dorsal/arxiv
View SchemaHigher-Dimensional Algebra II: 2-Hilbert Spaces
| Authors | John C. Baez |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9609018 |
| URL | https://arxiv.org/abs/q-alg/9609018 |
| Journal | Adv. Math. 127 (1997), 125-189. |
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, 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.
{
"annotation_id": "d2cfbdae-7ba1-4a42-a812-44bd90ac20c6",
"date_created": "2026-03-02T18:01:28.751000Z",
"date_modified": "2026-03-02T18:01:28.751000Z",
"file_hash": "ac46249abf86e42e412e9d264fd85332e36769a40f2613e6ed5069ea5aa8079e",
"private": false,
"record": {
"abstract": "A 2-Hilbert space is a category with structures and properties analogous to\nthose of a Hilbert space. More precisely, we define a 2-Hilbert space to be an\nabelian category enriched over Hilb with a *-structure, conjugate-linear on the\nhom-sets, satisfying \u003cfg,h\u003e = \u003cg,f*h\u003e = \u003cf,hg*\u003e. We also define monoidal,\nbraided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we\ncall 2-H*-algebras, braided 2-H*-algebras, and symmetric 2-H*-algebras, and we\ndescribe the relation between these and tangles in 2, 3, and 4 dimensions,\nrespectively. We prove a generalized Doplicher-Roberts theorem stating that\nevery symmetric 2-H*-algebra is equivalent to the category Rep(G) of continuous\nunitary finite-dimensional representations of some compact supergroupoid G. The\nequivalence is given by a categorified version of the Gelfand transform; we\nalso construct a categorified version of the Fourier transform when G is a\ncompact abelian group. Finally, we characterize Rep(G) by its universal\nproperties when G is a compact classical group. For example, Rep(U(n)) is the\nfree connected symmetric 2-H*-algebra on one even object of dimension n.",
"arxiv_id": "q-alg/9609018",
"authors": [
"John C. Baez"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "Adv. Math. 127 (1997), 125-189.",
"title": "Higher-Dimensional Algebra II: 2-Hilbert Spaces",
"url": "https://arxiv.org/abs/q-alg/9609018"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "99c054b0-ca79-4875-8054-0618d8511c4b",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}