WWrapped sheaves
Christopher KuoFebruary 16, 2021
Abstract
We construct a sheaf-theoretic analogue of the wrapped Fukaya category in La-grangian Floer theory, by localizing a category of sheaves microsupported away fromsome given Λ ⊂ S ∗ M along continuation maps constructed using the Guillermou-Kashiwara-Schapira sheaf quantization.When Λ is a subanalytic singular isotropic, we also construct a comparisonmap to the category of compact objects in the category of unbounded sheavesmicrosupported in Λ, and show that it is an equivalence. The last statement canbe seen as a sheaf theoretical incarnation of the sheaf-Fukaya comparison theoremof Ganatra-Pardon-Shende. Contents ∞ -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Microlocal sheaf theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Constructible sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A Homogenous symplectic geometry and contact geometry 46B Quotients of small stable categories 48 a r X i v : . [ m a t h . S G ] F e b eferences 51 The microlocal sheaf theory of Kashiwara and Schapira [23] relates sheaves – which aretopological structures – on manifolds with the symplectic geometry of their cotangentbundles. The basic construction is the microsupport : to a sheaf F on a C manifold M ,one associates a conic closed subset SS( F ) in the cotangent bundle T ∗ M . The intersec-tion of this set with the zero section recovers the support supp( F ); the projectivizationSS ∞ ( F ) := (SS( F ) \ M ) / R > indicates the codirections along which the sheaf changes.A key indicator of the symplectic nature of the theory is the involutivity theorem [23,Thm. 6.5.4], which asserts that SS( F ) is always a singular coisotropic subset with respectto the canonical symplectic structure of T ∗ M . Under an appropriate tameness assump-tion, SS( F ) is a singular Lagrangian if and only if F is constructible, i.e., there exists astratification { X s } of M such that F | X α is a local system for all X s .Deeper relationships between microlocal sheaf theory and symplectic geometry be-gan to emerge in the mid 2000s. Nadler and Zaslow related constructible sheaves to‘infinitesimally wrapped’ Floer theory [32, 28]. Meanwhile, Tamarkin introduced purelysheaf theoretical methods into symplectic topology in his study of non-displaceability,a problem previously studied largely by Floer theoretic methods [39]. The subsequentGuillermou-Kashiwara-Schapira sheaf quantization of contact isotopies [18] — i.e. thehighly nonobvious statement that contact isotopies of S ∗ M act on sheaves on M — ledto a host of further incursions by sheaf theorists into symplectic topology [15, 16, 17, 37,36, 6, 35, 40, 4, 5, 20, 3] and vice versa [41, 42, 34]Meanwhile on the Floer theoretic side, Abouzaid and Seidel formulated a way to in-corporate contact dynamics into Fukaya categories for noncompact symplectic manifolds[2]. Their construction is roughly to localize a partially-defined infinitesimally wrappedFukaya category along ‘continuation morphisms’ associated to positive isotopies. Theresulting notion of wrapped Fukaya category (and its later ‘partially wrapped’ general-izations [38, 12, 11]) provides the correct mirrors to coherent sheaf categories on certainsingular spaces. While such wrapped categories are nontrivial to compute directly (thesimplest case was [1]), Nadler conjectured that they matched categories of compact ob-jects inside categories of unbounded sheaves with prescribed microsupport [29]. Thisconjecture was later established by the work of Ganatra, Pardon, and Shende [13]; as aresult, sheaf-theoretic methods (e.g. [29, 30, 9, 24, 10]) can be used to establish homo-logical mirror symmetry in these settings.While Nadler termed his category the ‘wrapped sheaves’, the name did not entirelyreflect its construction: there is no wrapping in their definition. In this paper we study thecategory which this name manifestly suits — the localization of the category of sheavesalong the continuation morphisms of [18] — and give an entirely sheaf-theoretic proofthat it is equivalent to Nadler’s category. Remark . Let us compare and contrast this article with the work [13]. In that article,the authors construct an equivalence between the partially wrapped Fukaya category ofa cotangent bundle, stopped along some subanalytic isotropic Λ, with the category ofcompact objects in the category of unbounded sheaves microsupported in Λ:Perf W ( T ∗ M, Λ) op ∼ = Sh Λ ( M ) c M = R . The purple arrow indicates thepositive Reeb direction and the red dots are the stop Λ. The green objects, from left toright, are a smooth curve conic at infinity in T ∗ R , the microsupport of a constant sheafon an open interval, and the microsupport of the same sheaf after being pushed to thestop.Their approach was to introduce an abstract axiomatic characterization (‘microlocalMorse theatre’) and verify that both sides satisfy it.Here is another approach. Begin from an equivalence of infinitesimally wrapped cate-gories as per [32, 28]. Now localize both sides along the continuation morphisms. Finally,show purely on the sheaf-theoretic side that the resulting localized category of sheavesis in fact equivalent to Sh Λ ( M ) c . See Figure 1 for an illustration. The present articleestablishes the last step in this argument.(While this route of proof would be logically independent of [12, 11, 13], the proof ofthe main theorem of the present article follows a strategy adapted from [13].)We turn to a more precise discussion of our results. Let us first discuss the continuationmaps in the sheaf theoretical setting. Recall that for a conic closed subset X ⊆ T ∗ M ,the inclusion ι ∗ : Sh X ( M ) (cid:44) → Sh( M ) of sheaves microsupported in X to all sheaves hasa left adjoint ι ∗ and right adjoint ι ! . Now consider F ∈ Sh( M × R ) as a one-parameterfamily of sheaves on M and set F a := F | M ×{ a } . An explicit description of ι ∗ given in[18, Proposition 4.8] shows when F satisfies the condition SS( F ) ⊆ T ∗ M × T ∗≤ R , thereis a continuation map F a → F b for a ≤ b . (See [39, 2.2.2] and [19, (77)] for the dualconstruction.) Now pick a contact form α on S ∗ M coorienting the contact structureinduced from the symplectic structure on T ∗ M . We say a C ∞ map Φ : S ∗ M × R → S ∗ M is an isotopy if the induced map φ t := Φ( • , t ) is a contactomorphism for all t ∈ R and φ = id S ∗ M . If Φ is a positive isotopy ( α ( ∂ t φ t ) ≥
0) then the corresponding GKS sheafkernel K (Φ) and hence its convolution K (Φ) ◦ F with F ∈ Sh( M ) will satisfy this conditionand hence admit continuation maps.Now fixed an open set Ω ⊆ S ∗ M . Homotopy classes of compactly supported isotopieswith fixed ends can be organized to an ∞ -category W (Ω) whose morphisms are given byconcatenating with positive isotopies. We refer this category as the category of positivewrappings . The discussion on continuation maps will imply that there is a wrappingkernel functor w : W (Ω) → Sh( M × M ) which sends isotopies to the end point of theGKS sheaf kernels and positive isotopies to continuation maps. One can use this functorto the define the infinite wrapping functors W ± (Ω) : Sh( M ) → Sh S ∗ M \ Ω ( M ) by sending F
3o the colimit colim Φ ( w (Φ) ◦ F ) or limit lim Φ ( w (Φ) ◦ F ) over Φ ∈ W (Ω). Geometrically,we push F with increasingly positive (resp. negative) isotopies and take colimit (resp.limit) over them. These functors give a geometric description for the adjoints of theinclusion ι ∗ : Sh S ∗ M \ Ω ( M ) (cid:44) → Sh( M ). Proposition 1.2.
Let ι ∗ : Sh S ∗ M \ Ω ( M ) (cid:44) → Sh( M ) denote the tautological inclusion.Then the functor W + (Ω) (resp. W − (Ω) ) is the left (resp. right) adjoint of ι ∗ . See subsection 3.2 for the proof.The main construction of this paper is the category of wrapped sheaves w sh Λ ( M )where M is a real analytic manifold and Λ is a closed subset in S ∗ M . It is a stablecategory defined by first collecting sheaves which have subanalytic singular isotropicmicrosupport away from Λ and those which are compactly isotopic to them in S ∗ M \ Λ,and then inverting continuation maps which come from positive isotopies satisfying similarconditions. One effect of this localization is that objects which can be connected throughan isotopy on S ∗ M \ Λ will be identified. We show that Hom-complexes in w sh Λ ( M ) canbe computed as colimits of Hom-complexes between ordinary sheaves over W ( S ∗ M \ Λ).Finally, when Λ is a subanalytic singular isotropic, by using the infinite wrapping functor W + ( S ∗ M \ Λ), we define a canonical comparison functor W +Λ ( M ) : w sh Λ ( M ) → Sh Λ ( M ) c .The main theorem of this paper is that W +Λ ( M ) is an equivalence. Theorem 1.3.
Let Λ ⊆ S ∗ M be a subanalytic singular isotropic. The comparison functor W +Λ ( M ) : w sh Λ ( M ) → Sh Λ ( M ) c is an equivalence. See subsection 5.3 for the proof.
Remark . Note that, unlike the analogous isomorphism in [13], our isomorphism isinduced by an explicit functor.Since all the above constructions are functorial on the inclusion of open sets of M , weobtain a precosheaf w sh Λ and we refer its objects as the wrapped sheaves . The corollaryof the above theorem is that this precosheaf is a cosheaf. Corollary 1.5.
Let Λ ⊆ S ∗ M be a subanalytic singular isotropic. The comparison mor-phism W +Λ : w sh Λ → Sh c Λ between precosheaves is an isomorphism. In particular, theprecosheaf w sh Λ is a cosheaf. The proof of Theorem 1.3 follows the same strategy as [13]. In short, subanalyticgeometry implies that, for a subanalytic singular isotropic Λ, there exists a C Whitneytriangulation S such that Λ is contained in N ∗∞ S := ∪ s ∈ S N ∗∞ X s . For this special case,the two categories are natural identified as Perf S , the category of perfect S -modules,and hence admit a preferred set of generators which are matched under W + N ∗∞ S ( M ). Wethen apply the nearby cycle technology developed in [31] to conclude that W + N ∗∞ S ( M )induces an equivalence on the Hom-complexes for these generators and hence finishedthe proof for this case. To conclude the theorem for the general case, we note thatthe construction on both sides are contravariant on Λ. Thus we study the fiber of thecanonical maps w sh N ∗∞ S ( M ) → w sh Λ ( M ) and Sh N ∗∞ S ( M ) c → Sh Λ ( M ) c , and show thatthey are generated by a sheaf-theoretical version of the linking disks and microstalks atthe smooth points of N ∗∞ S \ Λ. Finally, we show that W +Λ ( M ) matches those objects andthus conclude the general case. Convention.
Let Y n − be a contact manifold. We say a set f is a singular isotropicif there is a decomposition f = f subcrit ∪ f crit ⊆ Y for which f subcrit is closed and is containedin the smooth image of a second countable manifold of dimension < n −
1, and f crit is aLegendrian submanifold. 4 .1 Acknowledgements I would like to thank my advisor Vivek Shende for pointing out the main ideas of theconstructions performed in this paper as well as helpful guidance. I would also like tothank Germ´an Stefanich for helpful discussions concerning higher category theory. Thisproject was supported by NSF CAREER DMS-1654545.
We recall the categorical settings and sheaf theoretical tools we will be using in thispaper. Experts can skip this section. ∞ -categories We work in the higher categorical setting developed in [25] and [26]. The main advantagesfor this choice is that for the categories discussed in this paper, there is an abundance oflimits and colimits (in an appropriate sense) in this setting. As a result, many construc-tions and arguments can be performed formally as universal constructions which greatlysimplifies the situation. Because of the higher categorical nature of this paper, we willrefer an ∞ -category C simply as a category and when we need to emphasis that it is inparticular an ordinary category, we will refer it a 1-category.Recall that a presentable category is a category with certain cardinality assumptions.Roughly speaking, such categories are large enough to contain (small) colimits but iscontrolled by a small category. A main consequence of these assumptions is that theadjoint functor theorem holds. In addition, up to set-theoretic issues, the totality of suchcategories form a category itself which has nice properties concerning limits and colimits.We won’t consider the whole collection of such categories but a small portion of it whichsatisfies stronger finiteness conditions which we now recall. Definition 2.1.
Let C be a category. An object c ∈ C is compact if Hom( c, • ) preserve(small) filtered colimit. That is for any (small) filtered index category I and any functor X : I → C , the canonical morphismlim −→ I Hom( c, X i ) → Hom( c, lim −→ I X i )is an isomorphism. Here, we use the notation lim −→ I instead of colim I to emphasis the indexcategory I is filtered. Definition 2.2.
A category C is compactly generated if there exists a small subcategory C ⊆ C consisting of compact objects such that C is generated by C under filteredcolimits. That is, Ind( C ) ∼ = C where Ind denotes the ind-completion. Definition 2.3.
Let Cat denote the (very large) category of categories. We use Pr L ω todenote the (non-full) subcategory of Cat whose objects are compactly generated categoriesand morphisms are functors which preserve small colimits and compact objects. We alsouse denote cat ω the subcategory of Cat consisting of idempotent complete small categorieswhich admit finite colimits whose morphisms are functors preserving finite colimits.5 roposition 2.4. The funtor
Ind : cat ω → Pr L ω taking C to Ind( C ) , its ind-completion,is an equivalence whose inverse is given by the functor θ : Pr L ω → cat ω sending C to C c ,the subcategory of C consisting compact objects. Proposition 2.5.
The category Pr L ω and hence cat ω admits small colimits, which canbe computed in Cat as limits by passing to right adjoints. Here we use the fact that amorphism F : C → D in Pr L ω is a left adjoint since it preserves colimits. Classically, one use the theory of triangulated categories to encode homological infor-mation of sheaves on topological spaces. These categories are 1-categories with structuresand can be used to remember a (small) portion of homotopies. However, limits and col-imits are scarce in this setting. For example, the non-zero morphism e : Z / → Z / D ( Z ) does not have a kernel. Hence, we use the theory of stable categories instead. Definition 2.6.
A category C is pointed if there exists a zero object 0, i.e., an objectwhich is both initial and final.A sequence X → Y → Z in a pointed category C is a fiber (resp. cofiber ) sequence ifthe diagram X Y Z is a pullback/pushforward. In this case, we say X (resp. Z ) is the fiber (resp. cofiber )of the corresponding morphism. Definition 2.7.
A pointed category C is stable if fibers and cofibers exist and a diagramas above is a fiber sequence if and only if it is a cofiber sequence. We say a functor F : C → D between stable categories is exact if F preserves finite limits and finitecolimits. Note in a stable category C finite limits are the same as finite colimits sopreserving one kind means preserving the other. Example 2.8.
A stable category C admits a “ shifting by 1” automorphism [1] : C → C which can be defined by X (cid:55)→ cof( X → − −
1] : C → C which can defined by X (cid:55)→ fib(0 → X ). Example 2.9.
Let C be a stable category. For X , Y ∈ C , the direct sum X ⊕ Y in C can be computed as cof( Y [ − −→ X ) = fib( Y −→ X [1]).We will use the following lemma: Lemma 2.10.
Let C be a stable category and X , Y , Z , X (cid:48) , Y (cid:48) , and Z (cid:48) ∈ C . Assume wehave the following commutative diagram X Y ZX (cid:48) Y (cid:48) Z (cid:48) α β γ uch that each row is a fiber sequence. Let X (cid:48)(cid:48) = cof( α ) , Y (cid:48)(cid:48) = cof( β ) and Z (cid:48)(cid:48) = cof( γ ) be the corresponding cofibers of the vertical maps. Then there exist a canonical fibersequence X (cid:48)(cid:48) → Y (cid:48)(cid:48) → Z (cid:48)(cid:48) . In particular, for f : X → X and g : X → X , we have afiber sequence cof( f ) → cof( f ◦ f ) → cof( f ) . This special case is usually referred asthe octahedral axiom in the triangulated category setting.Proof. The goal is to show that Z (cid:48)(cid:48) is the cofiber of ( X (cid:48)(cid:48) → Y (cid:48)(cid:48) ). Recall that cofibersare computed as a colimit of the diagram I = [ · ← · → · ]. For example, the object Z is computed as the colimit given by the diagram 0 ← X → Y . Now we consider thefollowing diagram. 0 Y Y (cid:48) X X (cid:48) ← Z → Z (cid:48) and taking the colimit againgives Z (cid:48)(cid:48) . Similarly, taking first the horizontal arrows and then the vertical arrows givescof( X (cid:48)(cid:48) → Y (cid:48)(cid:48) ). But colimits commute with each other and thus Z (cid:48)(cid:48) = cof( X (cid:48)(cid:48) → Y (cid:48)(cid:48) ).Now for the special case, we apply the above result to the commutative diagram X X X X f f ◦ f f Example 2.11.
Consider two short exact sequences in a Grothendieck abelian 1-category A and compatible maps between them.0 X Y Z X (cid:48) Y (cid:48) Z (cid:48) α β γ Recall that A naturally embeds into a stable category D ( A ). An application of the abovelemma implies a special case of the snake lemma in classical homological algebra.We denote Pr L ω,st and st ω the subcategory of Pr L ω and cat ω which consists of stablecategories. The property of being stable is compatible with the finiteness condition wediscussed earlier. In particular, the Ind-completion Ind( C ) of a (small) stable category C is stable . Similarly, the subcategory of compact objects C c of a compactly generatedstable category C is stable. 7 roposition 2.12. The equivalence
Ind : cat ω (cid:28) Pr L ω : θ restricts to the subcategoriesconsisting of stable categories Ind : st ω (cid:28) Pr L ω,st : θ. We recall some definitions and results of the microlocal sheaf theory developed in [23]with the above setting. Let Z -Mod denote the presentable stable category of modulesover Z . It can be modified as the dg category of (possibly unbounded) chain complexesof abelian groups with quasi-isomorphisms inverted. See for example [7]. This categoryis compactly generated and one usually denote the compact objects ( Z -Mod) c by Perf Z .When representing Z -Mod by chain complexes, Perf Z consists of objects which are quasi-isomorphic to bounded chain complexes consisting of finite rank abelian groups.For a topological space X , the category of presheaves (with integer coefficient) is thecategory PSh( X ) = Fun( Op opX , Z -Mod) of contravariant functor from the 1-category ofopen sets in X to Z -Mod. The Z -linear presentable stable category of sheaves Sh( X ) isthe reflexive subcategory of PSh( X ) consisting of those presheaves F which turn colimitsin Op X to limits in Z -Mod. In more concrete terms, F is a sheaf if for any open cover U ofan open set U ⊆ X , the canonical map F ( U ) ∼ −→ lim U I ∈ C ( U ) F ( U I ) is an isomorphism, i.e.,the sections on U can be computed as the totalization of the sections of the correspondingˇCech nerve C ( U ). Recall that reflexive means the inclusion Sh( X ) (cid:44) → PSh( X ) admits aleft adjoint by sheafification ( • ) † . Thus, the inclusion is limit-preserving and colimits canbe computed as the sheafification of the colimits in PSh( X ). Since PSh( X ) admits limitsand colimits, so does Sh( X ).Now we recall the six-functor formalism and the interaction among them. First, thereis a symmetric monoidal structure (Sh( X ) , ⊗ ) on Sh( X ) induced from ( Z -Mod , ⊗ ). Theunit of this tensor product is the sheaf Z X which is the sheafification of the presheaf( U (cid:55)→ Z ) whose restrictions are given by the identity id Z . For a fixed sheaf F , thefunctor ( • ) ⊗ F given by tensoring with F has a right adjoint H om( F, • ) which gives theinternal Hom of Sh( X ). The global section of this sheaf H om is the Hom-complex, i.e.,Γ( X ; H om( G, F )) = Hom(
G, F ) ∈ Z -Mod for any F, G ∈ Sh( X ).Let f : X → Y be a continuous map. There is a pushforward functor f ∗ : Sh( X ) → Sh( Y ) induced by pulling back open sets f − : Op Y → Op X , V (cid:55)→ f − ( V ). This functoradmits a left adjoint f ∗ : Sh( Y ) → Sh( X ) and the adjunction ( f ∗ , f ∗ ) is usually referredas the star pullback/pushforward . When X and Y are both locally compact Hausdorffspaces, there is another pair of adjunction ( f ! , f ! ) such that f ! : Sh( X ) → Sh( Y ) and f ! :Sh( Y ) → Sh( X ). This adjunction is usually referred as the shriek pullback/pushforward . Example 2.13.
When i : Z (cid:44) → X is a subset of X , one usually use F | Z to denote i ∗ F for F ∈ Sh( X ) and call it the restriction of F on Z . Example 2.14.
