### n-Curvature

#### Posted by Urs Schreiber

The concept of $n$-curvature of $n$-transport - and the nature of “fake” curvature.

An abstract definition of $n$-curvature, suitable for use with the notion of $n$-transport ($\to$) is given here

$\;\;\;$ Curvature.

Apart from the definitions, this text contains just the baby example of the curvature of a Lie group valued 1-transport, reproducing the notion of a Lie-algebra valued 1-form $A$, its curvature 2-form $F_A = dA + A \wedge A$ and the corresponding Bianchi identity $d_A F_A = 0$.

The first nontrivial example, that played a great role in motivating these abstract definitions, is that of curvature of a 2-transport with values in a strict Lie 2-group. The general concept of $n$-curvature, as described in the above pdf, explains why the $G_2$-valued 2-transport described before ($\to$) is just a special case of what one would more generally want to understand under principal 2-transport with values in a Lie 2-group.

This is explained and worked out here:

$\;\;\;$ $\Sigma(\mathrm{Inn}(G_2))$-2-Transport.

In particular, the nature of nonvanishing “fake curvature” in the context of 2-transport is clarified by this.

A quick way to derive these results at the differential level is to use FDA techniques ($\to$). Those who know how to use these to describe morphisms of Lie $n$-algebras may find the FDA-version of the above $\Sigma(\mathrm{Inn}(G_2))$-2-transport at the end of this file:

$\;\;\;$ FDA Laboratory.

(The discussion about this curvature topic seems to be going on here.)

## Re: n-Curvature

08 19 06

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