# Symplectic microgeometry I: micromorphisms

@article{Cattaneo2009SymplecticMI, title={Symplectic microgeometry I: micromorphisms}, author={Alberto S. Cattaneo and Benoit Richard Umbert Dherin and Alan D. Weinstein}, journal={Journal of Symplectic Geometry}, year={2009}, volume={8}, pages={205-223} }

We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a symmetric monoidal category, which is a version of the “category” of symplectic manifolds and canonical relations obtained by localizing them around Lagrangian submanifolds in the spirit of Milnor’s microbundles.

## 21 Citations

Symplectic microgeometry III: monoids

- Mathematics
- 2011

We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and Lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid…

Symplectic microgeometry II: generating functions

- Mathematics
- 2011

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an…

Formal Lagrangian Operad

- MathematicsInt. J. Math. Math. Sci.
- 2010

It turns out that the semiclassical part of Kontsevich's deformation of () is aDeformation of the trivial symplectic groupoid structure of .

Symplectic microgeometry, IV: Quantization

- MathematicsPacific Journal of Mathematics
- 2021

We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the…

On Lie algebroids, L1 algebras, and the homotopy Poisson structure on shifted conormal bundles of coisotropic submanifolds

- Mathematics
- 2019

In this semester research project, we will quickly review Lie algebroids, L1 algebras and related structures. We will also elaborate general aspects of the theory of supergeometry. Finally, we will…

Relational symplectic groupoids and Poisson sigma models with boundary

- Mathematics
- 2013

We introduce the notion of relational symplectic groupoid as a way to integrate Poisson manifolds in general, following the construction through the Poisson sigma model (PSM) given by Cattaneo and…

Double Groupoids and the Symplectic Category

- Mathematics
- 2017

We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects…

Quantizations of Momentum Maps and G-Systems

- Mathematics
- 2012

In this note, we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on Rd. This family of quantizations…

G-Systems and Deformation of G-Actions On Rd

- Mathematics
- 2014

Given a (smooth) action of a Lie group G on Rd we construct a DGA whose Maurer-Cartan elements are in one to one correspondence with some class of defomations of the (induced) G-action on the ring of…

Integration of Exact Courant Algebroids

- Mathematics
- 2012

In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids,
and we show that the differentiation procedure from [32] inverts our integration.

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