ar X iv : q - a lg / 9 60 90 18 v 2 2 2 O ct 1 99 6 Higher - Dimensional Algebra II : 2 - Hilbert Spaces
نویسنده
چکیده
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 ∗-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*-algebras, braided 2-H*-algebras, and symmetric 2-H*-algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized Doplicher-Roberts theorem stating 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.
منابع مشابه
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تاریخ انتشار 2008