## 2–Groups, Trialgebras and their Hopf Categories of Representations (2005)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Pfeiffer052–groups,trialgebras,

author = {Hendryk Pfeiffer},

title = {2–Groups, Trialgebras and their Hopf Categories of Representations},

year = {2005}

}

### OpenURL

### Abstract

### Citations

512 |
Foundations of Quantum Group Theory
- Majid
- 1995
(Show Context)
Citation Context ... category of cocommutative Hopf algebras. We call this a cocommutative trialgebra for reasons that are explained below. For general background on coalgebras, bialgebras and Hopf algebras, we refer to =-=[37]-=-. Definition 3.1. The functor k[\Gamma ] : Grp ! cocHopfAlgk is defined as follows. It associates with each group G its group algebra k[G]. This is the free vector space over the set G equipped with t... |

289 | Quantum Groups and their Representations - Klimyk, Schmuedgen - 1998 |

210 |
C*-algebras and Operator Theory
- Murphy
- 1990
(Show Context)
Citation Context ... Theorem 4.18.s34 2-Groups, trialgebras and Hopf categories A.1 Gel'fand representation theory For background on C\Lambda -algebras and for the proofs of the results summarized here, see, for example =-=[44]-=-. There is a category comUnC\Lambda Alg whose objects are commutative unital C\Lambda -algebras and whose morphisms are unital \Lambda -homomorphisms. This category has all finite coproducts. In parti... |

182 |
O.Yu.Viro State Sum Invariants of 3-Manifolds and Quantum 6j-symbols Topology 31
- Turaev
- 1992
(Show Context)
Citation Context ...from suitable associative algebras [18], there are two alternative algebraic structures in order to construct (2 + 1)- dimensional TQFTs: suitable Hopf algebras [4, 5] or suitable monoidal categories =-=[19, 20]-=-. Both types of structure are related [21]: the category of corepresentations of a Hopf algebra is a monoidal category and, conversely, under some conditions, the Hopf algebra can be Tannaka-Kre^in re... |

131 |
Tannakian categories
- Deligne, Milne
- 1982
(Show Context)
Citation Context ... 0 ; ! 0 ) be rigid monoidal categories over V and [F; i] : (C; !) ! (C 0 ; ! 0 ) (5.4) be a monoidal functor over V. Then [F; i] is a rigid monoidal functor over V. Proof. Standard, see, for example =-=[40]-=-. Proposition 5.6. Let V be a symmetric monoidal category, (C; !) and (C 0 ; ! 0 ) be symmetric monoidal categories over V and [F; i] : (C; !) ! (C 0 ; ! 0 ) be a monoidal functor over V. Then [F; i] ... |

101 |
Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases
- Crane, Frenkel
- 1994
(Show Context)
Citation Context ...eatures that appear in the corresponding dimension. Motivated by the construction of topological quantum field theories (TQFTs), Crane has introduced the concept of categorification, see, for example =-=[2, 3]-=-. Categorification can be viewed as a systematic replacement of familiar algebraic structures that are modelled on sets by analogues that are rather modelled on categories, 2-categories, and so on. Ca... |

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 ...ategories of these trialgebras in analogy to monoidal categories which appear as the representation categories of Hopf algebras.s2-Groups, trialgebras and Hopf categories 3 ffl Kapranov and Voevodsky =-=[6, 7]-=- have introduced braided monoidal 2-categories and 2vector spaces, a categorified notion of vector spaces, and shown that they are related to the Zamolodchikov tetrahedron equation. In the context of ... |

49 |
An introduction to Tannaka duality and quantum groups
- Joyal, Street
- 1991
(Show Context)
Citation Context ... symmetric Hopf categories and to establish a generalization of Tannaka–Kreǐn duality between commutative cotrialgebras and symmetric Hopf categories. Standard references on Tannaka–Kreǐn duality are =-=[40, 41, 42]-=-. We are aiming for an equivalence of categories, and it is difficult to find such a result explicitly stated in the literature. We follow the presentation of [43] which comes closest to our goal. In ... |

