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Abstract

Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods
of manifolds in shifted symplectic stacks, and serve as target spaces for
twisted variants of classical AKSZ topological field theory. In this paper, we
classify zero-, one- and two-shifted symplectic algebroids and their higher
gauge symmetries, in terms of classical geometric "higher structures", such as
Courant algebroids twisted by $\Omega^2$-gerbes. As applications, we produce
new examples of twisted Courant algebroids from codimension-two cycles, and we
give symplectic interpretations for several well known features of higher
structures (such as twists, Pontryagin classes, and tensor products). The
proofs are valid in the $C^\infty$, holomorphic and algebraic settings, and are
based on a number of technical results on the homotopy theory of $L_\infty$
algebroids and their differential forms, which may be of independent interest.