Gerbe
المؤلف:
Brylinski, J
المصدر:
Loop Spaces, Characteristic Classes and Geometric Quantization. Boston, MA: Birkhäuser, 1993.
الجزء والصفحة:
...
11-5-2021
2072
Gerbe
There are no fewer than two closely related but somewhat different notions of gerbe in mathematics.
For a fixed topological space 
, a gerbe on 
can refer to a stack of groupoids 
on 
satisfying the properties
1. ![X= union <span style=]()
{U:G(U)!=emptyset}" src="https://mathworld.wolfram.com/images/equations/Gerbe/Inline5.gif" style="height:15px; width:128px" /> for subsets 
open, and
2. given objects 
, any point 
has a neighborhood 
for which there is at least one morphism 
in 
.
The second definition is due to Giraud (Brylinski 1993). Given a manifold 
and a Lie group 
, a gerbe 
with band 
is a sheaf of groupoids over 
satisfying the following three properties:
1. Given any object 
of 
, the sheaf 
of automorphisms of this object is a sheaf of groups on 
which is locally isomorphic to the sheaf 
of smooth 
-valued functions. Such a local isomorphism 
is unique up to inner automorphisms of 
.
2. Given two objects 
and 
of 
, there exists a surjective local homeomorphism 
such that 
and 
are isomorphic. In particular, 
and 
are locally isomorphic.
3. There exists a surjective local homeomorphism 
such that the category 
is non-empty.
Clearly, the notion of a gerbe's band is fundamental for the second definition; though not explicitly mentioned, the band of a gerbe 
defined by the first definition is also important (Moerdijk 2002). According to Brylinski, gerbes whose bands 
corresponds to a Lie group 
are significant in that they give rise to degree-2 cohomology classes in 
, a fact utilized by Giraud in his study of non-abelian degree-2 cohomology.
REFERENCES:
Brylinski, J. Loop Spaces, Characteristic Classes and Geometric Quantization. Boston, MA: Birkhäuser, 1993.
Moerdijk, I. "Introduction to the Language of Stacks and Gerbes." 2002. https://arxiv.org/abs/math/0212266.
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