### Baković on *Bigroupoid 2-Torsors*

#### Posted by Urs Schreiber

Igor Baković – whose work I had mentioned here – has finished his thesis:

Igor Bakovič
*Bigroupoid 2-torsors*

(pdf)

He defines and classifies $G$-principal bundles for $G$ a *bigroupoid* (weak 2-groupoid), as bigroupoids $P \to X$ over base space $X$ – internal to some exact category (a topos, in particular, say of generalized smooth spaces).

Igor relates bigroupoid 2-torsors to the simplicial definition of $n$-torsors for any $n$ by Duskin and Glenn. Glenn essentially defined actions of $\infty$-groupoids (modeled as Kan complexes) in terms of the corresponding action $\infty$-groupoids (recalled as definitions 15.4, 15.5 in Igor’s thesis). Igor realizes this action-$n$-groupoid interpretation (section 13), for $n=2$, by first introducing the concept of an *action bigroupoid* (theorem 13.2) encoding the weak quotient of a bigroupoid action on some category.

Then he shows, theorem 15.3, that the nerve of the action bigroupoid encoding the action of the bigroupoid $G$ on a $G$-principal bundle is indeed a 2-bundle (2-torsor) in the sense of Duskin-Glenn.

Along the way he generalizes the tangent 2-categories from [Roberts-S.] (p. 114) and identifies our inner automorphism 3-group $INN(G) = \mathbf{E}G$ as, indeed, the action bigroupoid of the 2-group $G$ acting on itself.

Here is the summary section of Igor’s thesis:

**The summary**

In this thesis we follow two fundamental concepts from the higher dimensional algebra, 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 dimension $n=2$, Mauri and Tierney, and more recently Baez and Bartels from a different point of view, defined less general 2-torsors with 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 $n$. These actions are simplicial maps which are exact fibrations in dimensions $m \geq n$, 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 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 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.

## Re: Baković on Bigroupoid 2-Torsors

A part of Igor’s thesis concentrated on the connection (via “Duskin nerve” functor from bicategories to simplicial sets) to the Glenn-Duskin 2-torsors in simplicial context, in adapted form (mentioning also some sources of examples) appeared at the arXiv today:

The simplicial interpretation of bigroupoid 2-torsors