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# Journal of Symplectic Geometry

## Volume 20 (2022)

### Number 3

### Contact categories of disks

Pages: 665 – 759

DOI: https://dx.doi.org/10.4310/JSG.2022.v20.n3.a3

#### Authors

#### Abstract

In the first part of the paper we associate a pre-additive category $\mathcal{C}(\Sigma)$ to a closed oriented surface $\Sigma$, called the *contact category* and constructed from contact structures on $\Sigma \times [0, 1]$. There are also $\mathcal{C}(\Sigma, F)$, where $\Sigma$ is a compact oriented surface with boundary and $F \subset \partial\Sigma$ is a finite oriented set of points which bounds a submanifold of $\partial\Sigma$, and universal covers $\widetilde{\mathcal{C}}(\Sigma)$ and $\widetilde{\mathcal{C}}(\Sigma, F)$ of $\mathcal{C}(\Sigma)$ and $\mathcal{C}(\Sigma, F)$. In the second part of the paper we prove that the universal cover of the contact category of a disk admits an embedding into its “triangulated envelope.”

K.H. is supported by NSF Grants DMS-0805352, DMS-1105432, DMS-1406564, and DMS-154914.

Y.T. is supported by NSFC 11601256 and 11971256.

Received 6 June 2017

Accepted 19 April 2021

Published 28 February 2023