## Higher-Dimensional Algebra II: 2-Hilbert Spaces (1996)

Venue: | Adv. Math |

Citations: | 43 - 13 self |

### BibTeX

@ARTICLE{Baez96higher-dimensionalalgebra,

author = {John C. Baez},

title = {Higher-Dimensional Algebra II: 2-Hilbert Spaces},

journal = {Adv. Math},

year = {1996},

volume = {127},

pages = {125--189}

}

### Years of Citing Articles

### OpenURL

### 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 hfg; hi = hg; f hi = hf; hg i. 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...

### Citations

242 |
Braided tensor categories
- Joyal, Street
- 1983
(Show Context)
Citation Context ...t describing this, but the reader can fill in the details using the ideas described in HDA0 and the many references therein. Especially relevant is the work of Freyd and Yetter [11], Joyal and Street =-=[15]-=-, and Reshetikhin and Turaev [26, 28]. We discuss this relationship more carefully in the Conclusions. 3. Typical tangle in 2 dimensions The basic idea is to use tangles to represent certain morphisms... |

138 | Higher-dimensional algebra and topological quantum field theory
- Baez, Dolan
- 1995
(Show Context)
Citation Context ...y bother categorifying the notion of Hilbert space? As already noted, one motivation comes from the study of topological quantum field theories, or TQFTs. In the introduction to this series of papers =-=[2]-=-, we proposed that n-dimensional unitary extended TQFTs should be treated as n-functors from a certain n-category nCob to a certain n-category nHilb. Roughly speaking, the n-category nCob should have ... |

127 | Tannakian categories - Deligne, Milne - 1982 |

111 |
Roberts Why there is a field algebra with a compact gauge group describing the superselection strucure in particle physics Commun
- Doplicher, E
- 1990
(Show Context)
Citation Context ...fication of symmetric 2-H*-algebras. Doplicher and Roberts proved a theorem which implies that connected even symmetric 2-H*-algebras are all equivalent to categories of compact group representations =-=[8, 9]-=-. Here and in all that follows, by a `representation' of a compact group we mean a finite-dimensional continuous unitary representation. Given a compact group G, let Rep(G) denote the category of such... |

107 |
Coherence for tricategories
- Gordon, Power, et al.
- 1995
(Show Context)
Citation Context ... Voevodsky [17] have defined the notion of a weak monoidal structure on a strict 2-category, which should be sufficient for the purpose at hand. On the other hand the work of Gordon, Power and Street =-=[14]-=- gives a fully general notion of weak monoidal 2-category, namely a 1-object tricategory. This should also be suitable for studying the tensor product on 2Hilb, though it might be considered overkill.... |

104 |
Roberts: A new duality theory for compact groups
- Doplicher, E
- 1989
(Show Context)
Citation Context ...metric 2-H*-algebras arise from certain triangular Hopf algebroids --- for example, groups. In Section 6 of this paper we concentrate on the symmetric case. Generalizing the Doplicher-Roberts theorem =-=[8]-=-, we prove that all symmetric 2-H*-algebras are equivalent to categories of representations of `compact supergroupoids'. If a symmetric 2-H*-algebra is `purely bosonic', it is equivalent to a category... |

90 |
2-categories and Zamolodchikov tetrahedra equations,” in: Algebraic groups and their generalizations: quantum and infinite-dimensional methods
- Kapranov, Voevodsky
- 1994
(Show Context)
Citation Context ...theory analogous to the role played by the ring C of complex numbers in Hilbert space theory. Thus we expect 2-Hilbert spaces to be `module categories' over Hilb, as defined by Kapranov and Voevodsky =-=[17]-=-. An important part of our philosophy here is that C is the primordial Hilbert space: the simplest one, upon which the rest are modelled. By analogy, we expect Hilb to be the primordial 2-Hilbert spac... |

60 |
Higher Algebraic Structures and Quantization
- Freed
(Show Context)
Citation Context ...tion --- as opposed to one of the form x = x --- is better understood as an isomorphism, or more generally an equivalence. In their work on categorification in topological quantum field theory, Freed =-=[10]-=- and Crane [5] 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 ... |

53 | Higher-dimensional algebra I: braided monoidal 2-categories. Available as ftp://math.ucr.edu/pub/baez/bm2cat.ps.Z
- Baez, Neuchl
(Show Context)
Citation Context ... g. We denote the identity morphism of an object x either as 1 x or, if there is no danger of confusion, simply as x. We refer to our earlier papers on higher-dimensional algebra as HDA0 [2] and HDA1 =-=[3]-=-. 2 H*-Categories Let Hilb denote the category whose objects are finite-dimensional Hilbert spaces, and whose morphisms are arbitrary linear maps. (Henceforth, all Hilbert spaces will taken as finite-... |

52 |
Quantum Groups, Quantum Categories and Quantum Field Theory
- Fröhlich, Kerler
- 1993
(Show Context)
Citation Context ... the category of tilting modules of a quantum group when the parameter q is a suitable root of unity [4]. Categories very similar to our braided 2-H*-algebras have been studied by Frohlich and Kerler =-=[12]-=- under the name `C*- quantum categories'; our definitions differ only in some fine points. A good example of a symmetric 2-H*-algebra is the category of finite-dimensional continuous unitary represent... |

