Bigroupoid 2-torsors

"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."