Density of fibers for the filtered Fukaya category of $T^*N$

DENSITY OF FIBERS FOR THE FIL TERED FUK A Y A CA TEGOR Y OF T ∗ N STÉPHANE GUILLERMOU, CLA UDE VITERBO, AND BINGYU ZHANG A B S T R A C T . W e answer a question of Biran and Cornea about the density of iterated cones of ﬁbers in the Fukaya categor y of a cotangent bundle. W e pro ve that indeed if we take a dense set of basepoints, the iterated cones of the cotangent ﬁbr es are dense in the Filter ed Fukaya category . 1. Intr oduction 2 2. Ackno wledgements 4 3. Categories 4 4. Quantization 7 4.1. Continuation morphisms 8 4.2. Quantization functor 11 4.3. Properties of sheaf quantization functor 12 5. S trategy of proof 14 6. Distances on the category of sheaves 16 6.1. Sheaf distance for a continuous norm 16 6.2. Some properties of the distance 19 6.3. Main example: Small wrapp ing of the diagonal 23 7. Proof of the main r esult 25 7.1. Some r esults on the projector 25 7.2. Some lemmata on iterated c ones 28 7.3. Proof of Theor em 5.2 29 Appendix A. I nterleaving Rouquier dimension 30 Refer ences 32 Date : Thursday 26 th F ebruar y , 2026. Also supported b y COSY (ANR-21-CE40-0002).The third author was suppor ted by the N ovo N ordisk F ounda- tion grant NNF20OC0066298 and VILL UM FONDEN, VILLUM Inv estigator grant 37814. 1 DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 2 1. I N T R O D U C T I O N Abouzaid pro ved in [Abo11] that the wrapped Fukaya categor y of a cotangent bundle is generated by one cotangent ﬁber . In the ﬁltered case , this can not happen: the Hamiltonian perturbation of a Lagrangian is isomorphic to the original Lagr angian only when they geo- metrically coincide . H owev er the ﬁlter ed Fukaya category comes with a notion of interleaving distance and P aul Biran [Bir24]asked whether the iterated cones on the cotangent ﬁbers gen- erate a dense subcategory , in the sense that any Lagrangian is arbitrarily close to an iterated cone of cotangent ﬁbers. W e refer to [Amb25] for a construction of the ﬁltered Fukaya cate- gory and [BCZ] for its persistence structure , which yields a notion of distance. Similar density considerations had been evoked in [F uk21]. W e shall use the ﬁltered Fukaya categor y constructed in [Amb25]. I ts objects are the com- pact Lagrangians so it does not contain the cotangent ﬁbers. T o make sense of the question of density of iterated cones of the ﬁbers, we shall embed this categor y in its module categor y through the Y oneda embedding. Let us int roduce some notations . Let N be a connected closed manifold and D T ∗ N its unit ball cotangent bundle (with r espect to a Riemannian metric) with Liouville for m λ . W e de- note b y F ( D T ∗ N ) the ﬁltered F ukaya categor y whose objects ar e compact (exact) Lagr angian branes, that is pairs ( L , f L ) where L is a closed exact Lagr angian contained in D T ∗ N and f L is a primitive of λ | L . M orphisms from L to L ′ are given by the Floer cochains F C ∗ ( L , L ′ ). The space of Floer cochains is ﬁltered once we have primitives f L , f L ′ of λ | L , λ | L ′ : an intersection point x ∈ L ∩ L ′ has ﬁltration degr ee f L ′ ( x ) − f L ( x ). W e let Y : F ( D T ∗ N ) − → M od ﬁl ( F ( D T ∗ N )) be the Y oneda embedding. H er e Mod ﬁl ( C ) is the set of functors from C to the categor y of ﬁlter ed chain complexes, and the Y oneda morphism is given by associating to L the functor L ′ 7→ F C ∗ ( L ′ , L ; t ) wher e F C ∗ ( L ′ , L ; t ) is the set of Floer chains of action greater than t . F or simplicity we denote by F C ∗ ( L , L ′ ) the Floer chain complex with its ﬁltration being understood, so the Y oneda embedding will send L to L ′ 7→ F C ∗ ( L ′ , L ). Let V ( x , a ) be the ﬁber at x with primitive a of λ | V x = 0. This Lagrangian brane V ( x , a ) does not belong to F ( D T ∗ N ) but it makes