51 | Clock and category: is quantum gravity algebraic
- Crane
- 1995
(Show Context)
Citation Context ...osed to one of the form x = x --- is better understood as an isomorphism, or more generally an equivalence. In their work on categorification in topological quantum field theory, Freed [10] and Crane =-=[5]-=- 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 precise... |

50 |
Representation Theory. A First
- Fulton, Harris
- 1996
(Show Context)
Citation Context ...a x ) = \Sigmaf b x Since b xs= \Sigma1 xsdepending on whether x, and thus x , is even or odd, we have f y = \Sigmaf . ut This result is well-known if H is a category of compact group representations =-=[13]-=-. Here one may also think of the morphism f : x ! x as a conjugate-linear intertwining operator j: x ! x. The condition that f = \Sigmaf y is then equivalent to the condition that j 2 = \Sigma1 x . On... |

49 |
An introduction to Tannaka duality and quantum groups
- Joyal, Street
- 1991
(Show Context)
Citation Context ... x and y. 5.1 The balancing In the study of braided monoidal categories where objects have duals, it is common to introduce something called the `balancing'. The balancing can treated in various ways =-=[12, 16, 26]-=-. For example, one may think of it as a choice of automorphism b x : x ! x for each object x, which is required to satisfy certain laws. While very important in topology, this extra structure seems so... |

29 |
A Guide to Quantum Groups (Cambridge
- Chari, Pressley
- 1994
(Show Context)
Citation Context ...ut Next we turn to braided and symmetric 2-H*-algebras. A good example of a braided 2-H*-algebra is the category of tilting modules of a quantum group when the parameter q is a suitable root of unity =-=[4]-=-. Categories very similar to our braided 2-H*-algebras have been studied by Frohlich and Kerler [12] under the name `C*- quantum categories'; our definitions differ only in some fine points. A good ex... |

14 |
Braided compact monoidal categories with applications to low dimensional topology
- Freyd, Yetter
- 1989
(Show Context)
Citation Context ...l be quite sketchy about describing this, but the reader can fill in the details using the ideas described in HDA0 and the many references therein. Especially relevant is the work of Freyd and Yetter =-=[11]-=-, Joyal and Street [15], and Reshetikhin and Turaev [26, 28]. We discuss this relationship more carefully in the Conclusions. 3. Typical tangle in 2 dimensions The basic idea is to use tangles to repr... |

11 |
Basic Concepts of Enriched Category
- Kelly
(Show Context)
Citation Context ... for a Hilbert space takes values in C . Since we are treating Hilb as the categorification of C , the hom-functor for a 2-Hilbert space should take values in Hilb rather than Set. In technical terms =-=[18]-=-, this suggests that a 2-Hilbert space should be enriched over Hilb. To summarize, we expect that a 2-Hilbert space should be some sort of category with 1) a zero object, 2) binary coproducts, and 3) ... |

5 |
Structure theorems for a special class of Banach algebras
- Ambrose
- 1945
(Show Context)
Citation Context ... of interest in topological quantum field theory have simple algebraic descriptions. For example, knot theorists are familiar with the category of framed oriented 1-dimensional cobordisms embedded in =-=[0; 1]-=- 3 . We would call these `1-tangles in 3 dimensions'. They form not merely a category, but a braided monoidal category. In fact, they form the `free braided monoidal category with duals on one object'... |

4 |
personal communication
- Dolan, Corp
- 1985
(Show Context)
Citation Context ...l: ('F ) \Delta (F ffl) = 1 F ; (G') \Delta (fflG) = 1 G ; become equivalent to those for ffl and ' : (ffl G) \Delta (G' ) = 1 G ; (F ffl ) \Delta (' F ) = 1 F ; by taking duals. ut As noted by Dolan =-=[7]-=-, it is probably quite generally true in n-categories that duality for j-morphisms allows us to turn `left duals' of (j \Gamma 1)-morphisms into `right duals' and vice versa. This should give the theo... |

4 |
there is a algebra with a compact gauge group describing the superselection structure in particle physics
- Doplicher
- 1990
(Show Context)
Citation Context ...ication of symmetric 2-H*-algebras. Doplicher and Roberts proved a theorem which implies that connected even symmetric 2-H*-algebras are all equivalent to categories of compact group representations [=-=8, 9]-=-. Here and in all that follows, by a `representation' of a compact group we mean asnite-dimensional continuous unitary representation. Given a compact group G, let Rep(G) denote the 46 category of suc... |

1 |
personal communication. 61
- Dolan
- 1989
(Show Context)
Citation Context ...he triangle equations for ι and ǫ: (ιF) · (Fǫ) = 1F, (Gι) · (ǫG) = 1G, become equivalent to those for ǫ∗ and ι∗ : (ǫ ∗ G) · (Gι ∗ ) = 1G, (Fǫ ∗ ) · (ι ∗ F) = 1F, by taking duals. ⊓⊔ As noted by Dolan =-=[7]-=-, it is probably quite generally true in n-categories that duality for j-morphisms allows us to turn ‘left duals’ of (j − 1)-morphisms into ‘right duals’ and vice versa. This should give the theory of... |