43 | Higher-dimensional algebra VI. Lie 2-algebras
- Baez, Crans
(Show Context)
Citation Context ... [26], and Elgueta [27], for example, employ the 2-vector spaces of Kapranov and Voevodsky [6] (semisimple Abelian categories) whereas Forrester-Barker [28] uses the 2-vector spaces of Baez and Crans =-=[13]-=- (internal categories in the category of vector spaces or, equivalently, 2-term chain complexes of vector spaces). All these authors exploit the fact that one can associate with each 2-group a 2- cate... |

43 |
Aspects of topoi
- Freyd
- 1972
(Show Context)
Citation Context ... only up to (unique) isomorphism, we refer to Remark A.9 in the Appendix. For generic finitely complete C, the 1-category underlying Cat(C) is studied in the context of essentially algebraic theories =-=[30]-=-, going back to the work of Lawvere on functorial semantics [27]. For more details and references, see, for example [31]. The theory of categories Th(Cat) is the smallest finitely complete category th... |

42 |
Toposes, Triples, and Theories. Grundlehren der mathematischen Wissenschaften
- Barr, Wells
- 1985
(Show Context)
Citation Context ...y underlying Cat(C) is studied in the context of essentially algebraic theories [30], going back to the work of Lawvere on functorial semantics [27]. For more details and references, see, for example =-=[31]-=-. The theory of categories Th(Cat) is the smallest finitely complete category that contains objects C0, C1 ∏ and morphisms s,t: C1 → C0, ı: C1 → C0 and ◦: C1 t s C1 → C1 such that the relations (2.8)–... |

41 | Higher Gauge Theory
- Baez, Schreiber
- 2005
(Show Context)
Citation Context ...ted examples of trialgebras [8, 9] in the spirit of Crane–Frenkel and explained how they are related to integrability in 2 + 1 dimensions [10]. • Baez, Lauda, Crans, Bartels, Schreiber and the author =-=[11,12,13,14,15,16]-=- have used 2groups in order to find generalizations of fibre bundles and of gauge theory. Yetter [17] has used 2-groups in order to construct novel TQFTs, generalizing the TQFTs that are constructed f... |

40 |
Invariants of piecewise-linear 3-manifolds, hep-th
- Barrett, Westbury
- 1996
(Show Context)
Citation Context ...from suitable associative algebras [18], there are two alternative algebraic structures in order to construct (2 + 1)- dimensional TQFTs: suitable Hopf algebras [4, 5] or suitable monoidal categories =-=[19, 20]-=-. Both types of structure are related [21]: the category of corepresentations of a Hopf algebra is a monoidal category and, conversely, under some conditions, the Hopf algebra can be Tannaka-Kre^in re... |

39 | Involutory Hopf algebras and 3-manifold invariants
- Kuperberg
- 1991
(Show Context)
Citation Context ... Some examples of higher-dimensional algebraic structures that are relevant in the context of the present article, are the following. ffl Three-dimensional TQFTs can be constructed from Hopf algebras =-=[4, 5]-=-. In order to find four-dimensional TQFTs, Crane and Frenkel [2] have introduced the notion of a Hopf category, a categorification of the notion of a Hopf algebra. Roughly speaking, this is a monoidal... |

37 |
Lattice Topological Field Theory in two Dimensions, Commun.Math.Phys
- Fukuma, Hosono, et al.
- 1994
(Show Context)
Citation Context ...eferred algebraic structures that guarantee the consistency and triangulation independence of these state sums. Whereas (1 + 1)-dimensional TQFTs can be constructed from suitable associative algebras =-=[18]-=-, there are two alternative algebraic structures in order to construct (2 + 1)- dimensional TQFTs: suitable Hopf algebras [4, 5] or suitable monoidal categories [19, 20]. Both types of structure are r... |

33 |
Rivano, Catégories Tannakiennes
- Saavedra
- 1972
(Show Context)
Citation Context ...functor, the condition (5.3) that [IC; id!] is a symmetric monoidal functor over Vectk, implies that IC : C ! M H is a symmetric monoidal functor. Since IC is part of an equivalence of categories, by =-=[39]-=-, Chapter I, Proposition 4.4.2, IC forms a tensor equivalence, i.e. there exist a monoidal functor JC : M H ! C and monoidal natural isomorphisms j : 1C ) ICJC and " : JCIC ) 1D. The inverse of [IC; i... |

