetric 2-H*-algbebras, and we describe the relation between these and tangles in two, three, and four dimensions, respectively. We prove a generalized Doplicher Roberts theorem starting that every symmetric 2-H*-algebra is equivalent to the category Rep(G) of continuous unitary finite-dimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact Abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2-H*algebra on one even object of dimension n.
1997 Academic Press
1. Introduction
A common theme in higher-dimensional algebra is ``categorification'': the formation of (n+1)-categorical analogs of n-categorical algebraic structures. This amounts to replacing equations between n-morphisms by specified (n+1)-isomorphisms, in accord with the philosophy that any interesting equation as opposed to one of the form x=x is better understood as an isomorphism, or more generally as an equivalence.
In their work on categorification in topological quantum field theory, Freed [10] and Crane have, in an informal way, used the concept of a ``2-Hilbert space'': a category with structures and properties analogous to those of a Hilbert space. Our goal here is to define 2-Hilbert spaces precisely and begin to study them. We concentrate on the finite-dimensional case, as the infinite-dimensional case introduces extra issues that we article no. AI971617 125