Consider a closed set i : Z (cid:44) → X and and an open set j : U (cid:44) → X . Inthese cases, we have i ∗ = i ! and j ∗ = j ! . In addition, the functors i ∗ , j ∗ , j ! are fullyfaithful with the corresponding adjoints being a left inverse. Example 2.15.
Let x ∈ X be a point and denote i x : { x } (cid:44) → X the inclusion of thepoint. For F ∈ Sh( X ), we call the Z -module F x := i ∗ x F the stalk of F at x . An keyproperty of the stalks is that a morphism G → F is an isomorphism if and only if itinduces isomorphism G x → F x on the stalk for all x ∈ X .8ow fixed a topological space X . We see that taking integer coefficient sheaves itselfforms a presheaf Sh in Pr R st : For an open set U ⊆ X , we assign the category Sh( U ). For aninclusion of open sets i U,V : U (cid:44) → V , we assign the pullback functor i ∗ U,V : Sh( V ) → Sh( U ). Proposition 2.16.
The presheaf
Sh : Op X → Pr R st is a sheaf.Proof. Let U be an open set of X and U an open cover of U . The functorlim U I ∈ C ( U ) i ∗ U I ,U : Sh( U ) → lim U I ∈ C ( U ) Sh( U I )is an equivalence and has an inverse lim U I ∈ C ( U ) Sh( U I ) → Sh( U ) which is given by( F U I ) U I ∈ C ( U ) (cid:55)→ colim U I ∈ C ( U ) ( i U I ,U ) ! F U I . We mention some computational tools for integer coefficient sheaves. First, the cate-gories of sheaves has a structure of base change . (See [21] for the exact statement in thehigher categorical setting.)
Theorem 2.17.
Consider a pullback diagram of locally compact Hausdorff spaces, X (cid:48) Y (cid:48) X Yff (cid:48) g (cid:48) g . There is an equivalence g ∗ f ! ∼ −→ f (cid:48) ! g (cid:48)∗ . The push/pull functors satisfy some compatibility property with ⊗ and H om. Welist a few which we will use: Proposition 2.18.
Let f : X → Y be a continuous map between locally compact Haus-dorff spaces. Then:1. f ∗ ( F ⊗ G ) ∼ −→ f ∗ F ⊗ f ∗ G , for F , G ∈ Sh( Y ) ,2. ( f ! G ) ⊗ F ∼ −→ f ! ( G ⊗ f ∗ F ) , for F , G ∈ Sh( Y ) ,3. f ! H om( G, F ) ∼ −→ H om( f ∗ F, f ! F ) , for F , G ∈ Sh( Y ) . We recall the excision fiber sequences. Let X be a locally compact Hausdorff space, i : Z (cid:44) → X be a close set, and j : U = X \ Z (cid:44) → X be its open complement, then j ∗ i ∗ = 0and there are fiber sequences j ! j ! F → F → i ∗ i ∗ F, i ! i ! F → F → j ∗ j ∗ F where the arrows are the units/counits of the shriek/star adjunction pairs. Such a triple(Sh( X ) , Sh( Z ) , Sh( U )) is usually referred as a recollement in homological algebra. When9 is locally closed, one denotes F Z = i ! i ∗ F and Γ Z ( F ) = i ∗ i ! F for F ∈ Sh( X ). Thus, onecan write the above fiber sequences as F U → F → F Z , Γ Z ( F ) → F → Γ U ( F ) . Let a X : X → {∗} denote the projection to a point. We use a X to denote thepullback a ∗ X A of A ∈ Sh( {∗} ) = Z -Mod. When X is a manifold and A is an abeliangroup regarded as a chain complex concentrated as 0, a standard representative of A X is the singular cochains ( U (cid:55)→ C ∗ ( U ; A )) with A coefficient. When Z ⊆ X is a locallyclosed subset of X , we abuse the notation and write A Z for both the sheaf in Sh( Z ) orits shriek pushforward. Definition 2.19.
We call A Z the constant sheaf on Z with stalk A . In general, we saya sheaf F is a locally constant sheaf or local system if there exists an open cover U suchthat F | U is constant for U ∈ U and we use Loc( X ) denote the subcategory spanned bysuch sheaves. Example 2.20.
Let i : Z (cid:44) → X be a local closed subset and F ∈ Sh( X ). By the aboveformulae in Proposition 2.18, F ⊗ Z Z = F ⊗ i ! i ∗ Z X = i ! ( i ∗ F ⊗ i ∗ Z X ) = i ! i ∗ ( F ⊗ Z X ) = F Z .We will consider sheaves on Sh( M × B ) for some manifolds M and B , and regardthem as B -family of sheaves on Sh( M ). Let p : M × B → M be the projection Thefollowing general consideration provides a criterion to determine when such a B -family isconstant: Let f : X → Y be a continuous map between locally compact Hausdorff space.Denote Sh f ( X ) the subcategory of Sh( X ) consists of objects F satisfying the condition F | f − ( y ) ∈ Loc( f − ( y )) for all y ∈ Y . We note that f ∗ G ∈ Sh f ( X ) for G ∈ Sh( Y ). Proposition 2.21.
Assume there is an increase sequence of closed subsets { X n } suchthat X n ⊆ Int( X n +1 ) , X = ∪ n X n , and f n := f | X n is proper with contractible fibers. Then,the adjunction f ∗ : Sh f ( X ) (cid:28) Sh( Y ) : f ∗ is an equivalence of categories. Corollary 2.22.
Let M , B be manifolds and assume B is contractible. Let p : M × B → M denote the projection. Then F ∈ Sh( M × B ) is of the form p ∗ G for some G ∈ Sh( M ) if and only if F | { x }× B is locally constant for all x ∈ M . In this case, G = p ∗ F . We also consider set-theoretic invariants associated to sheaves.
Definition 2.23.
Let X be a topological space and F ∈ Sh( X ). The support of a sheaf F is defined to be the closed subsetsupp( F ) = { x ∈ X | F x (cid:54) = 0 } . Example 2.24.
Let i : Z ⊆ X be a closed subset. The pushforward i ∗ identifies Sh( Z )as the subcategory of Sh( X ) consisting of sheaves F whose support supp( F ) ⊆ Z iscontained in Z .Now let M be a C α -manifold where α ∈ Z > ∪ {∞ , ω } . The term ‘microlocal’ usuallyrefers to ‘local’ in the cotangent bundle T ∗ M . One central lemma in microlocal sheaftheory is the noncharacteristic deformation lemma: Lemma 2.25 ( [23, Proposition 2.7.2] , [33, Theorem 4.1] ) . Let X be a Hausdorff space, F ∈ Sh( X ) . Let { U s } s ∈ R be a family of open subsets of X . We assume a) for all t ∈ R , U t = (cid:83) s
The microsupport of a sheaf F is defined to be the closure of the locusof differentials of the C functions at their cohomological F -critical points. That is,SS( F ) = (cid:91) φ ∈ C ( M ) { ( x, ξ ) | t ∈ R , ( i ∗ φ,t F ) m (cid:54) = 0 , ξ = dφ x } . Although the microsupport is defined as a C -invariant, it’s sufficient to check asmaller class of functions. Proposition 2.27.
The microsupport of a sheaf F is the same as the closure of the locusof differentials of the C α functions at their cohomological F -critical points. That is, SS( F ) = (cid:91) φ ∈ C α ( M ) { ( x, ξ ) | t ∈ R , ( i ∗ φ,t F ) m (cid:54) = 0 , ξ = dφ x } . It is straightforward to see the microsupport is conic and closed, and its intersectionwith the zero section SS( F ) ∩ M = supp( F ) recovers the support. The involutivitytheorem [23, Theorem 6.5.4] of Kashiwara and Schapira staes that SS( F ) is always asingular coisotropic. Since SS( F ) is conic, it can be recovered from supp( F ) and theprojectivization SS ∞ ( F ) := (SS( F ) \ M ) / R > . Definition 2.28.
For a conic closed subset X ⊆ T ∗ M , we use Sh X ( M ) to denote thesubcategory of sheaves consisting of F such that SS( F ) ⊆ X . Similarly, for a closedsubset X ⊆ S ∗ M , we use Sh X ( M ) to denote the subcategory of sheaves consisting of F such that SS ∞ ( F ) ⊆ X . Note the latter case is a special case of the former. For if X ⊆ S ∗ M is closed, Sh X ( M ) = Sh ( R > X ∪ M ) ( M ). Example 2.29.
Let M be a manifold. Being a local system is a microlocal condition.More precisely, Loc( M ) = Sh M ( M ). Example 2.30.
Let M = R n and γ be a closed convex cone with vertex at 0. One hasSS( Z γ ) ∩ T ∗ R n = γ ◦ where γ ◦ := { ξ ∈ ( R n ) ∗ | ξ ( v ) ≥ , v ∈ γ } is the dual cone. As acorollary, if M (cid:48) ⊆ M is a closed submanifold, then SS( Z M (cid:48) ) = N ∗ M (cid:48) is the normal bundleof M (cid:48) . 11ne might want to assign an invariant similar to stalks for points in ( x, ξ ) ∈ S ∗ M . Ingeneral, the object ( i ∗ φ,t F ) m depends on φ and is not an invariant associated to the point( x, ξ ). However, the situation is better when transversality condition is satisfied. Definition 2.31.
Fix a singular isotropic Λ ⊆ S ∗ M . Let f be a function defined onsome open set U of M . We say a point x ∈ U is a Λ-critical point of f if the graph of itsdifferential Γ df intersect R > Λ ∪ M at ( x, df x ). A Λ-critical point x is Morse if ( x, df x )is a smooth point of R > Λ ∪ M and the intersection Γ df ∩ Λ is transverse at ( x, df x ). Afunction f is Λ-Morse if all its Λ-critical point is Morse. Proposition 2.32 ([23, Proposition 7.5.3]) . Let Λ be a singular isotropic. Assume φ is Λ -Morse at a smooth point ( x, ξ ) ∈ Λ . For F ∈ Sh( X ) such that SS ∞ ( F ) ⊆ Λ in aneighborhood of ( x, ξ ) , the object ( i ∗ φ,t F ) m ∈ Z -Mod is, up to a shift, independent of φ . Definition 2.33.
Let Λ ⊆ S ∗ M be a singular isotropic and ( x, ξ ) ∈ Λ a smooth point.For F ∈ Sh Λ ( M ), we call functors µ ( x,ξ ) : Sh Λ ( M ) → Z -Mod of the form µ ( x,ξ ) F := ( i ∗ φ,t F ) m a microstalk functor where φ is any function satisfying the assumption in the last Propo-sition. Since this functor is well-defined up to a shift, we will abuse notation and call µ ( x,ξ ) F the microstalk of F at ( x, ξ ).Let X ⊆ T ∗ M be conic and closed, and let m ∈ Λ ⊆ ( T ∗ M \ X ) be a closed singu-lar isotropic. The definition of microsupport requires one to check ( i ∗ φ,t F ) m for generalfunctions φ . However, it’s sufficient to check the Morse ones when SS( F ) is Lagrangian. Proposition 2.34 ([13, Proposition 4.9]) . Let X and Λ be as above. Then Sh X ( M ) ⊆ Sh X ∪ Λ ( M ) is the fiber of all microstalk functors µ ( x,ξ ) for smooth Lagrangian points ( x, ξ ) ∈ Λ . In practice, it’s hard to compute the microsupport of a sheaf directly and it’s usuallysufficient to deduce desired conclusions by having an upper bound. Here we collect somestandard results for microsupport estimation.Let f : X → Y be a map between manifolds, we use the following notations T ∗ X X × Y T ∗ Y T ∗ YX Y (cid:3) df ∗ f π fπ X π Y where the square on the right is the pullback of the cotangent bundle T ∗ Y of Y along f and df ∗ is given fiberwisely by the adjoint of the differential df x : T x X → T f ( x ) Y . Let T ∗ X Y denote the set { ( x, α ) ∈ X × Y T ∗ Y | df ∗ x α = α ◦ df x = 0 } . Definition 2.35.
Let A be a conic closed subset of T ∗ Y . We say f is noncharacteristic for A if f − π ( A ) ∩ T ∗ X Y ⊆ X × Y Y . For a sheaf F ∈ Sh( X ), we say f is noncharacteristic for F if it’s the case for SS( F ).12 roposition 2.36. We have the following results:1. If F → G → H is a fiber sequence in Sh( X ) , then (SS( F ) \ SS( H )) ∪ (SS( H ) \ SS( F )) ⊆ SS( G ) ⊂ SS( F ) ∪ SS( H ) . This is commonly referred as the microlocal triangular inequalities.2. For F ∈ Sh( X ) , G ∈ Sh( Y ) , SS( F (cid:2) G ) ⊆ SS( F ) × SS( G ) .3. For f : X → Y and F ∈ Sh( X ) , if f is proper on supp( F ) , then SS( f ∗ F ) ⊆ f π (cid:0) ( df ∗ ) − SS( F ) (cid:1) .
4. For f : X → Y and F ∈ Sh( Y ) , if f is noncharacteristic for F , then SS( f ∗ F ) ⊆ df ∗ ( f − π (SS( F ))) and the natural map f ∗ F ⊗ f ! Z Y → f ! F is an isomorphism. If f is furthermoresmooth, the estimation is an equality.5. Let Z ⊆ X be closed. If SS( F ) ∩ N ∗ out ( Z ) ⊆ X , then SS( F Z ) ⊆ N ∗ in ( Z ) + SS( F ) . Similarly, let U ⊆ X be open. If SS( F ) ∩ N ∗ in ( U ) ⊆ X , then SS( j ! j ∗ F ) ⊆ N ∗ out ( U ) + SS( F ) .
6. For F and G ∈ Sh( X ) . If SS( F ) ∩ − SS( G ) ⊆ X , then SS( F ⊗ G ) ⊆ SS( F ) + SS( G ) .
7. For F and G ∈ Sh( X ) . If SS( F ) ∩ SS( G ) ⊆ X , then SS( H om( G, F )) ⊆ SS( F ) − SS( G ) . If moreover G is cohomological constructible, then the natural map H om( G, Z X ) ⊗ F → H om( G, F ) is an isomorphism. If furthermore F = ω M , i.e., when H om( G, F ) = D M ( G ) isthe Verdier dual, then SS( D M ( G )) = − SS( G ) . Sometimes, when the noncharacteristic condition is absent, there is still less refinedupper bounds for pullbacks.
Definition 2.37.
We define two constructions of closed conic subsets of cotangent bun-dles:1. Given closed conic subset
A, B ⊆ T ∗ X , we define A ˆ+ B to be the closed subsetconsisting of points ( x, ξ ) ∈ T ∗ X such that, in some local coordinate, there existsequence { ( x n , ξ n ) } in A and { ( y n , η n ) } in B such that x n , y n → x , ξ n + η n → ξ ,and | x n − y n || x n | →
0. 13. Let i : M (cid:44) → X be a closed submanifold and choose a local coordinate ( x, y, ξ, η )of T ∗ X such that M is given by { y = 0 } . Given a closed conic subset A ⊆ T ∗ X ,we define i ( A ) to be the closed subset of T ∗ M consisting of points ( x, ξ ) suchthat there exists { ( x n , y n , ξ n , η n ) } in A such that y n → x n → x , ξ n → ξ , and | y n || η n | → f : X → Y and closed conic A ⊆ T ∗ Y , f ( A ) can be defined as aspecial case of a more general construction which aslo includes A ˆ+ B as a special case.Moreover, the definition can be made free of choice of local coordinates using the techniqueof deformation to the normal cone . Also, notice that the above construction A ˆ+ B and i ( A ) contain A + B and di ∗ ( i − π (SS( F ))) as closed subsets. It can be shown that theyare equal when the noncharacteristic condition is satisfied. Proposition 2.38.
We have the following results:1. Let j : U (cid:44) → X be open and F ∈ Sh( U ) , then SS( j ∗ F ) ⊆ SS( F ) ˆ+ N ∗ in U.
2. Let i : M → X be a closed submanifold and F ∈ Sh( X ) , then SS( i ∗ F ) ⊆ i SS( F ) . By Example 2.29, Corollary 2.22, and (2) of Proposition 2.38, we conclude:
Lemma 2.39.
Let B be a contractible manifold and p : M × B → M be the projection.A sheaf F ∈ Sh ( M ) satisfies p ∗ p ∗ F ∼ −→ F if and only if SS( F ) ⊆ T ∗ M × B . We will use the notion of convolution . Let M i , i = 1 , , i < j , weuse M ij := M i × M j to denote the product and M similarly. Denote also p ij : M → M ij the projection. For A ⊆ T ∗ M and B ⊆ T ∗ M , we set A ◦ B = { ( x, ξ, z, ζ ) ∈ T ∗ M |∃ ( y, η ) , ( x, ξ, y, η ) ∈ A, ( y, η, z, ζ ) ∈ − B } . Note when A and B are Lagrangian correspondences, A ◦ B is the composite Lagrangiancorrespondence twisted by a minus sign on the second component. Write q ij : T ∗ M → T ∗ M ij to be the projection on the level of cotangent bundles. Then A ◦ B = q ( q − ( − B ) ∩ q − A ). Definition 2.40.
For F ∈ Sh( M ), G ∈ Sh( M ), we define the convolution to be F ◦ G := p ( p ∗ G ⊗ p ∗ F ) . Assume1. p is proper on M × supp( G ) ∩ supp( F ) × M ;2. 0 M × SS( G ) ∩ SS( F ) × M ∩ M × T ∗ M × M ⊆ M . F ◦ G ) ⊆ SS( F ) ◦ SS( G ) . More generally, let B be a manifold viewed as a parameter space. Regard F ∈ Sh( M × B ), G ∈ Sh( M × B ) as B -family of sheaves, one can similarly define the relativeconvolution F ◦ | B G ∈ Sh( M × B ). One noticeable difference for the microsupportestimation is that instead of T ∗ M ij and T ∗ M one has to consider T ∗ M ij × T ∗ B and T ∗ M × ( T ∗ B × B T ∗ B ) instead. Here × B is taken over the diagonal B (cid:44) → B × B . Alsothe projection r ij : T ∗ M × ( T ∗ B × B T ∗ B ) → T ∗ M ij × T ∗ B for the B -component is nowgiven by the first projection (with a minus sign) for ij = 12, the addition for ij = 13,and the second projection ij = 23. Otherwise the microsupport estimation is similar tothe ordinary case. The theory of constructible sheaves is based on the results of stratified spaces. Standardreference for stratified spaces are [14] and [27].A stratification S of X is a decomposition of X into to a disjoint union of locallyclosed subset { X s } s ∈ S . A set Y ⊆ X is said to be S -constructible if it is a union of stratain S . We assume, without further mention, that a stratification should be locally finiteand satisfies the frontier condition that X s \ X s is a disjoint union of strata in S . Inthis case, there is an ordering which is defined by s ≤ t if and only if X t ⊆ X s . Wealways implicitly chose this ordering when considering S as a poset. For example, we willconsider its linearization S -Mod := PSh( S op , Z -Mod). Definition 2.41.
A sheaf F is said to be S -constructible if F | X s is a local system for all s ∈ S . We denote the subcategory of Sh( X ) consisting of such sheaves by Sh S ( X ). Asheaf F is said to be constructible if F is S -constructible for some stratification S .For s ∈ S , we denote star( s ) the smallest S -constructible open set containing X s .Alternatively, star( s ) = (cid:96) t ≤ s X t . The poset S can be then identified with the subposet { star( s ) | s ∈ S } of Op X . Hence, there is a functor S -Mod (cid:44) → Sh S ( X ) induced by therestriction. A stratification is called a triangulation if X = | K | is a realization of somesimplicial complex K and S := {| σ || σ ∈ K } is given by the simplexes of K . We note thatwhen S is a triangulation, the functor S -Mod (cid:44) → Sh S ( X ) is an equivalence since each X s is contractible.Now let X be a C α manifold. We consider regularity conditions of a stratification. Definition 2.42.
Let M and N be locally closed C manifolds of R n , with N ⊂ M \ M .Consider sequences x n ∈ M and y n ∈ N such that x n , y n → y ∈ N and { T x n M } convergesto τ (in the corresponding Grassmannian). Assume also { R ( x n , y n ) } converges to l . Wesay the pair ( N, M ) satisfies the
Whitney condition if any such sequence satiesfies τ ⊇ l .For general pairs of C submanifold manifolds of a C α manifold X , we say the pair ( N, M )satisfies the Whitney condition if the above condition is satisfies on local charts.
