Lie $2$-algebroids are geometrised by linear Courant algebroids, while symplectic Lie $2$-algebroids correspond to mere Courant algebroids.
This talk begins by explaining these correspondences due to Li-Bland, Severa and Roytenberg, by establishing the underlying equivalence between -manifolds and metric double vector bundles. The latter yields a dictionary between graded geometric structures on -manifolds, like homological vector fields, Poisson and symplectic structures, and corresponding `classical geometric' structures on the corresponding metric double vector bundles.
Metric double vector bundles dualise to double vector bundles equipped with a (signed) involution. The latter can then be understood as $S_2$-symmetric double vector bundles -- recovering Pradines’ ‘inverse' symmetric double vector bundles.
Similarly, positively graded manifold of arbitrary degree $n$ are equivalent to $n$-fold vector bundles equipped with a (signed) $S_n$-symmetry. This talk explains more precisely how symmetric $n$-fold vector bundle cocycles are the same objects as [n]-manifold cocycles, and how symmetric vector bundles, which are indexed by $n$-cube categories, provide a new and insightful point of view on (positively) graded geometry.
This is the groundwork for understanding a possible geometrisation of Lie $n$-algebroids, like VB-Courant algebroids geometrise Lie $2$-algebroids, and Lie algebroids geometrise Lie $1$-algebroids.
This work is partly joint with Malte Heuer.