sense to deﬁne the ﬁltered complex F C ∗ ( L ′ , V ( x , a ) ) for any L ′ ∈ F ( D T ∗ N ) and this gives a module V ( x , a ) ∈ Mod ﬁl ( F ( D T ∗ N )). The action of F C ∗ ( L , L ′ ) on F C ∗ ( L ′ , V ( x , a ) ) is given b y the standard triangle product F C ∗ ( L , L ′ ) ⊗ F C ∗ ( L ′ , V ( x , a ) ) − → F C ∗ ( L , V ( x , a ) ) which respects the ﬁltr ation by [Amb25]. The module categor y M od ﬁl ( F ( D T ∗ N )) inherits an interleaving distance denoted b y γ . I t is also a pre-triangulated category . W e deﬁne the subcategor y 〈 V ( x 1 , a 1 ) , . . . , V ( x l , a l ) 〉 of M od ﬁl ( F ( D T ∗ N )) the subcategor y gener- ated by the V ( x i , a i ) ’ s as the subcategor y having for objects the iterated cones on the generators. N ow w e can state our main result. Theorem 1.1 (Density Theor em) . For any closed exact Lagrangian L ∈ F ( D T ∗ N ) and ε > 0 , ther e are points ( x i ) i ∈ {1,..., l } , real numbers ( a i ) i ∈ {1,..., l } and C ∈ 〈 V ( x 1 , a 1 ) , . . . , V ( x l , a l ) 〉 such that γ ( L , C ) < ε . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 3 The strategy of proof is to use the quantization functor constructed in [Vit19] which as- sociates a sheaf Q ( L ) on N × R with any closed exact Lagrangian brane L . In this way we embed F ( D T ∗ N ) into S h( N × R ), more precisely into the subcategory S h D T ∗ N ( N × R ) (actu- ally the T amarkin categor y T D T ∗ N ( N ) introduced later), formed b y the sheaves with reduced microsupport contained in D T ∗ N . The categor y Sh D T ∗ N ( N × R ) should be understood as a model of the wrapped Fukaya categor y in the ﬁltered setting. The ﬁber V ( x , a ) corresponds to the sheaf k { x } × [ a , ∞ ) ∈ Sh( N × R ), which does not belong to Sh D T ∗ N ( N × R ). W e let P ′ D T ∗ N be a projector which takes values in S h D T ∗ N ( N × R ) (it will be r ecalled in §5 - see [K u o23] where it is built as a w rapping functor) and we set W ( x , a ) = P ′ D T ∗ N ( k { x } × [ a , ∞ ) ). A slight modiﬁcation of [ Zha23, Thm. 4.4] sho ws that the endomorphism algebra of W ( x , a ) computes the homol- ogy of length-ﬁltered based loop spaces, which further clar iﬁes the role of Sh D T ∗ N ( N × R ) as “wr apped ﬁltered Fukaya category” . The sheaf categor y also inherits an interleaving distance that we denote b y γ s . N ow in the cat egor y Sh D T ∗ N ( N × R ) we can pro ve a more general r esult: Theorem 1.2 (Density Theorem) . Let F ∈ Sh D T ∗ N ( N × R ) such that for all x ∈ N the sheaf F ⊗ k { x } × R is a γ s -limit of constructible sheaves on R . Then, for any ε > 0 , ther e are points ( x i ) i ∈ {1,..., l } , real numbers ( a i ) i ∈ {1,..., l } and C ∈ 〈 W ( x 1 , a 1 ) , . . . , W ( x l , a l ) 〉 in Sh D T ∗ N ( N × R ) such that γ s ( F , C ) < ε . The idea is to use a ˇ Cech resolution of the constant sheaf k N ≃ C 0 − → C 1 − → · · · − → C m , wher e C i = L J ⊂ I , | J |= i + 1 k U c l J with U c l J = T j ∈ J U j and the U j ’ s are ε -small balls co vering N . Then F = F ⊗ k N × R is written as an iterated cone on the F ⊗ k U c l J × R . N o w F ⊗ k U c l J × R can be ap- pro ximated by F ⊗ k { x } × R for some x ∈ U c l J and F ⊗ k { x } × R (a sheaf on a line) can be written as (in general can be approximated b y) an iterated cone on the W ( x , a ) ’ s by constructibility . Fr om the argument, we kno w that the points ( x i ) i ∈ {1,..., l } can be taken from an a priori given dense subset of the base manifold N . Also , the pr oof may be viewed as a ﬁltered analogue of the sectorial descent of [GPS24], formulated in the language of sheaves. Remar k 1.3 . In fact, as in [ABC26] we pro ve something stronger than density , we pro ve what they call approximability : F or any ε > 0 we can ﬁnd objects X 1 , ..., X N such that any object in the categor y is at distance at most