32 | Spherical 2-categories and 4-manifold invariants, Adv
- Mackaay
- 1999
(Show Context)
Citation Context ... at once, should give a monoidal 2- category as the category of representations of the trialgebra. Some aspects of the Crane-Frenkel scenario have already been analyzed in greater detail. ffl Mackaay =-=[22]-=- has given a precise definition of suitable monoidal 2-categories and has shown that one can define an invariant of combinatorial 4-manifolds from it. So far, only few examples of these monoidal 2-cat... |

27 |
of Categorical Algebra I. Basic Category Theory. Cambridge Univ
- BORCEUX
- 1994
(Show Context)
Citation Context ... the term `2-category' for a strict 2-category as opposed to a bicategory. Our terminology for 2-categories, 2-functors, 2-natural transformations, 2-equivalences and 2- adjunctions is the same as in =-=[31]-=-. 2.2 Internal categories Many 2-categories that appear in this article, can be constructed by internalization. The concept of internalization goes back to Ehresmann [32]. Here we summarize the key de... |

24 | Higher-dimensional algebra V. 2-groups
- Baez, Lauda
(Show Context)
Citation Context ...e new structures that includes the strict and the commutative or symmetric special cases. We expect two sorts of generalizations beyond the present work. Firstly, the use of weak or coherent 2-groups =-=[12]-=- rather than strict ones. Weak 2-groups are modelled on bicategories whereas strict ones are modelled on 2-categories. Secondly, starting from ordinary groups alone, one can construct only commutative... |

20 |
Structure of topological lattice field theories in three dimensions
- Chung, Fukuma, et al.
- 1994
(Show Context)
Citation Context ... Some examples of higher-dimensional algebraic structures that are relevant in the context of the present article, are the following. ffl Three-dimensional TQFTs can be constructed from Hopf algebras =-=[4, 5]-=-. In order to find four-dimensional TQFTs, Crane and Frenkel [2] have introduced the notion of a Hopf category, a categorification of the notion of a Hopf algebra. Roughly speaking, this is a monoidal... |

20 | Structures and diagrammatics of four dimensional topological lattice field theories
- Carter, Kauffman, et al.
- 1999
(Show Context)
Citation Context ... generalization of Tannaka-Kre^in duality. ffl Crane and Frenkel [2] have presented examples of Hopf categories and proposed a 4manifold invariant based on Hopf categories. Carter, Kauffman and Saito =-=[24]-=- have studied this invariant for some Hopf categories. Again, with the rather limited set of examples of Hopf categories, all invariants studied so far, are homotopy invariants.s2-Groups, trialgebras ... |

20 |
Tannaka duality for arbitrary Hopf algebras, volume 66 of Algebra Berichte
- Schauenburg
- 1992
(Show Context)
Citation Context ...ces on Tannaka-Kre^in duality are [39-41]. We are aiming for an equivalence of categories, and it is difficult to find such a result explicitly stated in the literature. We follow the presentation of =-=[42]-=- which comes closest to our goal. In Appendix B, we summarize how the results of [42] can be employed in order to establish the desired equivalence of categories. In the present section, we just state... |

18 |
Higher gauge theory and a non-abelian generalization of p-form electromagnetism
- Pfeiffer
(Show Context)
Citation Context ...re constructed from the gauge theories of flat connections on a principal G-bundle where G is an (ordinary) group. One can verify that [17] is a special application of the generalized gauge theory of =-=[14]-=-. All these constructions have a common underlying theme: the procedure of categorification on the algebraic side and an increase in dimension on the topological side. Although it is plausible to conj... |

17 |
Introduction to the theory of structured categories
- Ehresmann
- 1966
(Show Context)
Citation Context ...adjunctions is the same as in [31]. 2.2 Internal categories Many 2-categories that appear in this article, can be constructed by internalization. The concept of internalization goes back to Ehresmann =-=[32]-=-. Here we summarize the key definitions and results. For more details and proofs, see, for example [31]. Definition 2.1. Let C be a finitely complete category.s2-Groups, trialgebras and Hopf categorie... |

