Principaloid bundles
Thomas Strobl, Rafał R. Suszek
TL;DR
Principaloid bundles introduce a principled generalization of principal bundles by using a Lie groupoid ${\mathscr G}\rightrightarrows M$ as the fibre with structure reduced to its bisections ${\mathbb B}$. A canonical shadow structure $\mathscr F$ over the base and a sitting-duck map $\mathscr D: \mathscr P\to\mathscr F$ organize the data so that $\mathscr P$ is a principal $\mathscr G$-bundle over $\mathscr F$. Connections are characterized as ${\mathscr G}$-invariant Ehresmann connections whose local data take values in the Lie algebroid $E$ of $\mathscr G$, with gluing laws generalizing the usual $Ad_g$ and Maurer–Cartan terms. The Atiyah–Ehresmann groupoid ${\rm At}(\mathscr P)$ fits into a short exact sequence with the adjoint bundle ${\rm Ad}(\mathscr P)$ and the pair groupoid ${\rm Pair}(\Sigma)$, providing a trident of actions linking automorphisms, gauge transformations, and the shadow structure. Gauge transformations and covariant derivatives are extended to this setting, preserving consistency across the principal, shadow, and Atiyah layers and enabling gauge-theoretic constructions beyond traditional principal bundles. The framework unifies ordinary principal bundles, associated bundles, and general fibre bundles while enabling new symmetry and descent phenomena in Lie groupoid contexts.
Abstract
We present a novel generalisation of principal bundles -- principaloid bundles: These are fibre bundles $π:P\to B$ where the typical fibre is the arrow manifold $G$ of a Lie groupoid $G\rightrightarrows M$ and the structure group is reduced to the latter's group of bisections. Each such bundle canonically comes with a bundle map $D:P\to F$ to another fibre bundle $F$ over the base $B$, with typical fibre $M$. Examples of principaloid bundles include ordinary principal $\underline G$-bundles, obtained for $G:=\underline G\rightrightarrows\bullet$, bundles associated to them, obtained for action groupoids $G:=\underline G\ltimes M$, and general fibre bundles if $G$ is a pair groupoid. While $π$ is far from being a principal $G$-bundle, we prove that $D$ is one. Connections on the principaloid bundle $π$ are thus required to be $G$-invariant Ehresmann connections. In the three examples mentioned above, this reproduces the usual types of connection for each of them. In a local description over a trivialising cover $\{O_i\}$ of $B$, the connection gives rise to Lie algebroid-valued objects living over bundle trivialisations $\{O_i\times M\}$ of $F$. Their behaviour under bundle automorphisms, including gauge transformations, is studied in detail. Finally, we construct the Atiyah-Ehresmann groupoid ${\rm At}(P)\rightrightarrows F$ which governs symmetries of $P$, this time mapping distinct $D$-fibres to one another in general. It is a fibre-bundle object in the category of Lie groupoids, with typical fibre $G\rightrightarrows M$ and base $B\times B\rightrightarrows B$. We show that those of its bisections which project to bisections of its base are in a one-to-one correspondence with automorphisms of $π$.