UC Berkeley

We investigate a collection of posets- combinatorial arboreal singularities- which are the strata posets of the arboreal singularities constructed by David Nadler. Nadler demonstrated that any Lagrangian skeleton admits a non-characteristic deformation into a skeleton with only arboreal singularities, suggesting that arboreal singularities form the basis for a combinatorial theory of Lagrangian skeleta.

In this document, we introduce a form of combinatorial data called a `cyclic structure', which is essentially codimension-one data with compatibility conditions in codimensions two and three. We develop a comprehensive theory of isomorphisms of combinatorial arboreal singularities and cyclic arboreal singularities (singularities equipped with a cyclic structure). We show that a cyclic structure determines (up to quasi-equivalence) a sheaf of dg-categories on a combinatorial arboreal singularity, and investigate combinatorial properties of this sheaf. We describe a class of `arboreal moves', which are local mutations of combinatorial arboreal spaces preserving this sheaf of categories. Finally, we discuss how this combinatorial picture is related to the geometric understanding of Lagrangian embeddings of arboreal singularities.