17 |
Group-like structures in categories
- Eckmann, Hilton
- 1962
(Show Context)
Citation Context ...Deltash2) ffi (h 0 1 \Deltash 0 2) = (h1 ffi h 0 1) \Deltas(h2 ffi h 0 2); (3.8) for h1; h2; h 0 1; h 0 2 2 H1, whenever the partial multiplication `ffi' is defined. A possible EckmannHilton argument =-=[38]-=- which would render both multiplications commutative and equal, is sidestepped by the same mechanism as in strict 2-groups: both multiplications can in general have different units.s2-Groups, trialgeb... |

16 |
TQFTs from homotopy 2-types
- Yetter
- 1993
(Show Context)
Citation Context ...egrability in 2 + 1 dimensions [10]. ffl Baez, Lauda, Crans, Bartels, Schreiber and the author [11-16] have used 2-groups in order to find generalizations of fibre bundles and of gauge theory. Yetter =-=[17]-=- has used 2- groups in order to construct novel TQFTs, generalizing the TQFTs that are constructed from the gauge theories of flat connections on a principal G-bundle where G is an (ordinary) group. O... |

15 | Categorical representation of categorical groups
- Barrett, Mackaay
(Show Context)
Citation Context ... topological groups to strict compact topological 2-groups. Other authors have explored alternative strategies for representing 2-groups on 2-vector spaces. Crane and Yetter [25], Barrett and Mackaay =-=[26]-=-, and Elgueta [27], for example, employ the 2-vector spaces of Kapranov and Voevodsky [6] (semisimple Abelian categories) whereas Forrester-Barker [28] uses the 2-vector spaces of Baez and Crans [13] ... |

13 | The equality of 3-manifold invariants
- Barrett, Westbury
- 1995
(Show Context)
Citation Context ... are two alternative algebraic structures in order to construct (2 + 1)- dimensional TQFTs: suitable Hopf algebras [4, 5] or suitable monoidal categories [19, 20]. Both types of structure are related =-=[21]-=-: the category of corepresentations of a Hopf algebra is a monoidal category and, conversely, under some conditions, the Hopf algebra can be Tannaka-Kre^in reconstructed from this category. According ... |

13 |
Representation Theory of Hopf Categories
- Neuchl
- 1997
(Show Context)
Citation Context ...fine an invariant of combinatorial 4-manifolds from it. So far, only few examples of these monoidal 2-categories have been constructed all of which are thought to give homotopy invariants. ffl Neuchl =-=[23]-=- has studied Hopf categories and their representations on certain monoidal 2-categories. So far, it is open whether one can find a good class of Hopf categories and the corresponding monoidal 2-catego... |

11 | Measurable categories and 2-groups
- Crane, Yetter
(Show Context)
Citation Context ...decomposition from compact topological groups to strict compact topological 2-groups. Other authors have explored alternative strategies for representing 2-groups on 2-vector spaces. Crane and Yetter =-=[25]-=-, Barrett and Mackaay [26], and Elgueta [27], for example, employ the 2-vector spaces of Kapranov and Voevodsky [6] (semisimple Abelian categories) whereas Forrester-Barker [28] uses the 2-vector spac... |

9 | Group objects and internal categories
- Forrester-Barker
- 2002
(Show Context)
Citation Context ...ory in the category of groups so that the techniques of Section 2.2 are available. Alternatively, strict 2-groups can be defined as group objects in the category of small categories, see, for example =-=[35]-=-. In order to define strict 2-groups, let us first recall the construction of finite limits in the finitely complete category Grp of groups. The terminal object is the trivial group feg with the trivi... |

9 |
crossed modules and internal categories in categories of groups with operations
- Porter, Extensions
- 1987
(Show Context)
Citation Context ...he objects of f2Grp := Cat(fGrp) are called strict finite 2-groups. Examples of strict 2-groups, their homomorphisms and 2-homomorphisms can be constructed from Whitehead's crossed modules as follows =-=[36]-=-. Definition 2.11. 1. A crossed module (G; H; \Lambda ; @) consists of groups G and H and group homomorphisms @ : H ! G and G ! Aut H; g 7! (h 7! g \Lambdash) that satisfy for all g 2 G, h; h0 2 H, @(... |