Remark . The Whitney condition can also be formulated without working on co-ordinates. Recall the normal bundle of the diagonal ∆ X (cid:44) → X × X can be identi-fied as the tangent bundle T X of X . The (real) blow-up Bl ∆ X ( X × X ) can be seenas a disjoint union P ( T X ) (cid:113) ( X × X \ ∆ X ) of the projective tangent bundle and the15ff-diagonal. Then we say ( N, M ) satisfies the Whitney condition if for any sequence( y n , x n ) ∈ N × M ⊆ Bl ∆ X ( X × X ) \ ∆ X such that T x n M → τ and ( y n , x n ) → l ∈ P ( T X ),we have l ⊆ τ .We say a stratification S is C k if each X s is C k locally closed manifold. A C k stratifi-cation S is a Whitney stratification if ( X t , X s ) satisfies the Whitney condition for s ≤ t .Let N ∗ S denote the union ∪ α ∈ S N ∗ X α of the conormals of the strata. The set N ∗ S isa singular conic Lagrangian in T ∗ M and the Whitney condition implies a weaker prop-erty that N ∗ S ⊆ T ∗ X is closed. We also use N ∗∞ S to denote the corresponding singularisotropic at the infinity. The main advantage of considering Whitney stratifications arethe following proposition. Proposition 2.44 ([13, Proposition 4.8]) . Let S be a C stratification. A sheaf F whichsatisfies SS ∞ ( F ) ⊆ N ∗∞ S is S -contructible, i.e., there exists an inclusion Sh N ∗∞ S ( M ) (cid:44) → Sh S ( M ) . When S is Whitney, the inclusion Sh N ∗∞ S ( M ) (cid:44) → Sh S ( M ) is an equivalence. The proposition is a corollary of the existence of inward cornerings defined by applyingthe following lemma which is also proved in the same paper.
Lemma 2.45 ([13, Proposition 2.3]) . Fix any ≤ p ≤ ∞ , and let S be a C p Whitneystratification of M . Fix a relatively compact S -constructible set Y . Let S Y := { s | X s ⊆ Y } denote the collection of strata consisting of Y and set N ∗ S Y := ∪ α ∈ S Y N ∗ X α , which isclosed in T ∗ X by the Whitney condition. Then there exists a decreasing family Y (cid:15) ofneighborhoods of Y such that as (cid:15) → ,1. N ∗ Y (cid:15) becomes contained in arbitrary small conic neighborhood of N ∗ Y (cid:15) ,2. N ∗ Y (cid:15) ∩ N ∗ S = ∅ . Definition 2.46.
For a relative compact S -constructible open set U , an inward cornering of U is a open set of the form U − (cid:15) := U \ ( ∂U ) (cid:15) . When (cid:15) > U − (cid:15) is a codimension 0 open submanifold whoseclosure U − (cid:15) is a compact manifold with corners. The family U − (cid:15) depends smoothly on (cid:15) .Its outward conormal N ∗∞ ,out U − (cid:15) remains disjoint from N ∗∞ S as (cid:15) changes, and convergesto N ∗∞ S uniformly as (cid:15) → N ∗∞ S for some C Whitney triangulation S . Proposition 2.47.
Let S be a C Whitney triangulation. Then, there is an equivalence Sh N ∗∞ S ( M ) = S -Mod and an identification of the generators is given by Z X s (cid:55)→ s where s is the indicator which is defined by s ( t ) = (cid:40) Z , t ≤ s. , otherwise . Thus, the subcategory of compact objects Sh N ∗∞ S ( M ) c consists of sheaves with compactsupport and perfect stalks.
16e will use such triangulations as a tool to study general isotropics Λ ⊆ S ∗ M . Thatis, we would like to consider the situation where there is a C Whitney stratification S such that Λ ⊆ N ∗∞ S . We recall one setting which fulfills such a requirement. Definition 2.48.
Assume M is C ω , i.e., real analytic. A subset Z of M is said to besubanalytic at x if there exists open set U (cid:51) x , compact manifolds Y ij ( i = 1 , , ≤ j ≤ N )and morphisms f ij : Y ij → M such that Z ∩ U = U ∩ N (cid:91) j =1 ( f j ( Y j ) \ f j ( Y j )) . We say Z is subanalytic if Z is subanalytic at x for all x ∈ M .A key feature of subanalytic geometry we need is the following existence lemma. Lemma 2.49 ([23, Corollary 8.3.22]) . Let Λ be a closed conic subanalytic isotropic subsetof T ∗ M . Then there exists a C ω Whitney stratification S such that Λ ⊆ N ∗ S . Combining with the above proposition, it implies a microlocal criterion for a sheaf F with subanalytic microsupport being constructible. Proposition 2.50.
Let F ∈ Sh( M ) and assume SS ∞ ( F ) is subanalytic. Then F isconstructible if and only if SS ∞ ( F ) is a singular isotropic. Another feature of subanalytic geometry is that relatively compact subanalytic setsform an o-minimal structure. Thus, one can apply the result of [8] to refine a C p Whitneystratification to a Whitney triangulation, for 1 ≤ p < ∞ . Lemma 2.51.
Let Λ be a subanalytic singular isotropic in S ∗ M . Then there exists a C Whitney triangulation S such that Λ ⊆ N ∗ S . Combining the above two results, we conclude:
Theorem 2.52.
Let F ∈ Sh( M ) and assume SS ∞ ( F ) is a subanalytic singular isotropic.Then F is S -constructible for some C Whitney triangulation S . Collectively, sheaves with the same subanalytic singular isotropic microsupport formsa category with nice finiteness property. Let Λ be a subanalytic singular isotropic in S ∗ M .Pick a Whitney triangulation S such that Λ ⊆ N ∗∞ S . Constructibility implies, for smoothpoints ( x, ξ ) in N ∗∞ S , the microstalk functor µ ( x,ξ ) : Sh N ∗∞ S ( M ) → Z -Mod preserves bothlimits and colimits. Combining with Proposition 2.34, we see that the inclusion ι ∗ : Sh Λ ( M ) (cid:44) → Sh N ∗∞ S ( M )admits a left adjoint ι ∗ and a right adjoint ι ! . Since Sh N ∗∞ S ( M ) = S -Mod is compactlygenerated, we have: Proposition 2.53.
Let Λ be a subanalytic singular isotropic in S ∗ M . The category Sh Λ ( M ) is compactly generated. roof. Fix a Whitney triangulation S such that Λ ⊆ N ∗∞ S . Recall that S -Mod =Ind(Perf S ) is compactly generated. For F ∈ Sh Λ ( M ), there exists F i ∈ Sh N ∗∞ S ( M )such that ι ∗ F = lim −→ F i . Thus F = ι ∗ ι ∗ F = ι ∗ ι ∗ lim −→ F i = lim −→ ι ∗ F i . Now note that ι ∗ F i is compact in Sh Λ ( M ) since ι ∗ (cid:97) ι ∗ (cid:97) ι ! and the left adjoint of leftjoint preserves compact objects.Now recall from Proposition 2.16, the presheaf Sh in Pr R st of integer coefficient sheavesis itself a sheaf. Since a set can be recovered from its intersections with an open cover,the same argument shows that the assigement U (cid:55)→ Sh Λ ( U ) forms a sheaf Sh Λ in Pr R ω,st .By Proposition 2.5 and Proposition 2.12, passing to left adjoints turns Sh Λ to a cosheafin Pr L ω,st , and taking compact objects further turns it to a cosheaf in st ω . That is, Proposition 2.54.
The precosheaf Sh c Λ : Op M → st ω is a cosheaf. For an inclusion of subanalytic singular isotropics Λ ⊆ Λ (cid:48) , by picking a Whitneytriangulation S such that Λ (cid:48) ⊆ N ∗∞ S , a similar consideration as above shows that theinclusion Sh Λ ( M ) (cid:44) → Sh Λ (cid:48) ( M ) has both a left and a right adjoint. Thus, Proposition 2.55.
Passing to left adjoint, the inclusion Sh Λ ( M ) (cid:44) → Sh Λ (cid:48) ( M ) induces acanonical functor Sh Λ (cid:48) ( M ) c (cid:16) Sh Λ ( M ) c between compact objects. By applying the left adjoint, we see that the microstalk functor µ ( x,ξ ) : Sh Λ ( M ) → Z -Mod is tautologically corepresented by the compact object µ L ( x,ξ ) ( Z ) ∈ Sh Λ ( M ) c . Fur-thermore, in case of an inclusion Λ ⊆ Λ (cid:48) , these corepresentatives are sent to corepresenta-tives of the restricted functor on subcategories under the canonical functor Sh Λ (cid:48) ( M ) c → Sh Λ ( M ) c . Tautologically, the corepresentative of µ ( x,ξ ) for smooth points ( x, ξ ) ∈ Λ (cid:48) \ Λvanishes under this functor. By Proposition 2.34, the converse is also true :
Proposition 2.56 (Theorem 4.13 of [13]) . Let Λ ⊆ Λ (cid:48) be subanalytic isotropics andlet D µ Λ (cid:48) , Λ ( T ∗ M ) denote the fiber of the canonical functor Sh Λ (cid:48) ( M ) c → Sh Λ ( M ) c . Then D µ Λ (cid:48) , Λ ( T ∗ M ) is generated by the corepresentatives of the microstalk functors µ ( x,ξ ) forsmooth Legendrian points ( x, ξ ) ∈ Λ (cid:48) \ Λ . Let (
X, ω, Z ) be a Liouville manifold and α := ι Z ω be the Liouville form. Consideran isotopy of Lagrangian submanifolds conic at infinity L t , t ∈ [0 , L t is positive if α ( ∂ t ∂ ∞ L t ) ≥
0. Standard Floer theory implies there isan continuation element c ( L t ) ∈ HF ∗ ( L , L ). Recall that for any triple ( K , K , K ) oftransversally interacted Lagrangians, there exists a multiplication map µ : HF ∗ ( K , K ) ⊗ HF ∗ ( K , K ) → HF ∗ ( K , K ). Thus, for suitable K ’s, multiplying c ( L t ) induces a mapHF ∗ ( L , K ) → HF ∗ ( L , K ) which is usually referred as the continuation map and is akey ingredient for defining the wrapped Floer category.18 .1 Continuation maps We recall here the sheaf theoretical continuation maps studied in [18]. A dual constructioncan be found in [39] and [19]. In the sheaf theoretical setting, the object corresponds tothe continuation element is simply a morphism/map between sheaves. As a result, wesimply use the term continuation map to refer both the morphism between sheaves andthe induced map on the Hom-complex. Let ( t, τ ) denote the coordinate of T ∗ R and let T ∗≤ R = { τ ≤ } denote the set of non-positive covectors. Lemma 3.1.
Let N be a manifold, [ −∞ , ∞ ] be the compactification of R at the twoinfinities, p : N × R → N be the projection, j : N × R (cid:44) → N × [ −∞ , ∞ ] be the openinterior, and i ± : N × {±∞} (cid:44) → N × [ −∞ , ∞ ] be the closed inclusion at the infinities.Then for a sheaf F ∈ Sh T ∗ N × T ∗≤ R ( N × R ) , there are isomorphisms p ∗ F = i ∗− j ∗ F and p ! F [1] = i ∗ + j ∗ F identifying the two pushforwards as nearby cycles at the infinities.Proof. We first prove the case when supp( F ) ⊆ N × [ − C, C ] for some C ∈ R > . In thiscase, i ∗− j ∗ F = i ∗ + j ∗ F = 0 and p ∗ F = p ! F since p is proper on supp( F ). Let x ∈ N bea point. Base change implies ( p ∗ F ) x = Γ( { x } × R ; F | { x }× R ). Apply the microsupportestimation SS( f ∗ F ) ⊆ f (SS( F )) of Proposition 2.38 to the inclusion of the slice at x ,we obtain SS( F | { x }× R ) ⊆ T ∗≤ R so we reduce to the case N = {∗} . In this case, considerthe family of open sets { ( −∞ , t ) } t ∈ R } , the noncharacteristic deformation lemma, Lemma2.25, implies that Γ( R ; F ) ∼ −→ Γ(( −∞ , t ); F ) for all t ∈ R . Since supp( F ) is compact, thelatter is 0 for t << p ∗ F → i ∗− j ∗ F and i ∗ + j ∗ F → p ! F [1] functorial on F : Let j − : N × R (cid:44) → N × [ −∞ , ∞ ) denote the openembedding compactifying the negative end. For any G ∈ Sh( N × [ −∞ , ∞ )), there is afiber sequence j − ! j ∗− G → G → i − ! i ∗− G. Set G = j −∗ F and notice j ∗− j −∗ = id, we obtain the fiber sequence j − ! F → j −∗ F → i − ! i ∗− j −∗ F. Let p − : N × [ −∞ , ∞ ) → N denote the projection (and similarly for p + ). The canonicalmorphism p ∗ F → i ∗− j ∗ F is obtained by applying p −∗ to the above fiber sequence. Themorphism i ∗ + j ∗ F → p ! F [1] can be obtained similarly.Now recall that there is fiber sequence F N × ( −∞ , → F → F N × (0 , ∞ ) . (5) of Proposition 2.36 implies that both F N × ( −∞ , and F N × (0 , ∞ ) are contained in T ∗ N × T ∗≤ R . So it’s sufficient to prove the cases when supp( F ) ⊆ N × ( −∞ , C ] and supp( F ) ⊆ N × [ − C, ∞ ) for some C ∈ R > .We first prove the trivial cases: Assume supp( F ) ⊆ [ − C, ∞ ). We claim p ∗ F = 0 = i ∗− j ∗ F . One computes p ∗ F = p ∗ F [ − C, ∞ ) = p ∗ lim n →∞ F [ − C,n ] = lim n →∞ p ∗ F [ − C,n ] = 019y the case supp( F ) ⊆ N × [ − C, C ]. Similarly, by considering the colimit F ( −∞ ,C ] = colim n →∞ F ( − n,C ] , one can conclude p ! F = 0 = i ∗ + j ∗ F when supp( F ) ⊆ ( −∞ , C ].Now assume supp( F ) ⊆ ( −∞ , C ] and we claim p ∗ F = i ∗− j ∗ F . Consider again thefiber sequence j − ! F → j −∗ F → i − ! i ∗− j −∗ F. Apply p −∗ and notice that p − ! = p −∗ for these sheaves because of the compact supportassumption. Thus, we obtain the fiber sequence p ! F → p ∗ F → i ∗− j −∗ F. Since p ! F = 0 by the previous case, p ∗ F = i ∗− j −∗ F . The other isomorphism can beobtained similarly.In order to define the continuation map, we prove a prototype version of Theorem1.2. Proposition 3.2 ([18, Proposition 4.8]) . Let ι ∗ : Sh T ∗ N × T ∗≤ R ( N × R ) (cid:44) → Sh( N × R ) denote the standard inclusion. Then there exist left and right adjoints ι ∗ (cid:97) ι ∗ (cid:97) ι ! whichis given by convolutions ι ∗ F = Z { t (cid:48) >t } [1] ◦ F and ι ! F = lim r →∞ ( Z { t − r ≤ t (cid:48) ≤ r } ◦ F ) . Here ( t, t (cid:48) ) is the coordinate of R .Proof. To make the notation simpler, replace R by the open interval I = (0 , Z { t (cid:48) >t } ◦ F ) ⊆ T ∗ N × T ∗≤ I . Let π , π : N × I × I → N × I denote the pro-jection π ( x, t, t (cid:48) ) = ( x, t ) and π ( x, t, t (cid:48) ) = ( x, t (cid:48) ). Then Z { t (cid:48) >t } ◦ F = π [( π ∗ F ) N ×{ t (cid:48) >t } ].In order to estimate the effect of π on the microsupport, we need the map π to beproper on the support of the sheaf. Thus, let j : N × I × I (cid:44) → N × ( − , × I denotethe open inclusion, π (cid:48) : N × ( − , × I → N × I denote the projection, and we factorize Z { t (cid:48) >t } ◦ F to π (cid:48) ∗ j ! [( π ∗ F ) N ×{ t (cid:48) >t } ]. Before taking π (cid:48) ∗ , one observes, using (4) and (5)of Proposition 2.36 and (1) of Proposition 2.38, that none of the operations introducesnon-zero covectors on the second I -component to the microsupport except when taking( • ) N ×{ t (cid:48) >t } , covectors of the form (0 , σ, − σ ) for σ ∈ R > might be added to the cotangentfibers over the boundary { t (cid:48) = t } . ThusSS( Z { t (cid:48) >t } ◦ F ) ⊆ SS( π (cid:48) ∗ j ! [( π ∗ F ) N ×{ t (cid:48) >t } ]) ⊆ ( π (cid:48) ) π (cid:0) SS( j ! [( π ∗ F ) N ×{ t (cid:48) >t } ]) ∩ T ∗ N × ( − , × T ∗ I (cid:1) ⊆ T ∗ N × T ∗≤ I. For the right adjoint ι ! , we note thatlim r →∞ ( Z { t − r ≤ t (cid:48) ≤ r } ◦ F ) = lim r →∞ π [( π ∗ F ) N ×{ t − r ≤ t (cid:48) ≤ t } ]= lim r →∞ π ∗ [( π ∗ F ) N ×{ t − r ≤ t (cid:48) ≤ t } ]= π ∗ lim r →∞ [( π ∗ F ) N ×{ t − r ≤ t (cid:48) ≤ t } ]= π ∗ [( π ∗ F ) N ×{ t (cid:48) ≤ t } ] . Then one can argue as the left adjoint case. (Note the last term is different from Z { t (cid:48) ≤ t } ◦ F in general since limits do not commute with convolution.)20n sum, we’ve shown that there are functors Z { t (cid:48) >t } [1] ◦ ( • ) , Z { t (cid:48) ≤ t } ◦ ( • ) : Sh( M × I ) → Sh T ∗ N × T ∗≤ R ( N × R ) . In order to show that these are indeed adjunctions, it’s sufficient to show that the canon-ical morphisms Z ∆ I → Z { t (cid:48) >t } [1] and Z { t (cid:48) ≤ t } → Z ∆ I becomes isomorphism after convo-luting with sheaves in Sh T ∗ N × T ∗≤ R ( N × R ) since convoluting with Z ∆ I is the same as theidentity functor.Consider the fiber sequence Z { t (cid:48) >t } → Z { t (cid:48) ≥ t } → Z ∆ I . We have similarly Z { t (cid:48) ≥ t } ◦ F = π ( π ∗ F ) N ×{ t (cid:48) ≥ t } and a similar microsupport estimationimplies, before applying π , SS (cid:0) ( π ∗ F ) N ×{ t (cid:48) ≥ t } (cid:1) ⊆ T ∗ N × T ∗≤ I × T ∗ I . Thus, the lastlemma 3.1 implies π [( π ∗ F ) N ×{ t (cid:48) ≥ t } ] is the nearby circle of ( π ∗ F ) N ×{ t (cid:48) ≥ t } at ∞ along thefirst I -direction and it is 0. Thus F = Z ∆ I ◦ F ∼ −→ Z { t (cid:48) >t } [1] ◦ F . A similar argumentshows lim r →∞ ( Z { t − r ≤ t (cid:48) ≤ t } ◦ F ) ∼ −→ F for F with the same microsupport condition.Now let F ∈ Sh T ∗ N × T ∗≤ R ( N × R ) and, by the preceding lemma, F ∼ −→ Z { s (cid:48) >s } [1] ◦ F .Let a ∈ R and let i a : N (cid:44) → N × R denote the slice at a . Applying i ∗ a results theisomorphism i ∗ a F ∼ −→ Z ( −∞ ,a ) [1] ◦ F . Recall that for a ≤ b , there is a canonical morphism Z ( −∞ ,a ) [1] → Z ( −∞ ,b ) [1] induced by the open inclusion ( −∞ , a ) (cid:44) → ( −∞ , b ). Definition 3.3.
