Let $\Lambda \subset J^1(X)$ be Legendrian. In an earlier post, I gave a conceptual argument for why there should be a fully faithful morphism from the category $Aug(\Lambda)$ defined as in [NRSSZ] to the category $Shv_\Lambda(X \times \mathbb{R})$ of sheaves on $X \times \mathbb{R}$ with microsupport at infinity contained in $\Lambda$.
Here I will explain that the essential surjectivity of this morphism amounts to a question posed entirely in the augmentation category.
(And I don’t really think this final question is so hard.)
First let us arrive at the question in the augmentation category. The idea is to introduce first a derived augmentation category, then a notion of support, modeled after Nadler’s definition of singular support of an object in the infinitesimally wrapped Fukaya category [N].
Fix some setting in which Legendrian contact homology is defined, e.g. in a contact manifold $V = J^1(X)$ or maybe more generally in a contact manifold $V$ with some fixed filling $W$.
Then as in [NRSSZ], for $\Lambda \subset V$, one can define the augmentation category $Aug(\Lambda)$. Its objects are augmentations of $\Lambda$; morphisms are Reeb chords from $\Lambda$ to a push-off of itself, and there is an $A_\infty$ structure given by counting holomorphic disks.
More to the point here, one can define the “big” augmentation category of $V$: given any collection $\Lambda_1, \ldots, \Lambda_n$ of knots, one can use their union to define homs between augmentations of $\Lambda_i$ and $\Lambda_j$, and likewise the higher operations. Taking the appropriate limit, we arrive at the big augmentation category of $Aug(V)$ — its objects are pairs: a knot, plus an augmentation of the DGA of the knot; its morphisms are as above.
Remark. Essentially by definition, there is a fully faithful inclusion $Aug(\Lambda) \to Aug(V)$.
As $Aug(V)$ is an $A_\infty$ category, we can now consider the appropriate derived version, formed by taking twisted complexes (or whatever other version of stabilization is desired). We write this version as $DAug(V)$. Note that there is a fully faithful inclusion $Aug(V) \to DAug(V)$.
Definition. Consider an element $A \in DAug(V)$. The null locus $Null(A)$ is the union of all open sets $U$ such that for every Legendrian $\Lambda \subset U$, and every augmentation $\epsilon: A(\Lambda) \to k$, we have $Hom_{DAug(V)}(\epsilon, A) = 0$. The support of $A$ is the complement of the null locus.
Remark. This definition imitates Nadler’s definition of the singular support of an element of the infinitesimally wrapped Fukaya category of a Weinstein manifold, which in turn imitates the definition of Kashiwara and Schapira of the microsupport of a constructible sheaf.
Remark. I believe it can be shown that it is enough to consider unknots as test objects.
Definition. For a possibly singular Legendrian $\Lambda \subset V$, we define $DAug_\Lambda(V)$ as the full subcategory of $DAug(V)$ consisting of objects with support in $\Lambda$.
Remark. There is a fully faithful inclusion $Aug(\Lambda) \to DAug_\Lambda(V)$.
Conjecture: When $\Lambda$ is smooth (and maybe some assumptions on $V$, e.g. bounds subcritical domain or maybe is jet bundle), then the morphism $Aug(\Lambda) \to DAug_\Lambda(V)$ is essentially surjective.
Remark. If this conjecture is true, then $DAug_\Lambda(V)$ is a good definition of the augmentation category when $\Lambda$ is singular. If the conjecture is false, it seems from the discussion below that $DAug_\Lambda(V)$ is really the correct object, and the challenge is to gain some combinatorial access to it.
Now we will see how this implies an “augmentations are sheaves” result. Let $V$ be the boundary of a Weinstein domain $W$.
Definition. ([N, Def. 3.23]): given an element $L \in DFuk(W)$, its null locus is the union of all open sets such $U$ such that for all Lagragian branes $M$ supported on a collection of Lagrangians in $U$, one has $Hom(M, L) = 0$. The singular support is the complement of the null locus.
It can be shown that the singular support at infinity is a (possibly singular) Legendrian; for such a Legendrian $\Lambda$ we write $DFuk_\Lambda(W)$ for the subcategory of the Fukaya category.
Proposition. There is a morphism $DFuk_\Lambda(W) \to DAug_\Lambda(V)$. Under some assumptions on $W$, certainly satisfied when $W$ is a cotangent bundle of an open manifold, this map is fully faithful.
Proof. It is known that an honest smooth Lagrangian determines an augmentation of the DGA of the knot it ends on, compatible with the relevant structures. We have formally extended both sides to the derived setting, and then restricted both sides to things of a given support. The compatibility of the last step follows from the discussion in [N, Sec 3.7] that singular support at infinity can be tested using fillings of small unknots.
Full faithfulness can be tested on generating objects; where we know it by classical Floer theoretic results (here is where I have to look up what assumptions on $W$ are needed.)
Theorem. When $V = J^1(X)$, there is a fully faithful morphism from augmentations to sheaves.
Proof. I discussed this earlier; basically it is like taking family Floer homology with a Legendrian in $V$ which bounds the cotangent fibre in $T^*(X \times \mathbb{R})$.
Remark. I am still slightly confused about “$Q$ issues” in the above statement.
Corollary. When $V = J^1(X)$ and $W = T^*(X \times \mathbb{R})$, there are equivalences of categories
$$DFuk_\Lambda(W) \xrightarrow{\sim} DAug_\Lambda(V) \xrightarrow{\sim} Shv_{\Lambda}(X \times \mathbb{R})$$
Proof. The two morphisms are fully faithful, and their composition is the Nadler-Zaslow functor, which is an equivalence.
Remark. If we knew the above conjecture, we could replace the central term by $Aug(\Lambda)$. Conversely, when $X = \mathbb{R}$, we know from [NRSSZ] that the composition $Aug(\Lambda) \to DAug_\Lambda(\mathbb{R}^3) \to Shv_{\Lambda}(\mathbb{R}^2)$ is an equivalence. Thus from the corollary we can conclude that the conjectured equivalence $Aug(\Lambda) \xrightarrow{\sim} DAug_\Lambda(\mathbb{R}^3)$ holds in this case.
Remark. It is possible that one can make a Floer theoretic argument to show that the image of $DFuk_\Lambda(W)$ already lands in $Aug(\Lambda)$ rather than $DAug_\Lambda(V)$. In this case, we could again conclude the conjecture in the cotangent bundle setting. (However I suspect the same argument could be used directly to show $Aug(\Lambda) \cong DAug_\Lambda(V)$.)