8 |
Braided monoidal 2-categories and ManinSchechtman braid groups
- Kapranov, Voevodsky
- 1994
(Show Context)
Citation Context ...ategories of these trialgebras in analogy to monoidal categories which appear as the representation categories of Hopf algebras.s2-Groups, trialgebras and Hopf categories 3 ffl Kapranov and Voevodsky =-=[6, 7]-=- have introduced braided monoidal 2-categories and 2vector spaces, a categorified notion of vector spaces, and shown that they are related to the Zamolodchikov tetrahedron equation. In the context of ... |

7 | On second quantization of quantum groups
- Grosse, Schlesinger
(Show Context)
Citation Context ...f the integrability condition to three dimensions. In two dimensions, the integrability condition is the famous Yang–Baxter equation. • Grosse and Schlesinger have constructed examples of trialgebras =-=[8, 9]-=- in the spirit of Crane–Frenkel and explained how they are related to integrability in 2 + 1 dimensions [10]. • Baez, Lauda, Crans, Bartels and the author [11–15] have used 2-groups in order to find g... |

5 |
Higher dimensional group theory
- Brown
- 1982
(Show Context)
Citation Context ...onal algebra. Higher-dimensional algebraic structures have appeared in various areas of mathematics and mathematical physics. A prime example is the higher-dimensional group theory programme of Brown =-=[1]-=-, generalizing groups and groupoids to double groupoids and further on, in order to obtain a hierarchy of algebraic structures. The construction of these algebraic structures is motivated by problems ... |

5 |
A suggestion for an integrability notion for two dimensional spin systems, accepted for publication
- Grosse, Schlesinger
(Show Context)
Citation Context ...Yang-Baxter equation. ffl Grosse and Schlesinger have constructed examples of trialgebras [8, 9] in the spirit of Crane-Frenkel and explained how they are related to integrability in 2 + 1 dimensions =-=[10]-=-. ffl Baez, Lauda, Crans, Bartels, Schreiber and the author [11-16] have used 2-groups in order to find generalizations of fibre bundles and of gauge theory. Yetter [17] has used 2- groups in order to... |

5 |
Group-like structures in general categories
- Eckmann, Hilton
- 1962
(Show Context)
Citation Context ...terchange law, (h1 · h2 ) ◦ (h′ 1 · h′ 2 ) = (h1 ◦ h′ 1 ) · (h2 ◦ h′ 2 ), (3.8) for h 1 ,h 2 ,h ′ 1 ,h′ 2 ∈ H1, whenever the partial multiplication ‘◦’ is defined. A possible Eckmann– Hilton argument =-=[39]-=- which would render both multiplications commutative and equal, is sidestepped by the same mechanism as in strict 2-groups: both multiplications can in general have different units.20 2-Groups, trial... |

4 | Categorification
- Baez, Dolan
- 1998
(Show Context)
Citation Context .... We finally remark that we have never called Cat(\Gamma ) a categorification since it is not clear in which sense it can be reversed and whether this would correspond to a form of decategorification =-=[3]-=-. 2.3 Strict 2-groups Strict 2-groups form one of the simplest examples of higher-dimensional algebraic structures. Just as a group can be viewed as a groupoid with one object, every 2-group gives ris... |

4 | Representation theory of 2-groups on finite dimensional 2-vector spaces
- Elgueta
(Show Context)
Citation Context ...s to strict compact topological 2-groups. Other authors have explored alternative strategies for representing 2-groups on 2-vector spaces. Crane and Yetter [25], Barrett and Mackaay [26], and Elgueta =-=[27]-=-, for example, employ the 2-vector spaces of Kapranov and Voevodsky [6] (semisimple Abelian categories) whereas Forrester-Barker [28] uses the 2-vector spaces of Baez and Crans [13] (internal categori... |

4 |
Lawvere Functorial semantics of algebraic theories
- W
- 1963
(Show Context)
Citation Context ...rialgebras and commutative cotrialgebras. The application of such finite-limit preserving functors is the main theme of the present article. In the more sophisticated language of functorial semantics =-=[29]-=-, the key technique exploited in the present article is the study of models of the essentially algebraic theory of categories in various interesting base categories of familiar algebraic structures su... |

