Baković, Igor (2008): Bigroupoid 2-torsors. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics |

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**DOI**: 10.5282/edoc.9209

### Abstract

"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."

Item Type: | Theses (Dissertation, LMU Munich) |
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Subjects: | 500 Natural sciences and mathematics > 510 Mathematics 500 Natural sciences and mathematics |

Faculties: | Faculty of Mathematics, Computer Science and Statistics |

Language: | English |

Date of oral examination: | 27. June 2008 |

1. Referee: | Jurco, Branislav |

MD5 Checksum of the PDF-file: | c81bd70d094a96f0eda8fe9f65950bfb |

Signature of the printed copy: | 0001/UMC 17350 |

ID Code: | 9209 |

Deposited On: | 07. Nov 2008 09:15 |

Last Modified: | 24. Oct 2020 06:55 |