For F ∈ Sh T ∗ N × T ∗≤ R ( N × R ) and a ≤ b Set F x = i ∗ x F for x ∈ R .We define the continuation map c ( F, a, b ) : F a → F b to be the (homotopically unique)morphism c that makes the following diagram commute: F a F b Z ( −∞ ,a ) [1] ◦ F Z ( −∞ ,b ) [1] ◦ Fc The continuation map inherits various properties from Z ( −∞ ,a ) . For example, it com-poses in the sense that c ( F, a , a ) ◦ c ( F, a , a ) = c ( F, a , a )since the conical map Z ( −∞ ,a ) → Z ( −∞ ,a ) → Z ( −∞ ,a ) compose to Z ( −∞ ,a ) → Z ( −∞ ,a ) .Let p [ a,b ] : N × [ a, b ] → N denote the projection. If F | N × [ a,b ] = p ∗ [ a,b ] G is a pullbackfrom N for some G ∈ Sh( N ), one can identify F a = G = F b through the canonicalmap F → i a ∗ i ∗ a F . In this case, the continuation map c ( F, a, b ) is equivalent to thisidentification F a = F b .We consider the homotopical invariant property of the continuation map in the follow-ing setting. Let I and J be open intervals and let ( t, τ ) and ( s, σ ) be the correspondingcoordinates for their cotangent bundles. Let G ∈ Sh( M × I × J ) be a sheaf such thatSS( G ) ⊆ { τ ≤ } . For any x ∈ I , we use G t = x := G | M ×{ x }× J to denote the restrictionand similarly for G s = y , y ∈ J . Note by (2) of Proposition 2.38, the same condition21S( G s = y ) ⊆ { τ ≤ } holds. Assume further that there exists a ≤ b in I such thatSS( G t = a ), SS( G t = b ) ⊆ T ∗ M × J . By Lemma 2.39, this implies that there exist F a , F b ∈ Sh( M ) such that G t = a = p ∗ s F a and G t = b = p ∗ s F b where we use p s : M × J → M to denote the projection. Note that, for each y ∈ J , the restriction G s = y induces acontinuation map c ( G, y, a, b ) : F a → F b . Proposition 3.4.
The morphism c ( G, y, a, b ) is independent of y ∈ J .Proof. Since SS( G ) ⊆ { τ ≤ } , a family version of Proposition 3.2 implies G = Z ∆ I × J ◦ | J G ∼ −→ Z { s (cid:48) >s }× J [1] ◦ | J G is an isomorphism where ◦| J is the J -parametrized convolution. In particular, G t = a ∼ −→ Z ( −∞ ,a ) × J [1] ◦ | J G and thus there is a ( J -parametrized) continuation map c J ( G, a, b ) : G t = a → G t = b . For y ∈ J , let i y : M → M × J denote the inclusion of the slice at y . By Proposition2.18, there is equivalence i ∗ y ( K ◦ | J G ) = K | s = y ◦ G s = y for K ∈ Sh( I × J ). This impliesthat c J ( G, a, b ) restricts to i ∗ y c J ( G, a, b ) = c ( G s = y , a, b ). Hence, the i y ∗ (cid:97) i y ∗ adjunctioninduces a commuting diagram, G t = a G t = b i y ∗ i y ∗ G t = a i y ∗ i y ∗ G t = b c J ( G, a, b ) i y ∗ c ( G s = y , a, b )which is equivalent to p ∗ s F a p ∗ s F b i y ∗ F a i y ∗ F b c J ( G, a, b ) i y ∗ c ( G s = y , a, b ) . Since J is contractible, the horizontal arrows become isomorphism after applying p s ∗ . F a F b F a F b p s ∗ c J ( G, a, b ) c ( G s = y , a, b ) . That is, the continuation map c ( G s = y , a, b ) is equivalent to p s ∗ c J ( G, a, b ) for all y ∈ J . Remark . One can see from the proof that the continuation maps enjoy higher homo-topical independence. 22 .2 Sheaf theoretical wrappings
We specialize to the case when F comes from the Guillermou-Kashiwara-Schapira sheafquantization in this section. Recall that when M is a smooth manifold, its cotangentbundle admits a canonical symplectic structure ( T ∗ M, dα ). The Liouville form α is com-patible with the R > -action which freely acts on ˙ T ∗ M . Thus, there is an induced contactstructure on the cosphere bundle S ∗ M . It can be realized as a contact hypersurfaceof ˙ T ∗ M by picking a Riemannian metric. There is a dictionary between homogeneoussymplectic geometry of ˙ T ∗ M and contact geometry of S ∗ M . Thus, we will use them in-terchangeably when one language is more convenient. See subsection A for a more detaildiscussion. Definition 3.6.
Let M , B be a manifolds and I be an open interval containing 0. Wesay a C ∞ map Φ : S ∗ M × I × B → S ∗ M is a B -family contact isotopies if for each( t, b ) ∈ I × B , the map ϕ t,b := Φ( • , t, b ) is a contactomorphism and ϕ ,b = id S ∗ M for all b ∈ B .As remarked above, a B -family contact isotopies Φ corresponds to a B -family ofhomogeneous symplectic isotopies (of degree 1), which we abuse the notation and denoteit by Φ as well. For fixed b ∈ B , we let V Φ b denote the vector field generated by ϕ t,b . Since ϕ t,b is homogeneous, V Φ b is a Hamiltonian vector field with α ( V Φ b ) being its Hamiltonian.The latter is the function which evaluates to α ϕ t,b ( x,ξ ) ( ∂∂t ϕ ( t,b ) ( x, ξ )) at ϕ t,b ( x, ξ ). Proposition 3.7.
For each B -family homogeneous symplectic isotopies Φ , there is aunique conic Lagrangian submanifold Λ Φ in ˙ T ∗ ( M × M ) × T ∗ I × T ∗ B which is determinedby the equation T ∗ t,b ( I × B ) ◦ Λ Φ = Λ ϕ t,b where the later is { ( x, − ξ, φ t,b ( x, ξ )) | ( x, ξ ) ∈ ˙ T ∗ M } ,the twisted graph of ϕ t,b . More precisely, it is given by the formula Λ Φ = (cid:8)(cid:0) x, − ξ, ϕ t,b ( x, ξ ) , t, − α ( V Φ b )( ϕ t,b ( x, ξ )) , b, − α ϕ t,b ( x,ξ ) ◦ d (Φ ◦ i x,ξ,t ) b ( · ) (cid:1)(cid:9) (1) where the parameters run through ( x, ξ ) ∈ ˙ T ∗ M , t ∈ I , b ∈ B , and the map i x,ξ,t is theinclusion of B to the ( x, ξ, t ) -slice. We use the same notation Λ Φ to denote its projectionto S ∗ ( M × M × I × B ) which is a Legendrian submanifold. The following theorem of Guillermou-Kashiwara-Schapira is a categorification of themore classical statements of quantization which usually have operators as the quantizedobjects. The proof given there is the non-family case. Since the existence is proved byusing uniqueness to glue local existence and the local picture depends smoothly on thefamily J n , the same proof holds for the family version with minor modification. Theorem 3.8 ([18, Proposition 3.2]) . Let M be a manifold and Φ : S ∗ M × I × J n → S ∗ M be a J n -family of contact isotopies with parameter space J n where J is an openinterval. Then there exists a unique sheaf kernel K (Φ) ∈ Sh( M × M × I × J n ) such that SS ∞ ( K (Φ)) ⊆ Λ Φ and K (Φ) | t =0 = Z ∆ M × J n . Moreover, SS ∞ ( K (Φ)) = Λ Φ is simple along Λ Φ , both projections supp( K ) → M × I × J n are proper the composition is compatiblewith convolution in the sense that1. K (Ψ ◦ Φ) = K (Ψ) ◦ | I × J n K (Φ) ,2. K (Φ − ) ◦ | I × J n K (Φ) = K (Φ) ◦ | I × J n K (Φ − ) = Z ∆ M × I × J n .Here Φ − is the family isotopy given by Φ − ( • , t, b ) := φ − t,b . emark . The equality SS ∞ ( K (Φ)) = Λ Φ as well as a few other properties of K (Φ)followed by the uniqueness is explained in [15].We refer the above process of obtaining the sheaf kernel K (Φ) from a contact isotopyΦ as the Guillermou-Kashiwara-Schapira sheaf quantization or GKS sheaf quantizationin short. A corollary of this construction is that contact isotopies act on sheaves and theaction is compatible with the microsupport:
Corollary 3.10.
For a contact isotopy
Φ : S ∗ M × I → S ∗ M and a sheaf F ∈ Sh( M ) ,there is equality ˙SS( K (Φ) ◦ F ) = Λ Φ ◦ ˙SS( F ) . That is, if we set F t := ( K (Φ) ◦ F ) | M ×{ t } ,then SS ∞ ( F t ) = φ t SS ∞ ( F ) for t ∈ I . Furthermore, if F has compact support, then sodoes F t for all t ∈ I . We will consider the notion of wrapping for sheaves. Recall that in contact geometry,a wrapping is usually referring to a one-parameter deformation of Legendrian L t in acontact manifold Y . The wrapping is positive (resp. negative) if α ( ∂ t L t ) ≥ α ( ∂ t L t ) ≤
0) for some compatible contact form α . An exercise is that such a deformation L t can always be extended to a contact isotopy Φ on Y . Since deformations of singularisotropics are not yet available at this moment, we first consider globally defined contactisotopies on S ∗ M , and then use them to deform sheaves through GKS sheaf quantization. Remark . The term wrapping comes from the example of ˙ T ∗ ( R / Z ) with the isotopygiven by φ t ( x, ξ ) = (cid:40) ( x + t, ξ ) , ξ > , ( x − t, ξ ) , ξ < . In this paper, we will use the term positive/negative wrapping to mean a positive/negativeisotopy, a family of sheaves induced by such an isotopy, or the corresponding family ofsingular isotropics of those sheaves by taking SS ∞ ( • ). Since these two notions are dual toeach other, we will mainly work with positive isotopies and simply refer them as wrappings when the context is clear.We will define the category of positive wrappings whose (1)-morphisms will be givenby concatenation. In order to define concatenation easily, we assume that the isotopies are constant near the end points, and the interval I will be a closed interval from now on.This requirement doesn’t lose much information since for any positive contact isotopiesΦ : S ∗ M × [0 , → S ∗ M , one can always make it have constant ends through a homotopyof isotopies. For example, pick a non-decreasing C ∞ function ρ on R such that ρ | ( −∞ , / ≡ ρ | [2 / , ∞ ) ≡
1. Then an example of such a modification is given by ˜Φ( x, ξ, t, s ) =Φ( x, ξ, (1 − s ) t + sρ ( t )). By Proposition 3.4, they induce equivalent continuation maps andtwo such identifications can itself be identified by a similar consideration and so on. Thus,when we mention isotopies obtained through nature constructions such as integrated froma time-independent vector field, we will implicitly assume such a deformation procedure. Definition 3.12.
Let I = [ t , t ], J = [ s , s ] be two closed intervals. We use I J todenote the concatenated interval ( I (cid:113) J ) / { t ∼ s } . For isotopies Φ : S ∗ M × I → S ∗ M ,Ψ : S ∗ M × J → S ∗ M , the concatenation map Ψ S ∗ M × ( I J ) → S ∗ M is theisotopy which is given by(Ψ x, ξ, t ) = (cid:40) Φ( x, ξ, t ) , t ∈ I, Ψ(Φ( x, ξ, t ) , t ) , t ∈ I (cid:48) . I = J , one can also define the pointwise composition Ψ ◦ Φ : S ∗ M × I → S ∗ M by(Ψ ◦ Φ)( x, ξ, t ) = Ψ(Φ( x, ξ, t ) , t )) . Note that, up to a scaling, Ψ ◦ Φ and Ψ
Definition 3.13.
Let Ω ⊆ S ∗ M be an open subset. We say a contactomorphism ϕ : S ∗ M → S ∗ M is compactly supported on Ω if ϕ equals id S ∗ M outside a compact set C inΩ. Similarly, a contact isotopy Φ : S ∗ M × I × S ∗ M is compactly supported on Ω if ϕ t = idoutside a fixed compact set C in Ω for all t ∈ I . Definition 3.14.
We define the category W (Ω) of positive wrappings on Ω as follows: Anobject of W (Ω) is a pair ( ϕ, [Φ]) such that ϕ is a compactly supported contactomorphismand [Φ] is a homotopy class of compactly supported isotopies, defined on a closed interval I , realizing ϕ as Hamiltonian. Note the degenerate case I = {∗} is allowed. We willoften simply write ( ϕ, [Φ]) by Φ without emphasizing it’s a homotopy class through thepaper. A 1-morphism Ψ : [Φ ] → [Φ ] is a positive isotopy Ψ such that [Φ ] = [Ψ ].Composition of 1-morphisms is given by concatenation. For Ψ , Ψ : [Φ ] → [Φ ], a 2-morphism is a positive family of isotopy Θ : S ∗ M × I × J → S ∗ M which is constant nearthe end points on the J -direction such that Θ( • , t, s i ) = Ψ i ( • , t ) and Θ( • , t i , s ) = Φ i ( • ), i = 0 ,
1. Here t i and s i are the end points of I and J . An n -morphism will be a homotopybetween n − W (Ω) and we show such colimit is filtered.Recall that a 1-category C is filtered if,1. C is non-empty,2. for any X, Y ∈ C , there is Z ∈ C with morphisms X → Z and Y → Z , and,3. for any more morphism f, g : X → Y , there exist h : Y → Z such that h ◦ f = h ◦ g .This is the same as saying for any (ordered) n -simplex K , n ∈ [ − ,
1] and any functor F : K → C , there is an extension ˆ F on K (cid:46) , the n + 1 simplex obtained by adding a finalcone point to K . For example, we can realize a pair of morphisms f, g : X → Y as ahollowed triangle consisting of vertices X, X, Y and edges id X , f, g without the presenceof the face. A final cone point Z provides a morphism h : Y → Z for the edge between Y and Z . The existence of the three new faces and the fact that the only 2-morphism ina 1-category is the strict equality implies h ◦ f = h ◦ g . Definition 3.15.
A category C is filtered if for any simplex K and any functor F : K → C ,there is an extension ˆ F : K (cid:46) → C . Example 3.16.
Consider the case when K = S is the 2-sphere, or more precisely, when K = ∆ is the standard 2-simplex such that the base face has three vertices being a fixedobject X , three edges being id X , and the face being the trivial identification. This isessentially the situation that there are objects X , Y , a 1-morphism f : X → Y , with anon-trivial 2-automorphism T on f . The condition of C being filtered means that thereexist g : Y → Z such that the auto equivalence g ◦ T on g ◦ f is trivial, that is, g ◦ T = id g ◦ f . Proposition 3.17.
The category W (Ω) is filtered. roof. Similarly to classical algebraic topology, it’s sufficient to check the case when K = S n , the n -sphere, n = 0 , , · · · .When n = 0, we are given two homotopy classes of contact isotopies Φ and Φ with the same end point ϕ , and the goal is to find another contact isotopy Φ and twopositive contact isotopies Ψ and Ψ such that [Φ] = [Ψ ] = [Ψ ]. We firstnotice that, up to a rescaling, [Φ − ] = [Φ − ] = id. So it’s sufficient to modifyΦ − and Φ − by composing some Φ (cid:48) so that Φ (cid:48) ◦ Φ − and Φ (cid:48) ◦ Φ − are positive. Let H , H denote their Hamiltonians. Since Φ − and Φ − are compactly supported, thereexist compact set C ⊆ Ω such that H and H are zero outside C . Pick a positive real α such that α > max( | H | , | H | ), relative compact open sets U , V in S ∗ M such that C ⊂ U ⊆ U ⊆ V ⊆ Ω, and a bump function ρ such that ρ | U ≡ ρ ≡ V .The contact isotopy Φ (cid:48) generated by αρ will satisfy the requirement by the Leibniz rule.When n >
0, we are given a family of morphism Ψ θ : Φ → Φ parametrized by S n − such that [Φ ] = [Ψ θ ], and we have to show that, by possibly further concatenation,this family can be made to be null-homotopy through positive isotopies. By precomposingΦ − , we may assume there is an S n − -family of positive isotopy Ψ θ and a fixed (notnecessarily positive) isotopy Φ, such that, for each θ ∈ S n − , there exists a homotopyΣ θ : S ∗ M × I × [0 , → S ∗ M connecting Φ to Ψ θ . We can extend this map to a D n -family of isotopy Σ : S ∗ M × I × D n → S ∗ M by Σ( x, ξ, t, rθ ) = Σ θ ( x, ξ, t, r ) where wewrite elements in D n by r ∈ [0 ,
1] and θ ∈ S n − . Now the same compactness argumentas before shows that there is a positive isotopy Φ (cid:48) such that Φ (cid:48) ◦ Σ is positive.Let F : C → D be a functor. For any diagram p : D → E , the colimits colim C ( p ◦ F )and colim D F exist if either one exists. Thus, it’s well-defined to write the canonical mapcolim C ( p ◦ F ) → colim D F . Definition 3.18.
A functor F : C → D if cofinal if, for any diagram p : D → E , thecanonical map colim C ( p ◦ F ) → colim D F is an isomorphism.In the 1-categorical setting, a more classical notation is that a functor is cofinal if andonly if,1. for any d ∈ D , there exists c ∈ C and a morphism d → F ( c ),2. for any morphism f, g : d → F ( c ), there exist h : c → c (cid:48) such that F ( h ) ◦ f = F ( h ) ◦ g .An equivalent way of saying it is that the fiber product C × D d/ D is non-empty andconnected for all d ∈ D . Here, d/ D is the over category whose objects are morphisms ofthe form d → d (cid:48) and a morphism h : ( f : d → d (cid:48) ) → ( g : d → d (cid:48)(cid:48) ) is given by a morphism h : d (cid:48) → d (cid:48)(cid:48) such that h ◦ f = g , the fiber product is taken over the canonical projection d/ D → D by ( d → d (cid:48) ) (cid:55)→ d (cid:48) and F . Recall a 1-category is said to be connected if theassociated 1-groupoid (by formally inverting morphisms) is connected. The equivalenceof these definitions is the Quillen’s theorem A. In the ∞ -categorical setting it states: Theorem 3.19 (Quillen’s Theorem A) . A functor F : C → D is cofinal if and only ifthe fiber product C × D d/ D is contractible for any d ∈ D . Now consider the following construction: For n = 1 , , · · · , take a family of open setΩ n ⊆ S ∗ M such that Ω n ⊆ Ω n ⊆ Ω n +1 , ∪ n ∈ N Ω n = S ∗ M , and Ω n ⊆ S ∗ M is relativecompact. For n >
0, pick bump function ρ n such that ρ n ≤ ρ n +1 , ρ n | Ω n ≡ n , and vanishes26utside Ω n +1 . Let Φ n : S ∗ M × [0 , n ] → S ∗ M be the isotopy generated by ρ n . Since ρ ≤ ρ ≤ · · · ρ n ≤ · · · , there exists positive isotopy Ψ n : S ∗ M × [ n, n + 1] → S ∗ M suchthat the Φ n ’s and Ψ n ’s form a sequence id Ψ −→ Φ −→ · · · in W (Ω). That is, the abovedata organizes to a functor Φ : Z ≥ → W (Ω). Lemma 3.20.
The functor
Φ : Z ≥ → W (Ω) is cofinal.Proof. By Quillen’s Theorem A, we need to show that Z ≥ × W (Ω) (Φ /W (Ω)) is con-tractible. Let Φ ∈ W (Ω) and let H denote its Hamiltonian. Since Φ is compactly sup-ported there exist a compact set C ⊆ S ∗ M such that H vanishes outside C . Pick n largesuch that C ⊆ Ω n and max( H ) ≤ n . Then the factorization Φ n = (Φ n ◦ Φ − ) ◦ Φ providesan morphism Φ → Φ n since composition is homotopic to concatenation. Thus, the fiberproduct Z ≥ × W (Ω) (Φ /W (Ω)) is equivalent to { n ∈ Z ≥ | ρ n ≥ H } . Since ρ n +1 ≥ ρ n by ourconstruction, the latter is equivalent to the poset of integers larger than min { n : ρ n ≥ H } and is contractible.Now we quantize the above construction. For ( ϕ, [Φ]) ∈ W (Ω) where Φ is defined on M × [ t , t ], set w (Φ) := K (Φ) | t = t to be the restriction of the GKS sheaf quantization K (Φ i ) at the end point. We note that since we require the end point ϕ to be fixed,the sheaf w (Φ) ∈ Sh( M × M ) depends only on the homotopy class [Φ] by formula 1and Lemma 2.39. For a morphism Ψ : Φ → Φ , since Ψ is positive, the formula 1again implies there is a continuation map c (Ψ) : w (Φ ) → w (Φ ). Similarly, for a pair ofmorphism Ψ , Ψ : Φ → Φ , if there is a homotopy Θ between Ψ and Ψ , Proposition 3.4implies that K (Θ) provides an identification between the continuation maps c (Ψ ) , c (Ψ ) : w (Φ ) → w (Φ ). Definition 3.21.