4 |
Baez: Higher Yang–Mills theory
- C
- 2002
(Show Context)
Citation Context ...ted examples of trialgebras [8, 9] in the spirit of Crane–Frenkel and explained how they are related to integrability in 2 + 1 dimensions [10]. • Baez, Lauda, Crans, Bartels, Schreiber and the author =-=[11,12,13,14,15,16]-=- have used 2groups in order to find generalizations of fibre bundles and of gauge theory. Yetter [17] has used 2-groups in order to construct novel TQFTs, generalizing the TQFTs that are constructed f... |

4 |
Representations of crossed modules and Cat 1 -groups
- Forrester-Barker
- 2003
(Show Context)
Citation Context ...aces. Crane and Yetter [25], Barrett and Mackaay [26], and Elgueta [27], for example, employ the 2-vector spaces of Kapranov and Voevodsky [6] (semisimple Abelian categories) whereas Forrester-Barker =-=[28]-=- uses the 2-vector spaces of Baez and Crans [13] (internal categories in the category of vector spaces or, equivalently, 2-term chain complexes of vector spaces). All these authors exploit the fact th... |

4 | Measurable categories
- Yetter
(Show Context)
Citation Context ...senting 2-groups on 2-vector spaces. Barrett and Mackaay [25] and Elgueta [26], for example, employ the 2-vector spaces of Kapranov and Voevodsky [6] (semisimple Abelian categories), Crane and Yetter =-=[27,28]-=- use a measure theoretic refinement of these in order to include more interesting examples, whereas Forrester-Barker [29] uses the 2-vector spaces of Baez and Crans [13] (internal categories in the ca... |

3 | Yetter: A more sensitive Lorentzian state sum
- Crane, N
- 2003
(Show Context)
Citation Context ...iable four-manifolds that are finer than just homotopy invariants; thirdly to overcome open problems in the spin foam or state sum approach to the quantization of general relativity, see, for example =-=[30]-=-. 1.7 Outline This article is structured as follows. Section 2 fixes the notation. We summarize the definition of internal categories in finitely complete base categories and the definition of strict ... |

2 |
Categorified gauge theory: two-bundles
- Bartels
(Show Context)
Citation Context ...ted examples of trialgebras [8, 9] in the spirit of Crane–Frenkel and explained how they are related to integrability in 2 + 1 dimensions [10]. • Baez, Lauda, Crans, Bartels, Schreiber and the author =-=[11,12,13,14,15,16]-=- have used 2groups in order to find generalizations of fibre bundles and of gauge theory. Yetter [17] has used 2-groups in order to construct novel TQFTs, generalizing the TQFTs that are constructed f... |

1 |
On a trialgebraic deformation of the Manin
- Grosse, Schlesinger
(Show Context)
Citation Context ...the integrability condition to three dimensions. In two dimensions, the integrability condition is the famous Yang-Baxter equation. ffl Grosse and Schlesinger have constructed examples of trialgebras =-=[8, 9]-=- in the spirit of Crane-Frenkel and explained how they are related to integrability in 2 + 1 dimensions [10]. ffl Baez, Lauda, Crans, Bartels, Schreiber and the author [11-16] have used 2-groups in or... |

1 |
On second quantization of quantum groups
- Grosse, Schlesinger
- 2000
(Show Context)
Citation Context ...the integrability condition to three dimensions. In two dimensions, the integrability condition is the famous Yang-Baxter equation. ffl Grosse and Schlesinger have constructed examples of trialgebras =-=[8, 9]-=- in the spirit of Crane-Frenkel and explained how they are related to integrability in 2 + 1 dimensions [10]. ffl Baez, Lauda, Crans, Bartels, Schreiber and the author [11-16] have used 2-groups in or... |

1 | Higher Yang-Mills theory (2002). Preprint hep-th/0206130. [12 - Baez |

1 | Categorified gauge theory: two-bundles. Preprint math.CT/0410328. [16 - Bartels - 2004 |

1 |
Representations of crossed modules and cat -groups
- Forrester-Barker
- 2003
(Show Context)
Citation Context ...aces. Crane and Yetter [25], Barrett and Mackaay [26], and Elgueta [27], for example, employ the 2-vector spaces of Kapranov and Voevodsky [6] (semisimple Abelian categories) whereas Forrester-Barker =-=[28]-=- uses the 2-vector spaces of Baez and Crans [13] (internal categories in the category of vector spaces or, equivalently, 2-term chain complexes of vector spaces). All these authors exploit the fact th... |