On two-dimensional holonomy

Abstract:

We define the thin fundamental categorical group ${\mathcal P}_2(M,*)$ of a based smooth manifold $(M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$ and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps are rank-$n$ homotopic, when the rank of the differential of the homotopy between them equals $n$. Let $\mathcal {C}(\mathcal {G})$ be a Lie categorical group coming from a Lie crossed module ${\mathcal {G}= (\partial \colon E \to G,\triangleright )}$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal P}_2(M,*) \to \mathcal {C}(\mathcal {G})$, by using a notion of categorical connections, being a pair $(\omega ,m)$, where $\omega$ is a connection 1-form on $P$, a principal $G$ bundle over $M$, and $m$ is a 2-form on $P$ with values in the Lie algebra of $E$, with the pair $(\omega ,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.