Organizing the above construction, we obtain a functor w : W (Ω) → Sh( M × M ) sending an object Φ to the corresponding sheaf kernel w (Φ), a 1-morphismΨ : Φ → Φ to the continuation map c (Ψ) : w (Φ ) → w (Φ ) , and higher morphismsto higher equivalences of continuation maps. We will refer this functor as the wrappingkernel functor .For a sheaf F ∈ Sh( M ) and a contact isotopy Φ : S ∗ M × [0 , → S ∗ M , convolutingwith K (Φ) produce a sheaf K (Φ) ◦ F on Sh( M × [0 , F t on M parametrized by [0 , K (Φ) ◦ F ) | t =1 = K (Φ) | t =1 ◦ F = w (Φ) ◦ F functorial on Φ and F . When Ψ : Φ → Φ isa positive family of isotopy, we use c (Ψ , F ) : w (Φ ) ◦ F → w (Φ ) ◦ F to denote the inducedcontinuation map. To simplify the notation, we sometimes use F Φ to denote w (Φ) ◦ F .When there is no need to specify the isotopy, we simply write it as F w . Similarly, whenΨ is unspecified, we simply write c : F w → F w (cid:48) for the continuation map. We prove alocality property which we will use later. Proposition 3.22.
Let Φ , Φ : S ∗ M × I → S ∗ M be contact isotopies. If Φ = Φ foron SS ∞ ( F ) × I , then w (Φ ) ◦ F = w (Φ ) ◦ F . Similarly, let Ψ , Ψ be positive contactisotopies. If Ψ = Ψ on an open neighborhood Ω of SS ∞ ( F ) , then c (Ψ , F ) = c (Ψ , F ) .Proof. We abuse the notation and use Φ i to denote the corresponding homogeneoussymplectic isotopies. By convoluting with Φ − , it’s enough to assume Φ and show F Φ = F . We have SS( K (Φ ) ◦ F ) ⊆ SS( K (Φ )) ◦ SS( F ) ⊆ SS( F ) × I . Hence byProposition 2.39, F t is constant along constant on t . Similarly, let H i ≥ i , i = 0 ,
1. Let Ψ s be the homotopy of isotopies between Ψ andΨ generated by the Hamiltonian (cid:98) H ( x, ξ, t, s ) = (1 − s ) H ( x, ξ, t ) + sH ( x, ξ, t ). SinceΨ = Ψ on Ω , Ψ s | Ω is constant on s . Thus Proposition 3.4 applies to K (Ψ s ) ◦ F andwe conclude c (Ψ , F ) = c (Ψ , F ).The above construction defines a functor from Sh( M ) to [ W (Ω) , Sh( M )] since w (Φ) ◦ F is functorial on F . Further composing with the functor of taking limits and colimitsdefines functors W ± (Ω) : Sh( M ) → Sh( M ). Since the subcategory of W (Ω) consists ofobjects ( ϕ, [Φ]) such that there exists a positive isotopy Φ representing [Φ] is cofinal, wecan informally write the formula by W + (Ω) F = colim F → F w F w , W − (Ω) G = lim G w − → G G w − . With this definition, we can generalize Proposition 3.2 to Proposition 1.2.
Proof of Proposition 1.2.
Set Ω = S ∗ M \ X and let F ∈ Sh( M ). We first show thatfor any ( x, ξ ) ∈ Ω, ( x, ξ ) (cid:54)∈ SS ∞ ( W ± ( F )), i.e., for any function f defined near x suchthat f ( x ) = 0 and df x ∈ R > ξ , the restriction map ( W ± ( F )) x → Γ { f< } ( W ± ( F )) x is anisomorphism. Since the situation is local and df x (cid:54) = 0, by changing a coordinate, we mayassume f = x the first coordinate function near x = 0. Pick a family of open balls U i centered at x such that U i ⊇ U i +1 ⊇ U i +1 and ∩ i U i = { x } . The stalk Γ { f< } ( W ± ( F )) x can be computed by the colimit colim i Γ ( U i ∩ { x < } ; W ± ( F )).We first do the negative case. For each i , we take a small positive wrapping Φ i supposed in Ω such that w (Φ i ) ◦ Z U i ∩{ x < } = Z ˜ U i with 0 ∈ ˜ U i and ˜ U i shrinks to x as i → ∞ . For example, take U i × C i in Ω containing ( x, ξ ) where { C i } is a family of smallballs on the fiber direction with a condition similar to the { U i } . For each i , pick a bumpfunction ρ i on S ∗ M supported on U i × C i and equals 1 near ( x, ξ ). Take H i to be theHamiltonian associated to the Reeb flow with shrinking speed and modify it to ρ i H i .Finally, take Φ i to be the isotopy associated to ρ i H i .We compute,Γ (cid:0) U i ∩ { x < } ; W − ( F ) (cid:1) = lim W (Ω) Hom (cid:0) Z U i ∩{ x < } , w (Φ) ◦ F (cid:1) = lim W (Ω) Hom (cid:0) w (Φ i ) ◦ Z U i ∩{ x < } , w (Φ i ) ◦ w (Φ) ◦ F (cid:1) = lim W (Ω) Hom (cid:0) Z ˜ U i , w (Φ i ◦ | I Φ) ◦ F (cid:1) = Γ (cid:16) ˜ U i ; W − ( F ) (cid:17) . Here we use the fact that w (Φ i ) ◦ is an equivalence for the second equation. For the lastequation, we use the fact that negative wrappings of the form Φ i ◦ Φ is initial in W (Ω).Take i → ∞ and we conclude ( W − ( F )) x = Γ { f< } ( W − ( F )) x Now we turn to the positive case. We take the same family of U i , Φ i and ˜ U i , and28ompute,Γ (cid:0) U i ∩ { x < } ; W + ( F ) (cid:1) = Hom (cid:18) Z U i ∩{ x < } , colim W (Ω) ( w (Φ) ◦ F ) (cid:19) = Hom (cid:18) w (Φ i ) ◦ Z U i ∩{ x < } , w (Φ i ) ◦ (colim W (Ω) w (Φ) ◦ F ) (cid:19) = Hom (cid:18) Z ˜ U i ∩{ x < } , colim W (Ω) ( w (Φ i ◦ | I Φ) ◦ F ) (cid:19) = Γ (cid:16) ˜ U i ; W + ( F ) (cid:17) . We use the fact that w (Φ i ) ◦ is a left adjoint so it commutes with colimits for the thirdequation. Take i → ∞ and we get ( W + ( F )) x = Γ { f< } ( W + ( F )) x .From the above computation, we see that W ± (Ω) : Sh( M ) → Sh( M ) factorizes toSh X ( M ). Finally, we show that W + (Ω) (cid:97) ι ∗ (cid:97) W − (Ω). Take G ∈ Sh X ( M ) and F ∈ Sh( M ). We compute,Hom (cid:0) G, W − ( F ) (cid:1) = Hom (cid:18) G, lim W (Ω) w (Φ) ◦ F (cid:19) = lim W (Ω) Hom (
G, w (Φ) ◦ F )= lim W (Ω) Hom (cid:0) w (Φ − ) ◦ G, F (cid:1) = lim W (Ω) Hom (
G, F ) = Hom ( ι ∗ G, F ) . The second to last equality is implied by Proposition 3.22 by the fact that Φ is compactlysupported away from Λ ⊇ SS ∞ ( G ). A similar computation shows thatHom (cid:0) W + ( F ) , G (cid:1) = Hom ( F, ι ∗ G ) . Remark . Recall that, by the adjoint functor theorem [25, Corollary 5.5.2.9], thefollowing statements are equivalent to each other:1. The inclusion Sh X ( M ) (cid:44) → Sh( M ) has a left (resp. right) adjoint for closed subsets X ⊆ S ∗ M .2. The microsupport estimation SS( ⊕ i F i ) ⊆ ∪ i SS( F i ) (resp. SS( (cid:81) i F i ) ⊆ ∪ i SS( F i ))holds.The proof of the above statements are well-known among experts and a detailed discussioncan be found in [22, 2.7]. What we achieve here is a geometric description of the adjointfunctors and thus a potential recipe for computation. Aside from the prototype cases[39, 18, 19] mentioned in the introduction, special cases for such geometric descriptionscan be found in, for example, [24] in the setting of toric homological mirror symmetrywhich are defined by using the group structure of the torus and are crucial for matchingthe data with the coherent side. 29 The category of wrapped sheaves
In this section, we mimic the definition of wrapped Fukaya category W ( T ∗ M, Λ) anddefine the category of wrapped sheaves w sh Λ ( M ) using the techniques developed in thelast section. Let M be a real analytic manifold and Λ ⊆ S ∗ M be a closed subset. Let (cid:103) w sh Λ ( M ) be thesmall subcategory of Sh( M ) generated under finite colimits and retraction by sheaves ofthe form F Φ where F is a sheaf with compact support such that SS ∞ ( F ) is a subanalyticsingular isotropic and SS ∞ ( F ) ∩ Λ = ∅ , and Φ is a contact isotopy compactly supportedaway from Λ. To encode the wrappings, we take C Λ ( M ) := (cid:104) cof( c (Ψ , F )) | Ψ ∈ Mor( W ( S ∗ M \ Λ)) , F ∈ (cid:103) w sh Λ ( M ) (cid:105) to be the subcategory of Sh( M ) generated by the cofibers of the continuation maps. Definition 4.1.
We define the category of wrapped sheaves associated to ( M, Λ) to bethe quotient category w sh Λ ( M ) := (cid:103) w sh Λ ( M ) / C Λ ( M ):= cof (cid:16) C Λ ( M ) (cid:44) → (cid:103) w sh Λ ( M ) (cid:17) in Pr L ω,st . Remark . Localization identifies sheaves which are isotopic to each other: Let F c −→ F w → cof( c ) be a fiber sequence in (cid:103) w sh Λ ( M ) induced by a continuation map. Sincea quotient map is exact and cof( c ) = 0 in w sh Λ ( M ), the fiber sequence becomes F c −→ F w → c : F → F w is an isomorphism in w sh Λ ( M ). Now let Φ be any isotopycompactly supported away from Λ. By Proposition 3.17, Φ can be modified to be positiveby a further wrapping. That is, there exists Ψ : id → Φ (cid:48) and Ψ (cid:48) : Φ → Φ (cid:48) in W ( S ∗ M \ Λ)and thus there are continuation maps F c (Ψ ,F ) −−−−→ F Φ and F c (Ψ (cid:48) ,F ) −−−−→ F Φ (cid:48) . As a result, thetwo objects F and F Φ are isomorphic in w sh Λ ( M ).To simplify the notation, we will use Hom w to denote he Hom spaces of the localizedcategory when it’s clear in the context. As the localization is essentially surjective,we usually implicitly assume a preimage F ∈ (cid:103) w sh Λ ( M ) for objects in w sh Λ ( M ). ByProposition B.3, there are identificationsHom w ( X, Y ) = colim Y α −→ Y (cid:48) Hom(
X, Y (cid:48) ) = colim X (cid:48) β −→ X Hom( X (cid:48) , Y )where α and β run through morphisms whose cofibers cof( α ), cof( β ) are in C Λ ( M ). Wewill show that it’s enough to take the colimit over W ( S ∗ M \ Λ) in our case, which is thesame colimit over all continuation maps c : F → F w by cofinality. We begin with thecase of Homing out of objects in C Λ ( M ) and in this case Hom w vanishes. Lemma 4.3.
Let G ∈ C Λ ( T ∗ M ) and F ∈ (cid:103) w sh Λ ( M ) , we have colim c : F → F w Hom(
G, F w ) = 0 = Hom w ( G, F ) where F c −→ F w runs through all continuation maps. roof. We first consider G ∈ C Λ ( M ) which is built from iterated cones and shifts of cof( c )for some continuation map c . For such a G , we may assume G fits into a cofiber sequence H c −→ H Φ → G by induction. Apply Hom( • , F Φ (cid:48) ) and we obtain the cofiber sequenceHom( G, F Φ (cid:48) ) → Hom( H Φ , F Φ (cid:48) ) → Hom(
H, F Φ (cid:48) ) . Then one compute Hom( H Φ , F Φ (cid:48) ) = Hom( w (Φ) ◦ H, w (Φ (cid:48) ) ◦ F )= Hom( H, w (Φ − ) ◦ w (Φ (cid:48) ) ◦ F )= Hom( H, w (Φ − ◦ Φ (cid:48) ) ◦ F ) . Take colim F → F w (cid:48) over isotopies Φ (cid:48) of the form Φ ◦ Ψ which is cofinal and we obtain the fibersequence colim F → F w (cid:48) Hom(
G, F w (cid:48) ) → colim F → F w (cid:48) Hom(
H, F w (cid:48) ) ∼ −→ colim F → F w (cid:48) Hom(
H, F w (cid:48) )which implies colim F → F w (cid:48) Hom(
G, F w (cid:48) ) = 0.Now let G (cid:48) be a retract of the same G as above. Taking colim F → F w Hom( • , F w ) makescolim F → F w Hom( G (cid:48) , F w ) a retract of colim F → F w (cid:48) Hom(
G, F w (cid:48) ) = 0. Since the only retract of a zeroobject is a zero object, colim F → F w Hom( G (cid:48) , F w ) = 0. Proposition 4.4.
For
F, G ∈ (cid:103) w sh Λ ( M ) , we have Hom w ( G, F ) = colim F → F w Hom(
G, F w ) where F c −→ F w runs through all continuation maps.Proof. Consider any morphism α : G (cid:48) → G such that fib( α ) ∈ C Λ ( M ) which we willdenote it as G (cid:48) qis. −−→ G when the exact α is not relevant. Now take a continuation map c : F → F w and apply Hom( • , F w ) to the fiber sequence fib( α ) → G (cid:48) → G so we obtainthe fiber sequence Hom( G, F w ) → Hom( G (cid:48) , F w ) → Hom(fib( α ) , F w ) . Now recall that Hom w can be computed by either varying the first or the second factor.As a result, we can first take colimit over such α : G (cid:48) → G and we obtain the fibersequence Hom( G, F w ) → Hom w ( G, F w ) → colim G (cid:48) qis. −−→ G Hom(fib( α ) , F w ) . Then we take colimit over F → F w and getcolim F → F w Hom(
G, F w ) → colim F → F w Hom w ( G, F w ) → colim F → F w colim G (cid:48) qis. −−→ G Hom(fib( α ) , F w ) . Since colimits commute with each other, the above Lemma 4.3 impliescolim F → F w colim G (cid:48) qis. −−→ G Hom(fib( α ) , G (cid:48) ) = colim G (cid:48) qis. −−→ G colim F → F w Hom(fib( α ) , G (cid:48) ) = 0or, equivalentlycolim F → F w Hom(
G, F w ) ∼ −→ colim F → F w Hom w ( G, F w ) ∼ −→ Hom w ( G, F )since F → F w is an isomorphism in w sh Λ ( M ).31e note that the above construction is covariant on the open sets of M . For an openset U ⊆ M , we set Λ | U = Λ ∩ S ∗ U . We abuse the notation and use (cid:103) w sh Λ ( U ) to denote thecategory (cid:103) w sh Λ | U ( U ). When there is an inclusion of open sets U ⊆ V , objects in (cid:103) w sh Λ ( U )can be naturally regarded as objects in (cid:103) w sh Λ ( V ) since we require them to have compactsupport in U . We define C Λ ( U ) similarly and note that C Λ ( U ) ⊆ (cid:103) w sh Λ ( U ) ∩ C Λ ( V ). Thusthere is a canonical map w sh Λ ( U ) → w sh Λ ( V ). Definition 4.5.
The above construction defines a covariant functor w sh Λ : Op M → st ω ,i.e., a precosheaf with coefficient in small stable categories. We refer it as the precosheafof wrapped sheaves associated to Λ.Note also that this construction is contravariant on the closed set Λ. That is, ifΛ ⊆ Λ (cid:48) is an inclusion of closed subanalytic singular isotropic, there is a canonical map w sh Λ (cid:48) ( M ) → w sh Λ ( M ) by a similar consideration. In other words, there is a morphism w sh Λ (cid:48) → w sh Λ between precosheaves. Remark/Conjecture . Consider the case when Λ is a singular isotropic. Inspired byhomological mirror symmetry, Nadler defines in [29] a conic cosheaf µ Sh w Λ : Op T ∗ M → st ω through a purely categorical construction and term it as the cosheaf of wrapped microlocalsheaves . One main property of µ Sh w Λ is that its restriction to the zero section, the ‘wrappedsheaves’, is the cosheaf Sh c Λ discussed in Proposition 2.54. We will reserve the term‘wrapped sheaves’ for the geometrically constructed category w sh Λ through this paper.Corollary 1.5 of the main theorem asserts that these two cosheaves are the same after all.As a result, we expect to extend the construction w sh Λ to the cotangent bundle as well. We find a set of generators of w sh Λ ( M ) when Λ is a singular isotropic. We first provea special case of the K¨unneth formula which we will refer it as the stabilization lemma.Fix n ∈ R n . Let M be a real analytic manifold and Λ ⊆ S ∗ M be a closed subset. We setΛ st = (( R > Λ ∪ M ) × R n ) ∞ ⊆ S ∗ ( M × R n ) . Pick a small ball B ⊆ R n centered at 0. For F ∈ Sh( M ), by (6) of Proposition 2.36,there is microsupport estimationSS( F (cid:2) Z B ) ⊆ SS( F ) × N ∗ out ( B ) . As a result, exterior tensoring with Z B induces a functor • (cid:2) Z B : (cid:103) w sh Λ ( M ) → (cid:103) w sh Λ st ( M × R n ) . We claim that this functor induces a fully faithful functor on the quotient. We first recalla lemma.
Lemma 4.7.
Let C , D be stable categories, S and T be sets of morphism in C and D which are closed under composition and contain identities. Set C := (cid:104) cof( s ) | s ∈ S (cid:105) and D := (cid:104) cof( t ) | t ∈ T (cid:105) . Let F : C → D be a functor such that for all X s −→ X ∈ S ,there exists F ( X ) t −→ Y ∈ T such that t ◦ F ( s ) ∈ T . Then F | C factors through D and F descends to a functor ¯ F : ( C / C ) → ( D / D ) filling the commutative square with thequotient functors. roof. Let X s −→ X and F ( X ) t −→ Y ∈ T be as above. By the Lemma 2.10, there existsa fiber sequence F (cof( s )) → cof( t ◦ F ( s )) → cof( t ). Proposition 4.8 (The stabilization lemma) . The functor • (cid:2) Z B defined above descendsto a fully faithful functor on the quotient w sh Λ ( M ) (cid:44) → w sh Λ st ( M × R n ) which we will refer it as the stabilization functor.Proof. Let Φ : ˙ T ∗ M × I → ˙ T ∗ M be a positive homogenous symplectic isotopy and H = α (Φ ∗ ∂ t ) be its corresponding Hamiltonian. We note that since H is not definedon the entire T ∗ M , it is not always possible to extend it to ˙ T ∗ ( M × R n ) by setting thedependence on the second component to be constant. Nevertheless, H is defined on T ∗ M since it’s homogeneous of degree 1. Pick a bump function ρ on S ∗ ( M × R n ) suchthat (supp( H ) × T ∗ B ) ∞ ⊆ Int(supp( ρ )) so that H + ρ | ξ | > H >
0. Herewe use the same notation ρ to denote its pullback on ˙ T ∗ M . Then set ˜ H := (cid:112) H + ρ | ξ | and we denote its corresponding homogeneous isotopy on ˙ T ∗ ( M × R n ) to be ˜Φ.The above construction implies thatSS (cid:16) ( K (Φ − ) (cid:2) I Z ∆ R n × I ) ◦ I K ( ˜Φ) (cid:17) ⊆ SS( K (Φ − ) (cid:2) I Z ∆ R n × I ) ◦ I SS( K ( ˜Φ)) ⊆ { τ ≤ } since H ( x, ξ ) ≤ ˜ H ( x, ξ, t, τ ). Thus there is a continuation map Z ∆ M × R n → ( w (Φ − ) (cid:2) Z ∆ R n ) ◦ w ( ˜Φ)or equivalently w (Φ) (cid:2) Z ∆ R n → w ( ˜Φ)which precomposes with Z ∆ M × R n → w (Φ) (cid:2) I Z ∆ R n to Z ∆ M × R n → w ( ˜Φ). Thus, the lastlemma implies that the functor w sh Λ ( M ) → w sh Λ st ( M × R n )is well-defined.A similar argument implies that for any positive isotopy Ψ on ˙ T ∗ ( M × R n ) withHamiltonian ˜ H , there exists H on ˙ T ∗ M and ρ on ˙ T ∗ R n such that ˜ H ≤ (cid:112) H + ρ . Thusthe wrapping coming from the product is cofinal. Since B is contractible, there is anisomorphism Hom( F (cid:2) Z B , G (cid:2) Z ˜ B ) = Hom( F, G ) for any larger ball ˜ B in R n . ThisimpliesHom w sh Λ st ( M × R n ) ( G (cid:2) Z B , F (cid:2) Z B ) = colim ˜Φ ∈ W ( S ∗ ( M × R n ) \ Λ st ) Hom (cid:16) G (cid:2) Z B , ( F (cid:2) Z B ) ˜Φ (cid:17) = colim ˜Φ ∈ W ( S ∗ ( M × R n ) \ Λ st ) Hom (cid:0) G (cid:2) Z B , F Φ (cid:2) Z ˜ B (cid:1) = colim Φ ∈ W ( S ∗ M \ Λ) Hom (cid:0)
G, F Φ (cid:1) = Hom w sh Λ ( M ) ( G, F ) . Thus, the stabilization functor w sh Λ ( M ) (cid:44) → w sh Λ st ( M × R n ) is fully faithfull.33e first show that the category w sh ∅ ( T ∗ M ) = Perf Z is generated by one object.More precisely, we say an open set B ⊆ M is a ball if B is relative compact, contractibleand B is a closed disk. Now let B be a ball such that there exists an open chart U containing B . Since all such balls are smoothly isotopic to each other inside M , liftingsuch isotopies implies that the object Z B , where B is a such ball, is independent of thechoice of the exact ball. In order to show that Z B is a generator, we need a class ofauxiliary objects. Definition 4.9.
