Let $X$ be a manifold.  I am going to sketch some axiomatics the big augmentation category $\mathcal{A}(X)$ of Legendrians in the jet space $J^1(X)$.  These will certainly be satisfied by the “straightforward generalization” of the category we defined in the “augmentations are sheaves” paper.  I’ll show such a category always has a morphism to the sheaf category, and reduce showing that it’s fully faithful to a single calculation, which moreover is easy to make in the augmentation category.

Ok, anyway: the axioms

1. There’s a map from Objects($\mathcal{A}(X)$) to Legendrians in $J^1(X)$.  I will write the legendrian underlying $a \in \mathcal{A}(X)$ as $\underline{a}$
2. Contact isotopies of $J^1(X)$ act on $\mathcal{A}(X)$.  (Saying this properly may involve some higher categorical suffering.) Note in particular that there is an action of Reeb flow $\phi_t$ on the category.
3. For any $t > 0$, there is a canonical natural transformation $r_t: id \to \phi_t$, satisfying obvious compatibilities.  (On objects: $r_t : a \to \phi_{t} a$.)

Note that (2) and (3) together imply that the induced morphism $r_\epsilon: Hom(a, b) \to Hom(a, \phi_{\epsilon} b)$ is an isomorphism for sufficiently small $\epsilon$ so long as $\underline{a}, \underline{b}$ are disjoint.  We will require this holds with no assumption:

4. The canonical morphism $Hom(a, b) \to Hom(a, \phi_{\epsilon} b)$ is an isomorphism for sufficiently small $\epsilon$.

The sheaf category does in fact satisfy these axioms; probably this is proven already in Kashiwara-Schapira and if not it’s proven in [STZ].

To continue, let me introduce a standard unknot $U_{x,z}$ for each point $(x, z) \in X \times \mathbb{R} = J^0(X)$.  It’s defined by drawing a really microscopic unknot at like $(x, -\infty)$ and then taking the top of it and pulling it up to $(x, z)$.  Actually the only thing that really matters about this unknot is that, suppose you had some other knot $\Lambda$, whose front projection at $x$ is a bunch of z-constant slices.  Then there is one Reeb chord from $\Lambda$ to $U_{x,z}$ for each slice of $\Lambda$ below $z$ (!!!)

(I mentioned this guy $U_{x,z}$ before in the comments to a previous post.)

Anyway, now suppose

5. There is a canonical object on a standard unknot.  I’ll denote it $u_{x,z}$.

Now there’s a morphism to the sheaf category: we send an object $a$ to the sheaf $sh(a)$ characterized by

$$sh(a)_{(x,z)} = Hom(u_{x,z}, a)$$

This is a functor, because Hom is.  It defines a constructible sheaf by axiom (2) above — the point is that you can wiggle around the unknot $U_{x,z}$ inducing isomorphisms on Hom spaces, so long as it doesn’t bump into $\underline{a}$ — i.e., $sh(a)$ is locally constant in the complement of the front projection of $\underline{a}$, and in fact microsupported on $\underline{a}$. (The microsupport calculation uses axiom (4) above!)

(It is kind of cool to think about what has to happen when you try and move $u_{(x,z)}$ along $\underline{a}$ past a cusp…)

Now consider the morphism $Hom(a,b) \to Hom(a, \phi_t(b))$.  Note that this morphism can, by axiom (2), change only at finitely many instants — exactly the times corresponding to Reeb chords from $\underline{b}$ to $\underline{a}$.

Note that since $sh$ is a functor, we get a map

$$sh: Hom(a, b) \to Hom(sh(a), sh(b))$$

The whole point here is to show that this is an isomorphism.  We proceed by induction on the number of Reeb chords.  When there are no Reeb chords at all, then $b$ can be flowed until it is arbitrarily far from $a$; perhaps we have to make an axiom saying that in this case the Hom is zero.  So let $R$ be the length of the shortest chord.  By induction, the morphism $sh: Hom(a, \phi_{R+\epsilon} b) \to Hom(sh(a), sh(\phi_{R+\epsilon} b))$ is an isomorphism; by the five lemma, (and because we are dg or a-infinity and cones are functorial) it is enough to study the induced morphism

$$sh: Cone \big(Hom(a, \phi_{R-\epsilon} b) \to Hom(a, \phi_{R+\epsilon} b) \big) \to Cone \big(Hom(sh(a), sh(\phi_{R-\epsilon} b)) \to Hom(sh(a), sh(\phi_{R+\epsilon} b)) \big)$$

but in the microlocal rank one case [ok I should add some axiom to demand microlocal rank one; or maybe more to the point, meditate on how to eliminate it], this is a morphism between one dimensional vector spaces!  I have not yet reduced the checking that this is an isomorphism to a tautology (probably doing this requires thinking carefully about what microlocal rank 1 really means), but anyway checking it in the case of the actual augmentation category is a count of a single disk.

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As an aside, note that the above construction realizes the self-hom spaces in the sheaf category (or in any such category) as an iterated cone of one-dimensional vector spaces, one for each Reeb chord.  This, plus homological perturbation theory, gives an $A_\infty$ structure on the direct sum of these, or in other words, a dga on the tensor algebra of the dual.

I still feel like I’m missing something in this story — where the auxilliary data used to make a choice of homological perturbation is hiding, visavis the name of an augmentation…