Local Theory for 2-Functors on Path 2-Groupoids - CiteSeerX
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Local Theory for 2-Functors on Path 2-Groupoids - CiteSeerX
We infer that the path 2-groupoid is a 2-groupoid internal to the category of diffeological spaces. [SW11]. Diffeological spaces contain smooth manifolds as a full ...
Local Theory for 2-Functors on Path 2-Groupoids Urs Schreiber a and Konrad Waldorf b
arXiv:1303.4663v1 [math.GN] 19 Mar 2013
a
Faculty of Science, Radboud University Nijmegen Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands b
Fakult¨at f¨ ur Mathematik, Universit¨ at Regensburg Universit¨ atsstraße 31, 93053 Regensburg, Germany
Abstract In this article we discuss local aspects of 2-functors defined on the path 2-groupoid of a smooth manifold; in particular, local trivializations and descent data. This is a contribution to a project that provides an axiomatic formulation of connections on (possibly non-abelian) gerbes in terms of 2-functors. The main result of this paper establishes the first part of this formulation: we prove an equivalence between the globally defined 2-functors and their locally defined descent data. The second part appears in a separate publication; there we prove equivalences between descent data, on one side, and various existing versions of gerbes with connection on the other side.
Contents 1 Introduction
2
2 Locally Trivial 2-Functors and their Descent Data
Equivalence in the Direct Limit . . . . . . . . . . . . . . . . . . . . . . . . .
34
A Basic 2-Category Theory
37
References
47
1
Introduction
The present article is a contribution to the axiomatic formulation of connections on (possibly non-abelian) gerbes carried out by the authors in several articles [SW09, SW11, SW]. It is based on 2-functors / T F : P2 (M ) defined on the path 2-groupoid of a smooth manifold M , with values in some “target” 2-category T . The path 2-groupoid P2 (M ) is a strict 2-groupoid with objects the points of M , 1-morphisms certain homotopy classes of paths, and 2-morphisms certain homotopy classes of homotopies between paths. A typical example of a target 2-category is the 2category of algebras (over some fixed field), bimodules, and intertwiners. In that example, the algebra F (x) ∈ T associated to a point x ∈ M is supposed to be the fibre of the gerbe at x, the bimodule / F (y) F (γ) : F (x) associated to a path γ from x to y is supposed to be the parallel transport of the connection on that gerbe along the path γ, and the intertwiner F (γ1 )