We say an open set B ⊆ M is a stable ball if it is relative compact,contractible, and B has a smooth boundary in M .One can check that a stable ball is a ball up to a stabilization by the famous corollaryof the cobordism theorem. The following statements are Theorem 5.12 and Corollary5.13 in [13]. Theorem 4.10.
A stable ball of dimension ≥ with simply connected boundary is a ball. Corollary 4.11.
Let M be a stable ball. Then B × I k is a ball provided dim B + k ≥ and k ≥ .Proof. This is implied by a combination of the van Kampen Theorem and the Poinc´areduality for manifolds with boundary H k ( N, ∂N ) = H dim N − k ( N ) . Lemma 4.12.
Assume M is connected. The category w sh ∅ ( M ) is generated under finitecolimits and retractions by Z B for any small ball B .Proof. Let F ∈ w sh ∅ ( M ) be an object. By Remark 4.2, we may assume F is a sheafwith compact support, subanalytic isotropic microsupport, and perfect stalks. By Propo-sition 2.52, there is a Whitney triangulation T such that F is T -constructible. SinceSh N ∗∞ T ( M ) c = Perf T is generated under finite colimits and retractions by Z star( t ) for t ∈ T , we may assume F = Z star( t ) . We claimed that the object Z star( t ) is isomorphic to Z B for some small ball B ⊆ star( t ). Note the open set star( t ) is relatively compact andcontractible, however, star( r ) might not be a manifold with boundary and modificationneeds to be made.Apply the inward cornering construction in Definition 2.46 to U = star( s ), we obtaina family of star( s ) − (cid:15) depending smoothly on (cid:15) . When (cid:15) is small, the object Z star( s ) − (cid:15) in w sh Λ ( M ) is independent of (cid:15) so we abuse the notation and simply denote it by Z star( s ) − . Asthere is no stop restriction, the canonical map Z star( t ) − → Z star( t ) becomes an isomorphismin w sh ∅ ( T ∗ M ) through the positive wrapping obtained by taking (cid:15) →
0. The closurestar( t ) − is a manifold with corners when (cid:15) is small, i.e., the boundary star( t ) − can bemodified by the boundary of the inclusion [0 , ∞ ) k × R n − k ⊆ R n for some k ≥
1. ByExample 2.30, SS ∞ ( Z (0 , ∞ ) k × R n − k ) is smooth and Z (0 , ∞ ) k × R n − k can be wrapped to some Z V δ where V δ := { x ∈ R n | d ( x, [0 , ∞ ) k × R n − k ) < δ } by the Reeb flow. One can see fromthe local model that the boundary of V δ is smooth for small δ . Thus, we may furtherreplace Z star( t ) − by some Z U such that U is relative compact, contractible and has smoothboundary, i.e., a stable ball.Finally pick a ball B ⊆ U and consider the canonical morphism Z B → Z U inducedby the inclusion. Apply the stabilization lemma for n large and we see that we may34ssume U to be a ball as well. In this case, the canonical map Z B → Z U coincide withthe continuation map obtained by the standard Reeb flow and is an isomorphism. Thus,the original map is an isomorphism in w sh ∅ ( T ∗ M ) and we see that Z B generates.We assume for the rest of this section that Λ is a singular isotropic. To study gen-eration for the general case, we need the following lemma to perform general positionargument. Lemma 4.13 ([11, Lemma 2.2 and Lemma 2.3]) . Let Y n − be a contact manifold, and f = f subcrit ∪ f crit ⊆ Y be a singular isotropic.1. Let Λ ⊆ Y be a compact Lagrangian. Then Λ admits cofinal wrappings Λ (cid:32) Λ w with Λ w disjoint from f .2. Let Λ , Λ ⊆ Y be compact Legendrians disjoint from f . Consider the space ofpositive Legendrian isotopies Λ (cid:32) Λ . Then the subspace of isotopies which2.1. remains disjoint from f subcrit and2.2. intersect f crit only finitely many times, each time passing transversally at asingle point,is open and dense. For inclusion of singular isotropics Λ ⊆ Λ (cid:48) ⊆ S ∗ M , general position argument impliesthat the induced map w sh Λ (cid:48) ( M ) → w sh Λ ( M ) is always essentially surjective. In order tostudy the fiber of this map, we consider the following objects:Let Λ be a subanalytic singular isotropic and let ( x, ξ ) ∈ R > Λ be a smooth point.Consider a proper analytic Λ-Morse function f : M → R . We assume there exists (cid:15) > x is the only Λ-critical point over f − ([ − (cid:15), (cid:15) ]) with critical value 0, df x = ξ and f − ( −∞ , (cid:15) ) is relatively compact. By our assumption, both Z f − ( −∞ , ± (cid:15) ) are objects of (cid:103) w sh Λ ( M ). Definition 4.14.
A sheaf theoretical linking disk at ( x, ξ ) (with respect to Λ) is an object D ( x,ξ ) of the form cof( Z f − ( −∞ , − (cid:15) ) → Z f − ( −∞ ,(cid:15) ) )where the arrow is induced by the inclusion of opens f − ( −∞ , − (cid:15) ) ⊆ f − ( −∞ , (cid:15) ) givenby a function f with the above properties. Note that by scaling f with r ∈ R > , we seethat the object D ( x,ξ ) depends only on ( x, ξ )’s image in S ∗ M . Thus, we also use the samenotation D ( x,ξ ) for ( x, ξ ) ∈ S ∗ M . Remark . Since df (cid:54) = 0 over f − ([ − (cid:15), (cid:15) ]), the fibers f − ( t ) for t ∈ [ − (cid:15), (cid:15) ] are smoothsubmanifolds. Thus, the canonical map Z f − ( −∞ , − (cid:15) ) → Z f − ( −∞ ,(cid:15) ) is also given by thecontinuation map of the wrapping { N ∗∞ ,out f − ( −∞ , t ) } t ∈ [ − (cid:15),(cid:15) ] which passes through Λtransversely exactly once at ( x, [ ξ ]). Extend this wrapping to a global one Ψ. Since thereis no other intersection with Λ, we can decompose Ψ to Ψ + − so that Ψ ± do notintersect Λ and Ψ only moves points near ( x, ξ ). This way, we can see D ( x,ξ ) can bepresented as a cofiber induced by an expanding open half-plane.35hus, D ( x,ξ ) can also be presented as a cofiber induced by inclusions of small balls. Proposition 4.16.
Let Λ ⊆ Λ (cid:48) be subanalytic singular isotropics and let D w Λ (cid:48) , Λ ( M ) denotethe fiber of the canonical map w sh Λ (cid:48) ( M ) → w sh Λ ( M ) . Then D w Λ (cid:48) , Λ ( M ) is generated bythe sheaf theoretical linking disk D ( x,ξ ) for smooth Legendrian points ( x, ξ ) ∈ Λ (cid:48) \ Λ .Proof. Let F ∈ w sh Λ ( M ). We assume that SS ∞ ( F ) is a subanalytic isotropic and pick aWhitney triangulation T such that SS ∞ ( F ) ⊆ N ∗∞ T . Fixed a particular way to construct F out of sheaves of the form M star( t ) for some M ∈ Perf Z by taking finite steps of cofibersand use { F i } { i ∈ A } to denote those M star( t ) ’s which show up in these steps. Note that itis possible that their microsupport SS ∞ ( F i ) intersect Λ. However, we see from the proofof the last lemma, the microsupport SS ∞ ( F i ) of these F i ’s are a smooth Legendrians in S ∗ M . Thus, we can apply Lemma 4.13 and have SS ∞ ( F i ) ∩ Λ (cid:48) = ∅ for i ∈ A and the samecofiber sequences build F from these F i ’s so SS ∞ ( F ) ∩ Λ (cid:48) = ∅ as well by microsupporttriangular inequality (1) of Proposition 2.36. Similarly, application of Lemma 4.13 impliesthat we can assume F has a cofinal wrapping sequence F → F w → F w → · · · such thatSS ∞ ( F w n ) ∩ Λ (cid:48) = ∅ . This implies that the canonical map (cid:103) w sh Λ (cid:48) ( M ) / (cid:16) C Λ ( M ) ∩ (cid:103) w sh Λ (cid:48) ( M ) (cid:17) → w sh Λ ( M ) := (cid:103) w sh Λ ( M ) / C Λ ( M )is an equivalence. Thus, we can apply Lemma B.5 to the diagram C Λ (cid:48) ( M ) (cid:103) w sh Λ (cid:48) ( M ) w sh Λ (cid:48) ( M ) C Λ ( M ) ∩ (cid:103) w sh Λ (cid:48) ( M ) (cid:103) w sh Λ (cid:48) ( M ) w sh Λ ( M ) i pj q which implies that D w Λ (cid:48) , Λ ( M ) = (cid:16) C Λ ( M ) ∩ (cid:103) w sh Λ (cid:48) ( M ) (cid:17) / C Λ (cid:48) ( M ). Now C Λ ( M ) is thecategory generated by the cofibers cof( c (Ψ , F )) of continuation maps whose wrapping ψ t (SS ∞ ( F )) avoids Λ, and C Λ ( M ) ∩ (cid:103) w sh Λ (cid:48) ( M ) is generated by a similar constructionexcept we now only requires the end points to avoid Λ. The claim is that the quotientis generated by the sheaf theoretic linking disks D ( x,ξ ) for smooth Legendrian points( x, ξ ) ∈ Λ (cid:48) \ Λ.Lemma 2.10 implies that if H → H → H is a cofiber sequence, then cof( c (Ψ , H )) → cof( c (Ψ , H )) → cof(( c (Ψ , H )) is also a fiber sequence. Thus, it’s enough to assume F has smooth Legendrian microsupport by the discussion at the beginning of the proof.Let Ψ be a positive isotopy such that ψ t (SS ∞ ( F )) does not touch Λ and, by generalposition argument, we may assume ψ t (SS ∞ ( F )) touches Λ (cid:48) for finitely many times andtransversally through one point p ∈ Λ (cid:48) \ Λ each time. Decomposing Ψ to Ψ = Ψ k · · · so that passing happens once during the duration of each Ψ i . Since c (Ψ) = c (Ψ k ) ◦ · · · ◦ c (Ψ ), it’s sufficient to prove the case with only one such passing at ( x, ξ ) by inductionwith Lemma 2.10. 36et q ∈ SS ∞ ( F ) be the point so that the path ψ t ( q ) pass ( x, ξ ) ∈ Λ (cid:48) and U smallopen ball near q in S ∗ M . We again decompose Ψ to Ψ = Ψ + − such that there’sno passing happing during Ψ ± and Ψ only moves points in U . In this case, c (Ψ ± , F )are isomorphisms and we can further assume Ψ only moves points in U . Now set F t =( K (Ψ) ◦ F ) | M ×{ t } so SS ∞ ( F t ) = ψ t (SS ∞ ( F )). We use one last general position argumentto assume the front projection π ∞ : SS ∞ ( F t ) → π ∞ (SS ∞ ( F t )) is finite near ψ t ( U ) so π ( U ∩ SS ∞ ( F t )) ⊆ π ( U ) is a hyperplane. Thus, we reduce to the local picture defining D ( x,ξ ) discussed in the last Remark 4.15.We combine the above two results to deduce a generation result for a special case.Let S be a Whitney triangulation. For each stratum s ∈ S , we pick a small ball B s whichcentered at X s and contained in star( s ) such that N ∗∞ ,out B s ∩ N ∗∞ S = ∅ and consider Z B s ∈ w sh N ∗∞ ,out S ( M ). This is possible because of the Whitney condition. Again differentchoice of such small balls induces the same objects in w sh N ∗∞ S ( M ) since they are isotopicto each other in the base by isotopies respecting the stratification and the lifting isotopieson the microsupport won’t touch N ∗∞ ( S ). (See [27]) Proposition 4.17.
The set { Z B s } s ∈ S generates w sh N ∗∞ S ( M ) under finite colimits andretractions.Proof. Set S ≤ k = { s ∈ S | dim X s ≤ k } . We claim, when k < n − { Z B s } s ∈ S ≤ k plus Z B for any small ball B whose closure B is disjoint from any stratum of dimension ≤ k generates w sh N ∗∞ S ≤ k ( M ). To see this, we note that the case k = − k < n − w sh N ∗∞ S ≤ k ( M ) → w sh N ∗∞ S ≤ k − ( M ). We note that M \ ∪ s ∈ S k +1 X s is path connected by standard Hausdorffdimension theory since these X s ’s have codimension ≥
2. Thus, Z B is independent ofthe choice of B . By Proposition 4.16, the fiber of the above projection is generatedunder finite colimits and retraction by sheaf theoretic linking disks D ( x,ξ ) for ( x, ξ ) ∈ N ∗ S ≤ k +1 \ N ∗ S ≤ k . But D ( x,ξ ) can be written as the cofiber cof( Z B → Z B s ) by the localpicture mentioned in Remark 4.15. Finally, apply a similar argument to the projection w sh N ∗∞ S ( M ) = w sh N ∗∞ S ≤ n − ( M ) → w sh N ∗∞ S ≤ n − ( M ) implies the proposition.Recall in the proof of Lemma 4.12, we show that Z B s → Z star( s ) is an isomorphismin w sh ∅ ( M ). The late object is, however, not an object in w sh N ∗∞ S ( M ). Instead, weconsider the object Z star( s ) − where star( s ) − is a small inward cornering of star( s ) ofDefinition 2.46. Choose B s small so that B s ⊆ star( s ) − . We claim that the canonicalmap Z B s → Z star( s ) − is an isomorphism. By the Yoneda embedding, it is an isomorphismif the corresponding morphism Hom w ( • , Z B s ) → Hom w ( • , Z star( s ) − ) is an isomorphism aspresheaves on w sh N ∗∞ S ( M ). The following two statements are directly paralleled withProposition 5.18, Lemma 5.21, and Proposition 5.24 in [13]. Lemma 4.18.
For a S -constructible relatively compact open set U , we have Hom w ( Z B s , Z U − ) = (cid:40) Z star( s ) ⊆ U otherwiseProof. The construction of U − tautologically provides a cofinal sequence Z U − (cid:15) soHom w ( Z B s , Z U − ) = colim (cid:15) → Hom( Z B s , Z U − (cid:15) ) . s ) ⊆ U . Since B s ⊆ star( s ) has a non-zero distant from ∂U , it’s contained in U − (cid:15) for (cid:15) << Z . When star( s ) ∩ U = ∅ ,the Hom is clearly 0 so we assume s is a stratum on the boundary of U . In this case, oneneeds to refine the wrapping by adding the centered of the ball B s to the stratificationand consider the family U − (cid:15), − δ to conclude the result. Proposition 4.19.
The canonical map Z B s → Z star( s ) − is an isomorphism in w sh N ∗∞ S ( M ) .Proof. We proceed by induction on the codimension of s . When s has codimension zero,we may replacing M by star( s ) and it becomes Lemma 4.12.Now the previous lemma and the proposition 4.17 impliesHom( Z B s , Z star( t ) − ) = Hom( Z star( s ) − , Z star( t ) − ) = 0for t of strictly smaller codimension than s . By induction, Z B t ∼ −→ Z star( t ) − for such t ’s. Thelater generates a subcategory which contains the fiber of the projection w sh N ∗∞ S ( T ∗ M ) → w sh N ∗∞ S ≤ dim s ( T ∗ M ). This implies that it’s enough to show the isomorphism in the cate-gory w sh N ∗∞ S ≤ dim s ( T ∗ M ). This is a special case fo the following lemma applying to thecase Y = star( s ) − , X = s ∩ star( s ) − , and Z = t ∩ star( s ) − . Lemma 4.20.
Let X m ⊆ Y n be an inclusion of stable balls, with ∂X ⊆ ∂Y . Assumethere exists another stable ball (with corners) Z m +1 ⊆ Y n such that ∂Z is the unionof X with a smooth submanifold of ∂Y . Then the canonical map Z B (cid:15) ( x ) → Z Y is anisomorphism in w sh N ∗∞ X ( Y ) for any x ∈ X .Proof. Reduce to the case of balls by stabilization. Then the situation becomes that Y is a unit ball, X is the intersection of Y with a linear subspace, and Z is the intersectionof Y with a closed half-plane with the boundary being the linear subspace. The positiveisotopy which expands Z B (cid:15) ( x ) to Z Y is disjoint from N ∗∞ X . Corollary 4.21.
The set { Z star( s ) − } s ∈ S generates w sh N ∗∞ S ( M ) under finite colimits andretractions. Let M be a real analytic manifold and Λ ⊆ S ∗ M a subanalytic singular isotropic. Wedefine in this section a comparison functor W +Λ ( M ) : w sh Λ ( M ) → Sh Λ ( M ) c and show that it is an equivalence of category. Since such functors combine to a compar-ison morphism W +Λ : w sh Λ → Sh c Λ between precosheaves, the last statement will impliesthat w sh Λ is in particular a cosheaf for this case. Let Λ ⊆ S ∗ M be a closed subset. Recall from Proposition 1.2 that the inclusionSh Λ ( M ) (cid:44) → Sh( M ) has a left adjoint given by the positive infinite wrapping functor W + ( S ∗ M \ Λ) : Sh( M ) → Sh Λ ( M ) . F to the limiting object over increasingly positive wrap-pings. Since a continuation map c : F → F w tautologically becomes an isomorphismafter applying W + ( S ∗ M \ Λ), the functor W + ( S ∗ M \ Λ) vanishes on C Λ ( M ). We denotethe resulting functor on the quotient category by W +Λ ( M ) : w sh Λ ( M ) → Sh Λ ( M )and, when there’s no possibility of confusion, simply by W +Λ . In general, the categoryon the right hand side is much larger. For example, when Λ = S ∗ M and M is non-compact, W + S ∗ M ( M ) is the trivial inclusion { } (cid:44) → Sh( M ). So we restrict to the casewhen Λ is a subanalytic singular isotropic. We first notice that in this case the restrictionof W + ( S ∗ M \ Λ) on w sh Λ ( M ) takes image in the subcategory consisting of compactobjects. Lemma 5.1.
