engleski

Bigroupoid 2-torsors

In this thesis we follow two fundamental concepts from the higher dimensional algebra, the categorification and the internalization. From the geometric point of view, so far the most general torsors were defined in the dimension n=1, by 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 exact fibrations in dimensions greater then n, over special simplicial objects called 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, actions of bicategories and 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 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 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.

bigroupoid; torsor; action of bicategories; nonabelian cohomology

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