arXiv subjects

Stefano Ronchi

Publications and source records attributed to Stefano Ronchi.

4 recordsLinked to original sources

arXiv2026

Duals of higher vector bundles and cotangents of Lie 2-groupoids

In this thesis we define $n$-duals of VB $n$-groupoids over Lie $n$-groupoids and study their properties. For $n = 0$ this returns the dual vector bundle construction, while for $n = 1$ this returns Pradines's construction of the dual of a VB groupoid over a Lie groupoid, which includes the cotangent symplectic groupoid of Coste, Dazord and Weinstein. For $n = 2$, we propose a new construction that shows that VB 2-duals exist for VB 2-groupoids and they are VB 2-groupoids themselves. Their canonical dual pairings are nondegenerate up to homotopy in the same sense as shifted symplectic structures. In particular, we can apply this construction to the tangent of a Lie 2-groupoid and obtain a cotangent VB 2-groupoid (the 2-cotangent) which is canonically 2-shifted symplectic. We apply this in two ways: First, to characterize 2-shifted symplectic structures on a Lie 2-groupoid as Morita equivalences between its tangent and 2-cotangent groupoid. Second, to compute the 2-cotangent of a Lie 1-groupoid and show it is symplectic Morita equivalent to the bar construction of the 1-cotangent. Along the way, we develop the theory of $n$-duals for simplicial vector spaces, which covers the case where the base is a point. In this case, $n$-duals always exist, as they are defined by a mapping space construction. By a reformulation of the Eilenberg-Zilber theorem in terms of mapping spaces, we obtain that the canonical $n$-dual pairing is nondegenerate up to homotopy for all $n$-types.

math.DG↗

arXiv2025

Shifted Symplectic Geometry by Examples

These notes are intended to be an introduction to shifted symplectic geometry, targeted to Poisson geometers with a serious background in homological algebra. They are extracted from a mini-course given by the first author at the Poisson 2024 summer school that took place at the Accademia Pontaniana in Napoli.

math.SG↗

arXiv2024

Duals of Higher Vector Spaces

We introduce a notion of ``$n$-dual'' to a simplicial vector space for $n\ge 0$. Coming with it, there is a canonical pairing, which we show to be non-degenerate up to homotopy for homotopy $n$-types. As a result this notion of duality is reflexive up to homotopy for $n$-types. In particular the same properties hold for $n$-groupoid objects in vector spaces, whose $n$-duals are again such $n$-groupoid objects. We study this construction in the context of the Dold-Kan correspondence and we reformulate the Eilenberg-Zilber theorem, which classically controls monoidality of the Dold-Kan functors, in terms of internal homs. We compute explicitly the 1-dual of a groupoid object and the 2-dual of a 2-groupoid object in the category of vector spaces. As the 1-dual of a groupoid object, we recover its dual as a $\mathsf{VB}$ groupoid over a point.

math.DG↗

arXiv2023

Hamiltonian Lie algebroids over Poisson manifolds

We extend to Poisson manifolds the theory of hamiltonian Lie algebroids originally developed by two of the authors for presymplectic manifolds. As in the presymplectic case, our definition, involving a vector bundle connection on the Lie algebroid, reduces to the definition of hamiltonian action for an action Lie algebroid with the trivial connection. The clean zero locus of the momentum section of a hamiltonian Lie algebroid is an invariant coisotropic submanifold, the distribution being given by the image of the anchor. We study some basic examples: bundles of Lie algebras with zero anchor and cotangent and tangent Lie algebroids. Finally, we discuss a suggestion by Alejandro Cabrera that the conditions for a Lie algebroid $A$ to be hamiltonian may be expressed in terms of two bivector fields on $A^*$, the natural Poisson structure on the dual of a Lie algebroid and the horizontal lift by the connection of the given Poisson structure on the base.

math.SG↗