Let Λ be a subanalytic singular isotropic. For F ∈ w sh Λ ( M ) , the sheaf W +Λ ( M )( F ) is a compact object.Proof. Let F ∈ w sh Λ ( M ) and lim −→ F i be a filtered colimit in Sh Λ ( M ). We compute,Hom( W +Λ F, lim −→ F i ) = colim Φ ∈ W ( S ∗ M \ Λ) Hom( w (Φ) ◦ F, lim −→ F i )= colim Φ ∈ W ( S ∗ M \ Λ) Hom(
F, w (Φ − ) ◦ lim −→ F i )= colim Φ ∈ W ( S ∗ M \ Λ) Hom (cid:0) F, lim −→ ( w (Φ − ) ◦ F i ) (cid:1) = colim Φ ∈ W ( S ∗ M \ Λ) Hom( F, lim −→ F i ) = Hom( F, lim −→ F i ) . Here, we use the fact that Φ is supported away from Λ ⊇ SS ∞ ( F i ) so w (Φ − ) ◦ F i = F i by Lemma 3.22. Now pick a Whitney triangulation S such that F is S -constructibleand Λ ⊆ N ∗ S . In this case, the Hom can be computed in Sh N ∗∞ S ( M ) = S -Mod.Since Sh N ∗∞ S ( M ) c consists exactly objects with compact support and perfect stalks, F is compact in Sh N ∗∞ S ( M ). Thus Hom( F, lim −→ F i ) = lim −→ Hom(
F, F i ) is compact and so W +Λ F ∈ Sh Λ ( M ) c is compact.We note that this map is compatible with the precosheaf structure on both side. Lemma 5.2.
Let j : U ⊆ M be an open set. The restriction j ∗ : Sh Λ ( M ) → Sh Λ | U ( U ) has left and right adjoints which are given by W + ( S ∗ M \ Λ) ◦ j ! and W − ( S ∗ M \ Λ) ◦ j ∗ .Hence, taking left adjoint induces a functor W + ( S ∗ M \ Λ) ◦ j ! : Sh Λ | U ( U ) c → Sh Λ ( M ) c between compact objects.Proof. We use the fact that a left adjoint of a left adjoint preserves compact objects.Note when Ω ⊆ Ω (cid:48) , there is equivalence W + (Ω (cid:48) ) ◦ W + (Ω) = W + (Ω) ◦ W + (Ω (cid:48) ) = W + (Ω (cid:48) ). Thus, by the above lemma, there is commuting diagram for an inclusion ofopens j : U ⊆ V : w sh Λ ( U ) Sh Λ | U ( U ) c w sh Λ ( V ) Sh Λ | V ( V ) c W +Λ ( T ∗ U ) W +Λ ( T ∗ V ) j ! W + ( S ∗ V \ Λ) ◦ j ! . efinition 5.3. We call the morphism W +Λ : w sh Λ → Sh c Λ between precosheaves definedby the above diagram as the comparison morphism.Similarly, when Λ ⊆ Λ (cid:48) , recall the left adjoint of the inclusion Sh Λ ( M ) (cid:44) → Sh Λ (cid:48) ( M ) isgiven by W + ( S ∗ M \ Λ) and thus there is a commuting diagram w sh Λ (cid:48) ( M ) Sh Λ (cid:48) ( M ) c w sh Λ ( M ) Sh Λ ( M ) c W +Λ ( M ) W +Λ (cid:48) ( T ∗ M ) W + ( S ∗ M \ Λ) . One can see this is compatible with the corestrictions on both side. Thus, there is acommuting diagram in precosheaves with coefficient in Pr L ω,st : w sh Λ (cid:48) Sh c Λ (cid:48) w sh Λ Sh c Λ W +Λ W +Λ (cid:48) . The main theorem of this paper, Theorem 1.3, is that the comparison functor W +Λ ( M ) : w sh Λ ( M ) → Sh Λ ( M )is an equivalence. As a corollary, the comparison morphism W +Λ : w sh Λ → Sh c Λ is an isomorphism so w sh Λ is a cosheaf. For the rest of the section, we work with a fixed pair ( M, Λ) such that Λ ⊆ S ∗ M is a sub-analytic singular isotropic. We would like to study the effect of W +Λ on the Hom-complex.Since W +Λ is defined by a colimit, the conical map Hom w ( G, F ) → Hom( W +Λ G, W +Λ F )can be obtained from the following few steps. As a colimit, there is a canonical map limiting continuation map F w → W +Λ F for any wrapping w . This induces, for any otherwrapping w (cid:48) , a map between the Hom-complex Hom( G w (cid:48) , F w ) → Hom( G w (cid:48) , W +Λ F ). Sinceconvoluting with w (Φ) is an auto-equivalence on Sh( M ), there is a canonical mapHom( G, F w ) = Hom( G w (cid:48) , ( F w ) w (cid:48) ) → Hom( G w (cid:48) , W +Λ ( F )) . Take limit over w (cid:48) and then colimit over w , we obtain the map between Hom-complexHom w ( G, F ) = colim w Hom(
G, F w ) → lim w (cid:48) Hom( G w (cid:48) , W +Λ ( F )) = Hom( W +Λ ( G ) , W +Λ ( F )) . In short, we have the following lemma. 40 emma 5.4.
Running F and G through a set of generators of w sh Λ ( M ) . If the limitingcontinuation map F → W +Λ F becomes an isomorphism after applying Hom( G, • ) for allsuch G , then the canonical map on Hom -complex
Hom w ( G, F ) → Hom( W +Λ ( G ) , W +Λ ( F )) is an isomorphism for any F, G ∈ w sh Λ ( M ) . Pick any cofinal functor Ψ : N → W ( S ∗ M \ Λ) which corresponds to a sequence ofwrappings id Ψ −→ Φ −→ Φ → · · · . For convenience, we scale it so that Ψ i has domain S ∗ M × [ i, i + 1]. This sequence of positive family of isotopies patches to a positive isotopyΨ : S ∗ M × [0 , ∞ ) → S ∗ M whose restriction on S ∗ M × [ i, i +1] is Ψ i i . Note Ψ has a non-compact support by the cofinal criterion Lemma 3.20. By the GKS sheaf quantization,there is a sheaf kernel K (Ψ) on M × M × [0 , ∞ ) such that K (Ψ) | M × M × [ i,i +1] = K (Ψ i ).As before, for F ∈ (cid:103) w sh Λ ( M ), set F Ψ = w (Ψ) ◦ F and let F w n denote F Φ n = F Ψ | M ×{ n } the resulting sheaves under the wrapping Ψ. It’s enough to study the morphismHom( G, F w n ) → Hom( G, W +Λ ( F ))which is induced from the sequence of wrappings F → F w → · · · → F w n → · · · → W +Λ ( F ) . Definition 5.5.
Let X be a topological space, j : X × [0 , ∞ ) (cid:44) → X × [0 , ∞ ] and i : X × {∞} → X × [0 , ∞ ] be the inclusions as open and closed subset. We call thecomposition ψ = i ∗ ◦ j ∗ : Sh( X × [0 , ∞ )) → Sh( X ) the nearby cycle functor. Lemma 5.6.
The colimit W +Λ F can be computed as the nearby cycle at infinity of thesheaf F Ψ . That is, W +Λ F = ψF Ψ .Proof. Since Ψ n are cofinal, W +Λ ( F ) = colim n ∈ N F w n . By the construction above, for each n > F w n is given by Z { n } ◦ F Ψ ∼ −→ Z (0 ,n ) [1] ◦ F Ψ and the continuation map betweenthem is induced by Z (0 ,n ) → Z (0 ,m ) for m ≥ n . Since convolution commutes with colimit, W +Λ ( F ) = (colim n ∈ N Z (0 ,n ) [1]) ◦ F Ψ = Z (0 , ∞ ) [1] ◦ F Ψ = p ! F Ψ [1] where p : M × [0 , ∞ ) → M isthe projection. The latter is the same as ψF Ψ = i ∗ j ∗ F Ψ by Lemma 3.1.Inspired by the lemma above, we would like to realize the map Hom( G, F w n ) → Hom(
G, F Ψ ) as a limiting continuation map for some sheaf on M × [1 , ∞ ) whose restrictionon M × { n } , n ∈ Z > ∪ {∞} , are the corresponding Hom-complex. A natural candidateis the sheaf H om( p ∗ G, F Ψ ).We first recall some machinery from general microlocal sheaf theory. Let F ∈ Sh( M × J ) where J is an open interval with parameter t . We say that F is J -noncharacteristic ifthe inclusion i t : M ×{ t } (cid:44) → M × J is noncharacteristic for F for all t ∈ J . Equivalently, F is J -noncharacteristic if SS( F ) ∩ ( T ∗ M M × T ∗ J ) ⊆ T ∗ M × J M × J . As before, we compactify J to J + = J ∪{∞} to the right and let j : M × J (cid:44) → M × J + and i ∞ : M ×{∞} (cid:44) → M × J + denote the open and closed inclusions. Lemma 5.7.
Let F ∈ Sh( M × J ) be J -noncharacteristic. Let q : M × J → J be theprojection and assume q is proper on supp( F ) . Then,1. The natural morphism i ∗ t F [ − → i ! t F is an isomorphism for t ∈ J .2. The sheaf q ∗ F is locally constant. In particular, the global section Γ( M : i ∗ t F ) isconstant on t ∈ J . f we assume further that supp( ψF ) is compact, then the constancy of Γ( M ; i ∗ t F ) extendsto t ∈ J + .Proof. Let p : M × J → M denote the projection. The equivalence i ∗ t F [1] = i ! t F followsdirectly from (4) of Proposition 2.36 and the observation that since p ◦ i t = id, one has i ! t Z M × J = i ! t ◦ p ! Z M [1] = Z M [1] . Since q is proper on supp( F ), we have the microsupport estimation SS( q ∗ F ) ⊆ q π (SS( F ) ∩ T ∗ M M × T ∗ J ) ⊆ T ∗ J J by (3) of Proposition 2.36 and so q ∗ F is a local sys-tem. Since J is contractible, the local system q ∗ F is constant. The last statement is aconsequence of base change 2.17 and the fact that ∗ -pushforward sends locally constantsheaves on J to locally constant sheaves on J + . Lemma 5.8.
Let
F, G ∈ Sh( M × J ) be sheaves on M × J such that F and H om( G, F ) are J -noncharacteristic, and q is proper on supp( G ) and supp( ψG ) is compact. Then,1. The natural map i ∗ s H om( G, F ) → H om( i ∗ s G, i ∗ s F ) is an isomorphism.2. The Hom -complex
Hom( i ∗ s G, i ∗ s F ) is contant on s ∈ J and equals to Γ ( M ; ψ H om( G, F )) .Proof. Apply the last lemma and we compute, i ∗ s H om( G, F ) = i ! s H om( G, F )[ − H om( i ∗ s G, i ! s F )[ − H om( i ∗ s G, i ∗ s F ) . Since supp( H om( G, F )) ⊆ supp( G ) and q is proper on supp( G ), we apply the lastlemma and conclude that q ∗ H om( G, F ) is locally constant. Then base change 2.17 appliesand one can compute( q ∗ H om( G, F )) s = Γ( q − ( s ); i ∗ t H om( G, F )) = Γ( M ; H om( i ∗ s G, i ∗ s F )) = Hom( i ∗ s G, i ∗ s F )is constant. Because supp( ψG ) is compact, the constancy extends to Γ ( M ; ψ H om( G, F )).Since ∞ is a boundary point, we cannot conclude equivalence using transversality.Such situation is considered by Nadler and Shende in [31] in which they developed thetheory of nearby cycle to study the canonical mapΓ ( M ; ψ H om( G, F )) → Hom( ψG, ψF )which we recall now.
Definition 5.9 ([31, Definition 2.2]) . A closed subset X ⊆ S ∗ M is positively displaceablefrom legendrians (pdfl) if given any Legendrian submanifold L (compact in a neighbor-hood of X ), there is a 1-parameter positive family of Legendrians L s , s ∈ ( − (cid:15), (cid:15) ) (constantoutside a compact set), such that L s is disjoint from X except at s = 0. Definition 5.10 ([31, Definition 2.7]) . Fix a co-oriented contact manifold ( V , ξ ) andpositive contact isotopy η s . For any subset Y ⊆ V we write Y [ s ] := η s ( Y ). Given Y, Y (cid:48) ⊆ V we define the chord length spectrum of the pair to be the set lengths of Reebtrajectories from Y to Y (cid:48) : cls ( Y → Y (cid:48) ) = { s ∈ R | Y [ s ] ∩ Y (cid:48) (cid:54) = ∅ } we term cls ( Y ) := cl ( Y → Y ) the chord length spectrum of Y .42 efinition 5.11 ([31, Definition 2.9]) . Given a parameterized family of pairs ( Y b , Y (cid:48) b ) in S ∗ M over b ∈ B we say it is gapped if there is some interval (0 , (cid:15) ) uniformly avoided byall cl ( Y b → Y (cid:48) b ). In case Y = Y (cid:48) , we simply say Y is gapped. Definition 5.12 ([31, Definition 3.17]) . Given a subset X ⊆ T ∗ ( M × J ), we define its nearby subset as ψ ( X ) := Π( X ) ∩ T ∗ ( M × ( − , ∞ ]) | M ×{∞} . The main theorem for the nearby cycles is the following:
Theorem 5.13 ([31, Theorem 4.2]) . Let F , G be sheaves on M × J . Assume1. SS( F ) and SS( G ) are J -noncharacteristic;2. ψ (SS( F )) and ψ (SS( G )) are pdfl;3. The family of pairs in S ∗ M determined by (SS π ( F ) , SS π ( G )) is gapped for somefixed contact form on S ∗ M .Then Γ( M ; ψ H om( F, G )) → Hom( ψ ( F ) , ψ ( G )) is an isomorphism. Now we apply the theory of nearby cycle to the infinite wrapping functor.
Lemma 5.14.
Let F → F w → F w → · · · be a sequence in (cid:93) w sh Λ ( M ) as in Lemma 5.6.If for any conic open neighborhood U of Λ , there exists n such that SS( F w n ) ⊆ U . Thenthe sequence is cofinal.Proof. Pick a decreasing conic open neighborhood U ⊇ U ⊇ U ⊇ · · · ⊇ Λ of Λ suchthat U n ⊆ U n +1 and ∩ n U n = Λ. By taking a subsequence of the F w n ’s, we may assumeSS( F w n ) ⊆ U n for n ≥
1. Again by taking a subsequence of both the F w n ’s and U n ’s, wemay further assume U n +1 ∩ SS( F w n ) = ∅ . By the locality property 3.22, the continuationmap c : F w n → F w n +1 depends only on the value of Ψ on U n . Thus, we may modify Ψon T ∗ M \ U n − to satisfy the condition in Lemma 3.20. Since taking subsequence won’tchange the colimit, the original sequence is cofinal. Theorem 5.15.
Let F ∈ (cid:93) w sh Λ ( M ) . Assume there is a sequence of wrappings Φ −→ Φ → · · · which glues to a (non-compactly supported) positive contact isotopy Ψ : S ∗ M × [0 , ∞ ) → S ∗ M such that, for any neighborhood U of Λ , ψ s (SS ∞ ( F )) ⊆ U for s >> .Then for G ∈ (cid:93) w sh Λ ( M ) the canonical map, Hom(
G, F w n ) → Hom(
G, ψF Ψ ) is an isomorphism for n >> . Thus, the canonical map Hom w ( G, F ) → Hom( W +Λ G, W +Λ F ) is an isomorphism. roof. By Lemma 3.20, the sequence F w n is cofinal and W +Λ F is computed by colim n F w n .Thus, the first statement implies that W +Λ induces isomorphisms on the Hom-complex byLemma 5.4.Now note that SS ∞ ( G ) in S ∗ M is compact since supp( G ) is compact and the frontprojection π ∞ : S ∗ M → M is proper. Since a manifold is in particular a regular topolog-ical space, there exist open sets U and V containing Λ and SS ∞ ( G ) such that U ∩ V = ∅ .By restricting to n >>
0, we may assume ψ s (SS ∞ ( F )) ⊆ U is thus disjoint from SS ∞ ( G ),which implies that H om( p ∗ G, F Ψ ) is J -noncharacteristic. Lemma 5.8 then implies thatHom( G, F w n ) = Γ( M ; ψ H om( p ∗ G, F Ψ )).So it’s sufficient to check the conditions of Theorem 5.13 hold for the pair p ∗ G and F Ψ . The set SS( p ∗ G ) = SS( G ) × J is tautologically J -noncharacteristic. For F Ψ , werecall that ˙SS( F Ψ ) = Λ Ψ ◦ F = { (Ψ( x, ξ, t ) | ( x, ξ ) ∈ ˙SS( F ) , t ∈ [0 , ∞ ) } which implies that F Ψ is J -noncharacteristic.By picking a shrinking neighborhood V n of Λ, we see that the nearby set ψ (SS ∞ ( F Ψ ))is contained in Λ. One can pick a Whitney triangulation S such that Λ ⊆ N ∗∞ S . Similarly,up to an isotopy, there exists a Whitney triangulation T such that SS ∞ ( G ) ⊆ N ∗∞ T . Thesingular isotropics N ∗∞ S and N ∗∞ T are pdfl by Lemma 4.13.Now the same argument showing H om( p ∗ G, F Ψ ) is J -noncharacteristic implies thatthere is an N ∈ N such that ψ s (SS ∞ ( F )) ⊆ V for s ≥ N . Thus, when restricting to M × [ N, ∞ ), p ∗ G and F Ψ are microlocally disjoint and the gapped condition is tautologicallysatisfied. Corollary 5.16.
If there exists a set of generators S in (cid:93) w sh Λ ( M ) such that each F ∈ S admits a sequence F w n constructed in the above manner such that SS ∞ ( F w n ) is containedin arbitrary small neighborhood of Λ for n large, then the comparison functor W +Λ : w sh Λ ( M ) (cid:44) → Sh Λ ( M ) c is fully faithful. We first consider the special case when Λ = N ∗∞ S for some Whitney triangulation S . Theorem 5.17.
The comparison functor W + N ∗∞ S : w sh N ∗∞ S ( M ) → Sh N ∗∞ S ( M ) c is an equiv-alence.Proof. By Proposition 4.12 and Proposition 4.19, w sh S ( T ∗ M ) has { Z star( s ) − } as a set ofgenerators. Recall that Z star( s ) − is defined to be an unspecified inward cornering star( s ) − (cid:15) for small enough (cid:15) . As mentioned in Definition 2.46, the construction of star( s ) − (cid:15) ismade so that N ∗∞ ,out star( s ) − (cid:15) is disjoint from N ∗∞ S and is contained in arbitrary smallneighborhood of N ∗∞ S as (cid:15) →
0. Since SS( Z star( s ) − (cid:15) ) = N ∗∞ ,out star( s ) − (cid:15) , Theorem 5.15applies and W + N ∗∞ S is fully faithful. We see from the same generators Z star( s ) − that W + N ∗∞ S is essential surjective since { Z star( s ) } form a set of generators of Sh N ∗∞ S ( M ) c by Proposition2.47.To prove the general case, we have to match a special class of objects on both sides Lemma 5.18.
