Contact categories of disks

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 {\em 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."