The notion of a higher-order algebroid, as introduced by Jóźwikowski and Rotkiewicz in Higher-order analogs of Lie algebroids via vector bundle comorphisms (SIGMA 2018), generalizes the concepts of a higher-order tangent bundle $T^k M \to M$ and a (Lie) algebroid. This idea is based on a (vector bundle) comorphism approach to (Lie) algebroids and the reduction procedure of homotopies from the level of Lie groupoids to that of Lie algebroids. In brief, an alternative description of a Lie algebroid $(A, [ , ], \#)$ is a vector bundle comorphism κ, defined as the dual of the Poisson map from $T^*A$ to $T A^*$ (contraction with the Poisson bivector) associated with the Lie algebroid $A$. The framework of comorphisms has proven to be a suitable language for describing higher-order analogues of Lie algebroids from the perspective of the role played by (Lie) algebroids in geometric mechanics.

In my talk, I will introduce the notion of higher algebroids and, drawing from recent preprint [MR2024], explain how to uncover the classical algebraic structures that underpin the intricate description of higher-order algebroids via comorphisms. Specifically, in the case $k=2$, I will demonstrate a one-to-one correspondence between higher-order Lie algebroids and pairs consisting of a two-term representation (up to homotopy) of a Lie algebroid and a morphism to the adjoint representation of that algebroid.

[MR2024], Mikołaj Rotkiewicz, Exploring the Structure of Higher Algebroids, 2024, arXiv:2408.02194.