Augmentations are sheaves for Legendrian graphs

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legen-drian isotopy invariants: the augmentation category, a unital A_∞-category, which lifts the set of augmentations of the associated Chekanov-Eliashberg DGA, and a DG category of constructible sheaves on the front plane, with micro-support at contact infin-ity controlled by the (bordered) Legendrian graph. In other words, generalizing [21], we prove “augmentations are sheaves” in the singular case.