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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 TkMM 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 TA to TA (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.