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Results tagged with higher-category-theory
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user 3473
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
42
votes
Are higher categories useful?
I am very much used to these kind of questions. Are 2-categories useful? What can one prove using gerbes? Why should I care about stacks?
I think a funny way to react to these kind of questions, wit …
14
votes
Are higher categories useful?
B-Fields in string theory form a 2-category.
This has physical relevance, for example when string theories are glued together along defect lines on the worldsheet. Along a defect line, the two B-fie …
9
votes
Accepted
Is the first differential Pontryagin class a morphism of stacks?
Yes, every differential characteristic class is a stack morphism.
The point is that there exist universal differential characteristic classes. These are not easy to describe since they involve a not …
- 4,519
9
votes
Accepted
Long exact sequence of cohomology from 2-groups
$c$ is your crossed module, or 2-group, in a sense. Anything more concrete will depend on a choice of a cocycle description of the pointed set $[BH,B^3A]$.
For example, $[BH,B^3A]$ classifies central …
- 4,519
8
votes
Accepted
Categorifying the definition of a principal $G$ bundle
The definition you are looking for is precisely Def. 6.1.5 in:
Nikolaus, Thomas; Waldorf, Konrad, Four equivalent versions of nonabelian gerbes, Pac. J. Math. 264, No. 2, 355-420 (2013). ZBL1286.55006 …
- 4,519
3
votes
The Grothendieck plus construction for stacks of n-types
Thomas Nikolaus and Christoph Schweigert discuss the +-construction for $n=2$ in their paper Equivariance in Higher Geometry. They split it up into two steps (I think): first producing a pre-2-stack o …
- 4,519
2
votes
Accepted
Classification of principal G-bundles over a differentiable stack
They key insight is that the bicategory of differentiable stacks is equivalent to the bicategory of Lie groupoids, where the 1-morphisms are so-called bibundles, or Hilsum-Skandalis morphisms. Under t …
- 4,519
2
votes
Given a Lie $2$-group $G$ does every principal $G$ $2$-bundle admit a $2$-connection?
It depends on what version of connection on principal 2-bundles you consider. There are at least four versions:
Fake-flat connections, these are the ones that have a well-defined 2-dimensional parall …
- 4,519