Sheaves via augmentations of Legendrian surfaces

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Given an augmentation for a Legendrian surface in a 1-jet space, Λ⊂ J¹(M), we explicitly construct an object, F ∈ ShΛ, of the (derived) category from arXiv:1402.0490 of constructible sheaves on M× R with singular support determined by Λ. 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 ShΛ in the 2-dimensional case.