Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


A flat principal $G$-bundle over $X$ is determined by its holonomies, which are (after picking a trivialization) group homomorphisms $\pi_1(X)\rightarrow G$. The fiber of the bundle is not canonically identified with $G$, so these maps are only determined up conjugation by $G$. Equivalently these are gauge transformations at the basepoint where $\pi_1$ is evaluated.

Now let $H \overset{t}\rightarrow G$ present a 2-group. With respect to some trivialization, a flat 2-connection on a principal 2-bundle assigns an element of $G$ to a closed curve, and an element $h$ of $H$ to a surface bounding a curve $\gamma$ such that $\gamma$ gets $t(h)$. Flat means that $h$ only depends on the homotopy class of this surface (homotopies fixing $\gamma$).

Thus, I expect flat 2-connections are determined by a functor from the path 2-group of $X$ into $(H\rightarrow G)$. My question is : what is the degeneracy of this presentation? In other words, what is analogous to "up to conjugation in $G$" for flat 1-bundles?


share|improve this question
Conjugation in a group $G$ is merely a natural isomorphism between homomorphism with codomain $G$, thought of as a functor between one-object groupoids. This is in keeping with category-theoretical philosophy that things should only be determined up to isomorphism if there is really no good way to choose between a number of choices. –  David Roberts Aug 27 '12 at 7:12

1 Answer 1

up vote 11 down vote accepted

Let $\mathcal{P}$ be a principal 2-bundle with structure 2-group a crossed module $t:H \to G$. Then, the holonomy of a connection on $\mathcal{P}$ around a surface $\Sigma$ is a well-defined element $$\mathrm{Hol}_{\mathcal{P}}(\Sigma) \in H/[G,H],$$ where $[G,H]$ is the normal subgroup of $H$ generated by all elements of the form $h^{-1}\alpha(g,h)$, where $\alpha$ denotes the action of $G$ on $H$ of the crossed module. Above quotient generalizes the concept of "conjugacy class".

This is explained in detail in my paper with Urs Schreiber

see Example 5.7.

To which extend the elements $\mathrm{Hol}_{\mathcal{P}}(\Sigma) \in H/[G,H]$ characterize the 2-bundle $\mathcal{P}$ I cannot say.

Maybe you are willing to drop "flat" upon replacing "holonomy" by "parallel transport": even in the classical world it is true that every principal $G$-bundle with connection is characterized by its parallel transport, may it be flat or not.

The same statement remains true in the context of connections on 2-bundles: every principal 2-bundle with connection is characterized by its 2-transport. This is one of the main statements of my above-mentioned paper with Urs.

share|improve this answer
Thanks, Konrad. This is precisely what I was looking for. And yes, I see that I did not need to just talk about flat 2-connections and holonomy. I was just thinking towards my own problem :-). –  Ryan Thorngren Aug 27 '12 at 7:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.