Let Λ be a subanalytic singular isotropic and ( x, ξ ) ∈ Λ be a smooth point.For any F ∈ Sh( M ) such that SS( F ) is contained in Λ near ( x, ξ ) , there is an equivalence Hom( W +Λ D ( x,ξ ) , F ) = µ ( x,ξ ) F. hat is, the object W +Λ D ( x,ξ ) co-represents µ ( x,ξ ) .Proof. Recall D ( x,ξ ) is defined to be the cofiber of the canonical map Z f − ( −∞ , − (cid:15) ) → Z f − ( −∞ ,(cid:15) ) ) where f is a proper analytic function defined near x satisfying the followingconditions: There exists an (cid:15) >
0, so that f has only one Λ-critical point x over f − [ − (cid:15), (cid:15) ]with f ( x ) = 0, df x = ξ and f − ( −∞ , (cid:15) ) is relatively compact. Proposition 2.32 impliesthe function f defines a microstalk functor µ ( x,ξ ) , by µ ( x,ξ ) ( F ) := Γ { f ≥ } ( F ) x . As in Remark 4.15, the local picture of the fibers f − ( { t } ) for t ∈ [ − (cid:15), (cid:15) ] is a hyperplanenear x . Let Ψ denote any global extension of the wrapping N ∗∞ ,out f − ( −∞ , t ) for t ∈ [ − (cid:15), (cid:15) ]. If we modify Ψ to Ψ by multiplying a bump function supported near ( x, ξ ) on itsHamiltonian, the resulting wrapping ( Z f − ( −∞ ,(cid:15) ) ) Ψ will appear as expanding f − ( −∞ , (cid:15) )to some large open set where the expansion happens only near x in the ξ codirection.The cofiber cof( c (Ψ , Z f − ( −∞ ,(cid:15) ) ) can be seen as the cofiber induced by some small openneighborhood U and its open subset U ∩ Z f − ( −∞ ,(cid:15) ) .Since the graph Γ df does not intersection Λ except at ( x, ξ ), there is an isomoprhismΓ( f − ( −∞ , (cid:15) ); F ) = Γ( f − ( −∞ , F ) by the non-characteristic deformation lemma 2.25.Thus, Γ { f ≥ } ( F ) x can be computed as a colimitΓ { f ≥ } ( F ) x = colim Ψ Hom(cof (cid:0) Z f − ( −∞ , → ( Z f − ( −∞ , ) w (cid:1) , F )= colim Ψ Hom(cof (cid:0) Z f − ( −∞ ,(cid:15) ) → ( Z f − ( −∞ , ) w (cid:1) , F )by picking Ψ so that the corresponding U as above forms a neighborhood basis of x .Similarly, Hom(( Z f − ( −∞ , ) w , F ) can be replaced by Γ( f − ( −∞ , (cid:15) ); F ) such that the mapsare compatible with inclusions of the corresponding open sets. That is, we are takingcolimit over a constant and thusΓ { f ≥ } ( F ) x = colim Ψ Hom(cof (cid:0) Z f − ( −∞ ,(cid:15) ) → ( Z f − ( −∞ , ) w (cid:1) , F )= colim Ψ Hom(cof( Z f − ( −∞ , − (cid:15) ) → Z f − ( −∞ ,(cid:15) ) ) , F )= Hom(cof( Z f − ( −∞ , − (cid:15) ) → Z f − ( −∞ ,(cid:15) ) ) , F ) = Hom( D ( x,ξ ) , F )Finally, we recall that W +Λ is defined by the restriction of the left adjoint. Since leftadjoints preserve corepresentatives, we conclude that Γ { f ≥ } ( F ) x = Hom( W +Λ D ( x,ξ ) , F ).45 emark . Corepresentatives of the microstalk functors µ ( x,ξ ) : Sh Λ ( M ) → Z -Mod arefrequently considered since they often provide a preferred set of generators. For example,Zhou in [43] finds an explicit description of corepresentatives in the case of FLTZ skeletonfirst considered in [9], and uses it to match them with certain line bundles on the coherentside, which gives an explicit description to the equivalence proved in [24] through descentargument. A common recipe for finding such a description is to first find a sheaf F whichis constructed locally near x , and is thus not necessarily in Sh Λ ( M ), but still satisfiesthe identification Hom( F, • ) = µ ( x,ξ ) on Sh Λ ( M ). Then one constructs a one-parameterfamily of sheaves F t , t ∈ [0 , F = F , F ∈ Sh Λ ( M ), and Hom( F t , • ) remainsconstant as t varies. This lemma can be seen as a general recipe for such a constructionwhen subanalytic structure is presented. Proof of Theorem 1.3.
Pick a Whitney triangulation S such that Λ ⊆ N ∗∞ S . We use D wN ∗∞ S , Λ ( M ) denote to the subcategory in w sh N ∗∞ S ( M ) generated by the sheaf theoret-ical linking discs D ( x,ξ ) at Legendrian points of N ∗∞ S \ Λ and, similarly, D µN ∗∞ S , Λ ( M )the subcategory in Sh N ∗∞ S ( M ) generated by the corresponding microstalk representa-tives. By Proposition 2.56 and Proposition 4.16, they are the fiber of the projections w sh N ∗∞ S ( M ) → w sh Λ ( M ) and Sh N ∗∞ S ( M ) c → Sh Λ ( M ) c respectively. Thus, there is acommuting diagram D wN ∗∞ S , Λ ( M ) w sh N ∗∞ S ( M ) w sh Λ ( M ) D µN ∗∞ S , Λ ( M ) Sh N ∗∞ S ( M ) c Sh Λ ( M ) c W + N ∗∞ S W +Λ . The last lemma implies that the equivalence W + N ∗∞ S : w sh S ( T ∗ M ) ∼ −→ Sh S ( M ) c restricts to W + N ∗∞ S : D wN ∗∞ S , Λ ( M ) ∼ −→ D µN ∗∞ S , Λ ( M ) . Hence, Lemma B.5 implies that W +Λ is an equivalence as well. A Homogenous symplectic geometry and contact ge-ometry
We recall some facts about the homogenous symplectic geometry and contact geometryand how they are interchangeable with each other. We assume the contact manifolds inthis section are co-orientable. Let (
X, dα ) be a Liouville manifold and let Z denote itsLiouville vector field. We define a homogeneous symplectic manifold to be a Liouvillemanifold such that the Liouville flow induces a proper and free R > -action. In this case,the quotient X/ R > is a manifold. Definition A.1.
A subset Y ⊆ ( X, dα ) is conic if it is preserved under the R > -action. Proposition A.2.
A coisotropic submanifold Y is conic if and only if α | T Y dα = 0 .Proof. Y is coisotropic iff
T Y dα ⊆ T Y . Y is conic if and only if Z ( y ) ∈ T y Y for y ∈ Y which implies α y ( w ) = dα y ( Z ( y ) , w ) = 0 for all w ∈ T Y dα . Note this direction alwaysholds. On the other hand, the same equation implies that Z ( y ) ∈ T Y if α | T Y dα = 0.Since Y is coisotropic, Z ( y ) is in particular in T y Y .46 orollary A.3. A Lagrangian submanifold L ⊆ ( X, dα ) is conic if and only if α | L = 0 . Example A.4.
The Liouville vector field Z of the cotangent bundle T ∗ N can be writtenlocally by Z = (cid:80) ξ i ∂ ξ i where the ξ i ’s are the dual coordinates of local coordinates x i of N . The Liouville flow is given by Φ Zs ( x, ξ ) = ( x, e s ξ ) and the R > -action is simply themultiplication, r · ( x, ξ ) = ( x, rξ ) for r ∈ R > . Proposition A.5.
The one form α descends to a one form α on X/ R > .Proof. Let X denote the Liouville vector field associated to α (which is non-vanishingsince R > acts freely). By definition, α ( Z ) = ω ( Z, Z ) = 0 so it defines a section on(
T X/ (cid:104) Z (cid:105) ) ∗ = T ∗ ( X/ R > ). This is a contact form since on X , ι Z ω ∧ d ( ι Z ω ) n − = ι Z ω ∧ ( L Z ω ) n − = ι Z ω ∧ ω n − = n ι Z ω n − and T ( X/ R > ) can be identified as vectors transversalto Z . Example A.6.
The example we will study in this paper is the cotangent bundle awayfrom the zero section ˙ T ∗ V for some smooth manifold V . Pick a metric g and restrict theprojection p : ˙ T ∗ V → S ∗ V to { ( x, ξ ) | g x ( ξ, ξ ) = 1 } and denote it as p g . The map p g isa diffeomorphism because its domain is transversal to the R > -action and p g is clearlyone-to-one. Its inverse s : S ∗ V → ˙ T ∗ V provides S ∗ V a global contact form s ∗ α can .Note any such section gives the same contact structure but there might not be anycontactomorphism sending one contact form to another. A more intrinsic description ofthe contact structure is η [ x,ξ ] = ker ξ . Lemma A.7.
A homogeneous symplectomorphism ψ : ( X, dα ) → ( Q, dβ ) preserves theLiouville form, i.e., ψ ∗ β = α .Proof. let ψ be homogeneous and ψ ∗ dβ = dα . We denote the Liouville vector fields by Z and Y and the corresponding flow by φ Zt and φ Yt , t ∈ R . Since ψ is a homogeneoussymplectomorphism, we have ψ ( φ Zt ( x )) = φ Yt (Ψ( x )) for all x ∈ X . Differentiate theequation and evaluate at 0, we obtain that dψ x ( Z ( x )) = Y ( ψ ( x )), i.e., Y = ψ ∗ Z . So forany differential ( p -)form ν on Q ,( ψ ∗ ( ι Y ν )) ( v , · · · , v p − ) = ( ι Y ν )( ψ ∗ v , · · · , ψ ∗ v p − )= ν ( Y, ψ ∗ v , · · · , ψ ∗ v p − )= ( ψ ∗ ν )( Z, v , · · · , v p − )= ( ι Z (Ψ ∗ ν )) ( v , · · · , v p − ) . That is, ψ ∗ ◦ ι Y = ι Z ◦ ψ ∗ . In particular, ψ ∗ α = ψ ∗ ι Z dα = ι Y ψ ∗ dα = β . Proposition A.8.
A co-orientation preserving contactomorphism ϕ : ( N, ξ ) → ( P, η ) gives rise to a unique homogeneous symplectomorphism ˜ ϕ : SN → SP between theirsymplectizations. On the other hand, a homogeneous symplectomorphism ψ : ( X, dα ) → ( Q, dβ ) induces a contactomorphism on the contact quotient in Proposition A.5. Thesetwo constructions are inverse to each other if X and Q come from symplectization.Proof. Assume (
N, ξ ) and (
P, η ) are co-oriented by α and β . The equation dφ x ( ξ x ) = η ϕ ( x ) implies that ϕ ∗ β = hα for some h >
0. More precisely, let R be the Reeb vector fieldof α , then h = β ( ϕ ∗ R ). Define ˜ ϕ : ( N × R > , d ( tα )) → ( P × R > , d ( sβ )) by ˜ ϕ ( x, t ) =( ϕ ( x ) , ( h ( x )) − t ). Then ˜ ϕ ∗ sβ = t h ϕ ∗ β = tα so ˜ ϕ is a homogeneous symplectomorphism.Now assume there is another lifting ˜ ϕ (cid:48) . Since they both descend to ϕ , there is g > ϕ (cid:48) ( x, t ) = g ( x ) ˜ ϕ ( x, t ). But then tα = ( ˜ ϕ (cid:48) ) ∗ tβ = gϕ ∗ tβ = gtα so g ≡
1. Since ψ preserves the Liouville form, it’s clear that ψ descends to a contactomorphism on thequotient. And we also see the two constructions are inverse to each other when thehomogeneous symplectic manifolds are given by symplectization. Example A.9.
We consider the case when X = Q = ˙ T ∗ M the cotangent bundle awayfrom the zero section M . One can identify it as the symplectization of S ∗ M by pickinga metric g . Let ϕ : S ∗ M → S ∗ M be a co-orientation preserving contactomorphism andwe would like to lift it to a homogeneous symplectomorphism ˆ ϕ : ˙ T ∗ M → ˙ T ∗ M .We describe here how the identification intertwines with the construction in the propo-sition. Denote s the section of p : ˙ T ∗ M → S ∗ M which is given by the unit covectors.We claim that there is a (unique) section t : S ∗ M → ˙ T ∗ M so that ϕ ∗ ( t ∗ α can ) = s ∗ α can .(Note we cannot just require t = s ◦ ϕ − since this would implies id S ∗ M = ϕ − .) If such t exists, then t ∗ α can = ( ϕ − ) ∗ s ∗ α can = hs ∗ α can for some h ∈ C ∞ ( S ∗ M ; R > ) given by ϕ .So we simply define t : S ∗ M → ˙ T ∗ M by t = h · s where · is the R > action. Then we candefine ˆ ϕ : ˙ T ∗ M → ˙ T ∗ M by ˆ ϕ = √ g · ( t ◦ φ ◦ p ). One can compute thatˆ ϕ ∗ α can = √ g · ( p ∗ ◦ ϕ ∗ ◦ t ∗ α can )= √ g · ( p ∗ ◦ s ∗ α can )= √ g · ( s ◦ p ) ∗ α can = √ g · √ g α can = α can is symplectomorphic. Note that we use s ◦ p ( x, ξ ) = (cid:16) / (cid:112) g x ( ξ, ξ ) (cid:17) ( x, ξ ) for the secondto last equality.Now consider a family of isotopy ϕ t : S ∗ M → S ∗ M such that ϕ = id S ∗ M . Therequirement ( ϕ − t ) ∗ s ∗ α can = h t s ∗ α can ensures h t > h ≡
1. We can then lift ϕ t toa family of homogeneous symplectomorphism ˆ ϕ t : ˙ T ∗ M → ˙ T ∗ M by the above process.Since this process can be reserved, we see that there is a one-to-one correspondencebetween contact isotropy on S ∗ M and homogeneous isotropy on ˙ T ∗ M . Note that thefamily version of isotopies works similarly. B Quotients of small stable categories
We discuss quotients of small idempotent complete stable ∞ -categories. Our main ref-erence is section 5.5.7 of [25]. We first recall the definition of the following three (verylarge) categories, Pr R ω , Pr L ω , and cat. Definition B.1.
The category Pr R ω is the (very large) category consisting of compactlygenerated categories whose morphisms are functors preserving limits and filtered colimits.The category Pr L ω is the (very large) category consisting of compactly generated categorieswhose morphisms are functors preserving colimits and compact objects. The categorycat ω is the (very large) category consisting of small categories which are idempotentcomplete whose morphisms are simply functors.The category Pr R ω admits small limits which can be computed in Cat. The categoryPr L ω is equivalent to (Pr R ω ) op by passing to right adjoints which exist by the adjoint functortheorem. Because colimits turn to limits when passing to adjoints, Pr L ω admits small48olimits. Finally, there is equivalence of category Pr L ω ∼ = cat which is given by takingcompact objects C (cid:55)→ C c with inverse given by taking Ind-objects C (cid:55)→ Ind C . Theoperation of taking adjoints or Ind-completion are compatible with stable structures sowe can consider the subcategory Pr L ω,st and st ω of Pr L ω and cat ω which consist of the stableones.Take a small idempotent complete stable category C ∈ st ω and a collection of objects S . We would like to construct an associated localization C → C / S so that a morphism f : X → Y with cof( f ) ∈ S becomes an isomorphism in C / S . This localization canbe defined as a quotient in the following way. First, we take the stable subcategory (cid:104) S (cid:105) generated by S , and then take its idempotent completion which we will denote itby N = N ( S ). Abstract non-sense implies N is still stable and is embedded in C asthe subcategory of retracts of objects in (cid:104) S (cid:105) because C is idempotent complete. Let ι ∗ : N (cid:44) → C denote the inclusion. Definition B.2.
We define the quotient C /S of C by S as the cofiber cof ( ι ∗ ) taken incat ω and use j ∗ : C (cid:16) C /S to denote the projection. The category C /S is stable bythe description below so it is also the cofiber in st ω .In more concrete terms, we pass to the large categories by taking the ind-completionInd. The cofiber cof(Ind( ι ∗ )) can be computed as fib(Ind( ι ∗ ) R ) which is identified asthe subcategory (Ind( N )) ⊥ ,r consisting of objects right orthogonal to Ind( N ). Then,cof ( ι ∗ ) in cat is the compact objects the (cof(Ind( ι ∗ ))) c which is automatically idempotentcomplete. Note that since Ind( C ) → cof(Ind( ι ∗ )) preserves compact objects, there is thefactorization j ∗ : C (cid:16) C /S . We sketch a proof regarding the computation of the Homin C /S . Proposition B.3.
Let X , Y be objects in C . Then the Hom in C /S can be computedas a colimit, Hom C /S ( j ∗ X, j ∗ Y ) = colim Y → Y (cid:48) Hom C ( X, Y (cid:48) ) where the colimit runs through the morphism Y → Y (cid:48) whose cofiber is in N . Alternatively,we can compute the Hom-spaces by varying the first component, i.e., Hom C /S ( j ∗ X, j ∗ Y ) = colim X (cid:48) → X Hom C ( X (cid:48) , Y ) with cof( X (cid:48) → X ) ∈ N . To prove the proposition, we first look more closely into the construction. Begin withthe inclusion N ι ∗ (cid:44) −→ C , we translate to the category Pr L ω by taking Ind and obtainInd( N ) Ind( ι ∗ ) (cid:44) −−−→ Ind( C ) . Because Ind( ι ∗ ) preserves small colimits, it admits a right adjoint Ind( ι ∗ ) R . Lemma B.4.
Let ι ∗ : N (cid:44) → C be inclusion of idempotent complete small categories. For X ∈ C (cid:44) → Ind( C ) , the right adjoint of Ind( ι ∗ ) can be given by the formula Ind( ι ∗ ) R ( X ) = “ colim α : Z → X, Z ∈ N ” Z. Here we use the quotation “ colim ” to emphasis the colimit is taken formally in
Ind( N ) .Note that we do not require C or N to be stable for this lemma. roof. By definition, the formal colimit “ colim α : Z → X, Z ∈ N ” Z is an object of Ind( N ). Becausean object of Ind( N ) is of the form “ colim ” W over some filtered colimit by some objectsin N , it’s sufficient to show, for all W ∈ N ,Hom C ( ιW, X ) = Hom Ind( N ) ( W, “ colim α : Z → X, Z ∈ N ” Z ) . Since Ind( ι ) preserves compact objects, we computeHom Ind( N ) ( W, “ colim α : Z → X, Z ∈ N ” Z ) = colim α : Z → X, Z ∈ N Hom N ( W, Z )= Hom C ( W, X ) . Proof of the proposition B.3.
The cofiber cof(Ind( ι ∗ )) can be computed as fib(Ind( ι ∗ ) R )which is identified as the subcategory (Ind( N )) ⊥ ,r of objects right orthogonal to Ind( N ).Because we are in the stable setting, this gives us a fiber sequenceInd( ι ∗ )Ind( ι ∗ ) R → id → j ∗ j ∗ . Thus, for
X, Y ∈ C , one computesHom C /S ( j ∗ X, j ∗ Y ) = Hom C ( X, j ∗ j ∗ Y )= Hom C (cid:0) X, cof(Ind( ι ∗ )Ind( ι ∗ ) R Y → Y ) (cid:1) = Hom C (cid:18) X, cof(“ colim α : Z → Y, Z ∈ N ” Z → Y ) (cid:19) = colim α : Z → Y, Z ∈ N Hom C ( X, cof( Z → Y ))= colim Y β −→ Y (cid:48) , cof( β ) ∈ N Hom C ( X, Y (cid:48) ) . Here, we notice the last equation is simply changing the expression of the same colimit.To obtain the similar formula which we varies the first component, we notice that thereis equivalence ( C /S ) op = C op / N op because they satisfies the same universal property. Wethus compute Hom C /S ( j ∗ X, j ∗ Y ) = Hom ( C /S ) op ( j ∗ Y, j ∗ X )= Hom C op / N op ( j ∗ Y, j ∗ X )= colim X γ ←− X (cid:48) , cof( γ ) ∈ N op Hom C op ( Y, X (cid:48) )= colim X (cid:48) γ −→ X, cof( γ ) ∈ N Hom C ( X (cid:48) , Y ) . We will use the following ”snake lemma” for categories in the main text.
Lemma B.5.
Consider the following diagram in Pr L ω,st : C C D D D i pj qF F ¯ F where p and q are the quotient functor of the inclusion i and j , F is the restrictionof F which factors through D and ¯ F is the induced functor between the quotients. Let ι : fib( ¯ F ) (cid:44) → C denote the fiber of ¯ F , π : D (cid:16) cof( F ) the cofiber of F and ∂ :fib(¯( F )) → cof( F ) the functor given by the composition ∂ = π ◦ j R ◦ F ◦ p R ◦ ι . If F isan equivalence, then ∂ is an equivalence.Proof. For simplicity, we assume C = D and F is the identity so the diagram becomes,fib( ¯ F ) C C C D C D cof( F ) i pj qp R j R ιF ¯ Fπ We will prove that the functor θ := ι R ◦ p ◦ j ◦ π R is the inverse by showing that θ ◦ ∂ = id fib( ¯ F ) . The equation ∂ ◦ θ = id cof( F ) can be proved similarly. Recall that in thestable setting, the sequence C (cid:44) → C (cid:16) C comes with a fiber sequence of functors ii R → id C → p R p. First write out θ∂ as ι R pjπ R πj R p R ι . Apply this fact to C (cid:44) → D (cid:16) cof( F ), we see thereis a fiber sequence F F R → id → π R π. Apply j ◦ ( • ) ◦ j R and the fiber sequence becomes ii R → jj R → jπ R πj R . Further apply p ◦ ( • ) ◦ p R and the we see that pjj R p = pjπ R πj R p R since p ◦ i = 0. Thus, we can simplify θ∂ to ι R pjj R p R ι . Similar argument allows us to further simplify θ∂ to ι R pp R ι = ι R id C ι =id fib( ¯ F ) . References [1] Mohammed Abouzaid, Denis Auroux, Alexander Efimov, Ludmil Katzarkov, andDmitri Orlov. Homological mirror symmetry for punctured spheres.
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