Higher-Dimensional Algebra II. 2-Hilbert Spaces

AbstractA 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 ∗-structure, conjugate-linear on the hom-sets, 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 supergroupoidG. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform whenGis a compact Abelian group. Finally, we characterize Rep(G) by its universal properties whenGis a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2-H*-algebra on one even object of dimensionn