The shifted symplectic geometry of derived higher groupoids

Miquel Cueca; Florian Dorsch; Reyer Sjamaar; Chenchang Zhu

The shifted symplectic geometry of derived higher groupoids

Symplectic Geometry 2026-07-19 v1 Category Theory Differential Geometry

Authors: Miquel Cueca , Florian Dorsch , Reyer Sjamaar , Chenchang Zhu

Abstract

The main goal of this work is to introduce derived Lie n-groupoids and their shifted symplectic structures. We further define shifted lagrangian structures and prove that their composition is well defined under suitable conditions. As an application, we show that our framework incorporates several reduction procedures at critical values, including: classical Hamiltonian reduction, group valued moment maps, Poisson Lie group valued moment maps and Mikami-Weinstein for proper symplectic groupoids.

Keywords

derived Lie n-groupoids shifted symplectic structures lagrangian structures moment maps reduction Poisson Lie groups

Cite

@article{arxiv.2607.17362,
  title  = {The shifted symplectic geometry of derived higher groupoids},
  author = {Miquel Cueca and Florian Dorsch and Reyer Sjamaar and Chenchang Zhu},
  journal= {arXiv preprint arXiv:2607.17362},
  year   = {2026}
}

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71 pages