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Title:

Bigroupoid 2-tensors

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-03-27

Source:

http://www.irb.hr/korisnici/ibakovic/2torsbig.pdf

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text

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en

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In der vorliegenden Doktorarbeit werden zwei fundamentale Konzepte der höher dimensionalen

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Algebra, die Kategorifizierung und Internalisierung, verfolgt. Von der geometrischen

Algebra, die Kategorifizierung und Internalisierung, verfolgt. Von der geometrischen Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-06-25

Source:

http://edoc.ub.uni-muenchen.de/9209/1/Bakovic_Igor.pdf

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text

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en

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Title:

The classifying topos of a topological bicategory

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For any topological bicategory 2C, the Duskin nerve N2C of 2C is a simplicial space. We introduce the classifying topos B2C of 2C as the Deligne topos of sheaves Sh(N2C) on the simplicial space N2C. It is shown that the category of topos morphisms Hom(Sh(X), BC) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying ...

For any topological bicategory 2C, the Duskin nerve N2C of 2C is a simplicial space. We introduce the classifying topos B2C of 2C as the Deligne topos of sheaves Sh(N2C) on the simplicial space N2C. It is shown that the category of topos morphisms Hom(Sh(X), BC) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal C-bundles. As a simple consequence, the geometric realization |N2C | of the nerve N2C of a locally contractible topological bicategory 2C is the classifying space of principal 2C-bundles (on CW complexes), giving a variant of the result of Baas, Bökstedt and Kro derived in the context of bicategorical K-theory [1]. We also define classifying topoi of a topological bicategory 2C using sheaves on other types of nerves of a bicategory given by Lack and Paoli [13], Simpson [17] and Tamsamani [18] by means of bisimplicial spaces, and we examine their properties. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-26

Source:

http://arxiv.org/pdf/0902.1750v1.pdf

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text

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en

DDC:

514 Topology *(computed)*

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Title:

The classifying topos of a topological bicategory

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For any topological bicategory 2C, the Duskin nerve N2C of 2C is a simplicial space. We introduce the classifying topos B2C of 2C as the Deligne topos of sheaves Sh(N2C) on the simplicial space N2C. It is shown that the category of topos morphisms Hom(Sh(X), BC) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying ...

For any topological bicategory 2C, the Duskin nerve N2C of 2C is a simplicial space. We introduce the classifying topos B2C of 2C as the Deligne topos of sheaves Sh(N2C) on the simplicial space N2C. It is shown that the category of topos morphisms Hom(Sh(X), BC) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal C-bundles. As a simple consequence, the geometric realization |N2C | of the nerve N2C of a locally contractible topological bicategory 2C is the classifying space of principal 2C-bundles (on CW complexes), giving a variant of the result of Baas, Bökstedt and Kro derived in the context of bicategorical K-theory [1]. We also define classifying topoi of a topological bicategory 2C using sheaves on other types of nerves of a bicategory given by Lack and Paoli [13], Simpson [17] and Tamsamani [18] by means of bisimplicial spaces, and we examine their properties. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-26

Source:

http://arxiv.org/pdf/0902.1750v2.pdf

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text

Language:

en

DDC:

514 Topology *(computed)*

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The simplicial interpretation of bigroupoid 2-torsors

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Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk’s classification of regular Lie groupoids in foliation theory [34] to Waldmann’s work on deformation quantization [38]. For any such action we introduce an action bicategory, together with a can...

Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk’s classification of regular Lie groupoids in foliation theory [34] to Waldmann’s work on deformation quantization [38]. For any such action we introduce an action bicategory, together with a canonical projection (strict) 2-functor to the bicategory which acts. When the bicategory is a bigroupoid, we can impose the additional condition that action is principal in bicategorical sense, giving rise to a bigroupoid 2-torsor. In that case, the Duskin nerve of the canonical projection is precisely the Duskin-Glenn simplicial 2-torsor, introduced in [25]. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-26

Source:

http://arxiv.org/pdf/0902.3436v1.pdf

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text

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en

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Title:

Noncommutative gerbes and deformation quantization

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We define noncommutative gerbes using the language of star products. Quantized twisted Poisson structures are discussed as an explicit realization in the sense of deformation quantization. Our motivation is the noncommutative description of

We define noncommutative gerbes using the language of star products. Quantized twisted Poisson structures are discussed as an explicit realization in the sense of deformation quantization. Our motivation is the noncommutative description of Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-07-15

Source:

http://arxiv.org/pdf/hep-th/0206101v1.pdf

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text

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en

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Title:

The simplicial interpretation of bigroupoid 2-torsors

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Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk's classification of regular Lie groupoids in foliation theory, to Waldmann's work on deformation quantization. For any such action we introduce an action bicategory, together with a canonical pr...

Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk's classification of regular Lie groupoids in foliation theory, to Waldmann's work on deformation quantization. For any such action we introduce an action bicategory, together with a canonical projection (strict) 2-functor to the bicategory which acts. When the bicategory is a bigroupoid, we can impose the additional condition that action is principal in bicategorical sense, giving rise to a bigroupoid 2-torsor. In that case, the Duskin nerve of the canonical projection is precisely the Duskin-Glenn simplicial 2-torsor. ; Comment: preliminary version Minimize

Year of Publication:

2009-02-19

Document Type:

text

Subjects:

Mathematics - Category Theory ; Mathematics - Algebraic Topology

Mathematics - Category Theory ; Mathematics - Algebraic Topology Minimize

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Title:

Bigroupoid 2-torsors

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"In this thesis we follow two fundamental concepts from the {\it higher dimensional algebra}, the {\it categorification} and the {\it internalization}. From the geometric point of view, so far the most general torsors were defined in the dimension $n=1$, by {\it actions of categories and groupoids}. In the dimension $n=2$, Mauri and Tierney, and...

"In this thesis we follow two fundamental concepts from the {\it higher dimensional algebra}, the {\it categorification} and the {\it internalization}. From the geometric point of view, so far the most general torsors were defined in the dimension $n=1$, by {\it actions of categories and groupoids}. In the dimension $n=2$, Mauri and Tierney, and more recently Baez and Bartels from the different point of view, defined less general 2-torsors with the structure 2-group. Using the language of simplicial algebra, Duskin and Glenn defined actions and torsors internal to any Barr exact category $\E$, in an arbitrary dimension $n$. This actions are simplicial maps which are {\it exact fibrations} in dimensions $m \geq n$, over special simplicial objects called {\it n-dimensional Kan hypergroupoids}. The correspondence between the geometric and the algebraic theory in the dimension $n=1$ is given by the Grothendieck nerve construction, since the Grothendieck nerve of a groupoid is precisely a 1-dimensional Kan hypergroupoid. One of the main results is that groupoid actions and groupoid torsors become simplicial actions and simplicial torsors over the corresponding 1-dimensional Kan hypergroupoids, after the application of the Grothendieck nerve functor. The main result of the thesis is a generalization of this correspondence to the dimension $n=2$. This result is achieved by introducing two new algebraic and geometric concepts, {\it actions of bicategories} and {\it bigroupoid 2-torsors}, as a categorification and an internalization of actions of categories and groupoid torsors. We provide the classification of bigroupoid 2-torsors by {\it the second nonabelian cohomology} with coefficients in the structure bigroupoid. The second nonabelian cohomology is defined by means of the third new concept in the thesis, a {\it small 2-fibration} corresponding to an internal bigroupoid in the category $\E$. The correspondence between the geometric and the algebraic theory in the dimension $n=2$ is given by the Duskin nerve construction for bicategories and bigroupoids since the Duskin nerve of a bigroupoid is precisely a 2-dimensional Kan hypergroupoid. Finally, the main results of the thesis is that bigroupoid actions and bigroupoid 2-torsors become simplicial actions and simplicial 2-torsors over the corresponding 2-dimensional Kan hypergroupoids, after the application of the Duskin nerve functor." Minimize

Publisher:

Ludwig-Maximilians-Universität München

Year of Publication:

2008-06-27

Document Type:

Dissertation ; NonPeerReviewed

Subjects:

Fakultät für Mathematik ; Informatik und Statistik

Fakultät für Mathematik ; Informatik und Statistik Minimize

DDC:

514 Topology *(computed)*

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Title:

The classifying topos of a topological bicategory

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For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topo...

For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization |NB| of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties. ; Comment: accepted for a publication in "Homology, Homotopy and Applications" Minimize

Year of Publication:

2009-02-10

Document Type:

text

Subjects:

Mathematics - Category Theory ; Mathematics - Algebraic Topology ; 18D05 ; 18B25 ; 55Fxx

Mathematics - Category Theory ; Mathematics - Algebraic Topology ; 18D05 ; 18B25 ; 55Fxx Minimize

DDC:

514 Topology *(computed)*

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Title:

The classifying topos of a topological bicategory

Year of Publication:

2010

Source:

Homology, Homotopy and Applications, v.12, 279-300 (2010)

Homology, Homotopy and Applications, v.12, 279-300 (2010) Minimize

Document Type:

Article

Language:

en

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