Contact categories of disks

In the first part of the paper we associate a pre-additive category $\mathcal{C}(\mathrm{\Sigma })$ to a closed oriented surface $\mathrm{\Sigma }$, called the {\em contact category} and constructed from contact structures on $\mathrm{\Sigma }\times [0,1]$. There are also $\mathcal{C}(\mathrm{\Sigma },F)$, where $\mathrm{\Sigma }$ is a compact oriented surface with boundary and $F\subset \mathrm{\partial }\mathrm{\Sigma }$ is a finite oriented set of points which bounds a submanifold of $\mathrm{\partial }\mathrm{\Sigma }$, and universal covers $\tilde{\mathcal{C}}(\mathrm{\Sigma })$ and $\tilde{\mathcal{C}}(\mathrm{\Sigma },F)$ of $\mathcal{C}(\mathrm{\Sigma })$ and $\mathcal{C}(\mathrm{\Sigma },F)$. 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.”