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

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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 2torsors with the structure 2group. 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 ndimensional 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 1dimensional Kan hypergroupoid. One of the main results is that groupoid actions and groupoid torsors become simplicial actions and simplicial torsors over the corresponding 1dimensional 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 2torsors}, as a categorification and an internalization of actions of categories and groupoid torsors. We provide the classification of bigroupoid 2torsors 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 2fibration} 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 2dimensional Kan hypergroupoid. Finally, the main results of the thesis is that bigroupoid actions and bigroupoid 2torsors become simplicial actions and simplicial 2torsors over the corresponding 2dimensional Kan hypergroupoids, after the application of the Duskin nerve functor."
Item Type:  Thesis (Dissertation, LMU Munich) 

Subjects:  600 Natural sciences and mathematics > 510 Mathematics 600 Natural sciences and mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  27. June 2008 
1. Referee:  Jurco, Branislav 
Persistent Identifier (URN):  urn:nbn:de:bvb:1992092 
MD5 Checksum of the PDFfile:  c81bd70d094a96f0eda8fe9f65950bfb 
Signature of the printed copy:  0001/UMC 17350 
ID Code:  9209 
Deposited On:  07. Nov 2008 09:15 
Last Modified:  19. Jul 2016 16:25 