Engineering

Pure mathematics

Equivalence (formal languages)

Coherent sheaf

Sheaf

Chemical engineering

Knot (papermaking)

Unital

Mathematics

Mathematical analysis

Functor

Conjecture

Cohomology

Derived category

Algebra over a field

Infinity

Geometry & Topology

We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.

Augmentations are sheaves