Sheaves via augmentations of Legendrian surfaces

Given an augmentation for a Legendrian surface in a $1$-jet space, $Lambda subset J^1(M)$, we explicitly construct an object, $mathcal{F} in Sh_{Lambda}$, of the (derived) category from arXiv:1402.0490 of constructible sheaves on $Mtimes R$ with singular support determined by $Lambda$. In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in $1$-jet spaces that, based on arXiv:1608.02984 and arXiv:1608.03011, is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of arXiv:1402.0490 for $1$-dimensional Legendrian knots to obtain a combinatorial model for sheaves in $Sh_Lambda$ in the $2$-dimensional case.