Higher-Dimensional Algebra II: 2-Hilbert Spaces John C. Baez Department of Mathematics, University of California Riverside, California 92521 USA email:
[email protected] August 27, 1996 Abstract 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, h i h ∗ i h ∗i 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 continu- ous 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 prop- erties 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. 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 an equivalence.