ε from an iterated cone of the X k . Even better if we consider the set of all dir ect sums obtained from the X i , we need only n + 1-iterated cones (wher e n is the dimension of the base manifold). A related notion of complexity called the interleaving Rouquier dimension will be discussed in A ppendix A. Once we have the result for sheaves, we can come back to M od ﬁl ( F ( D T ∗ N )) using the quan- tization functor Q since sheaf quantization of closed exact Lagrangian satisﬁes the assump- tion of the Density Theorem, due to constructibility of Q ( L ) ⊗ k { x } × R ≃ F C ∗ ( V ( x ,0) , L ) as sheaves on R . In fact, the sheaf result is a little stronger i n the sense that it may apply to sheaves as- sociated with immersed Lagrangians or certain C 0 -Lagrangians . One should also notice that the F ukaya category may depend on perturbation data; howev er , the sheaf category does not. Then it tells that the sheaf distance bou nd gives a uniform bound of Fukaya categor y distance for all perturbation data. DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 4 T o implement this plan of proof we need to be mor e precise on the quantization functor Q . In [Vit19] the quantization functor is deﬁned on the Donaldson-Fukaya category of T ∗ N with value in the (classical) derived category of sheaves. H o wever the Donaldson-Fukaya category is not triangulated and we cannot state our result in this framework. F or this reason we will ﬁrst enhance Q to a functor deﬁned on the Fukaya category in a higher coherent way , and then we can extent Q to the categor y of modules o ver Fukaya category . R ecall that the start- ing point in [Vi t19] i s to ﬁrst deﬁne a pr esheaf whose sections on some open set U × ( −∞ , a ) is lim f F H ( Γ d f , L ) where f is a function running over the smooth functions greater than the characteristic function of U , rescaled by a . T o tur n this into a functor we need some functo- riality of homotopy limits. This may be tr ue in the framework of A ∞ -categories but we lack refer ences. So we will turn our categories into ∞ -categories, for which the appropriate r esults are av ailable in the literature . In the sequel and for simplicity , all coefﬁcients will be in Z /2 Z . 2. A C K N O W L E D G E M E N T S The second and third authors ﬁrst heard about the question of the density of cones on the ﬁbers in the cotangent bundle –and more generally of other Lagrangians in other symplectic manifolds– from a talk by P aul Biran in J une 2024 ([Bir24]). The approach he mentioned (still incomplete at the time) is radically different from the one given here and is no w ﬁnalized in [ABC26]. W e believe both proofs have their own merit. W e warmly thank P aul Biran for shar- ing these ideas on this occasion as well as G io vanni Ambrosioni and Octav Cornea for sev eral useful discussions. The thir d author thanks T atsuki Kuwagaki, Adrian P etr , and Vivek S hende for sharing their progress on their independent work regar ding similar considerations. H e thanks them as well as Laur ent Côté and Zhen Gao for helpful discussions . 3. C AT E G O R I E S W e ﬁrst introduce some notations. F or a dg-category C , its dg ner ve is a simplicial set N d g ( C ) deﬁned as follo ws: N d g ( C ) is the simplicial set such that an n -simplex f ∈ N d g ( C ) n consists of the following data f = ({ X i } 0 ≤ i ≤ n , { f I }), where the X i are objects in C ; and for each order ed subset I = { i 0 > i 1 > · · · > i k } ⊆ [ n ] having at least two elements, f I : X i k → X i 0 is a degree k − 1 map 1 satisfying the coherent cocycle condition (3.1) ∂ f I = k − 1 X a = 1 ( − 1) a ( f { i 0 > i 1 >···> i a } ◦ f { i a >···> i k } − f I \{ i a } ). The structural maps are deﬁned as follows: a nondecreasing function α : [ m ] → [ n ] induces α ∗ : N d g ( C ) m → N d g ( C ) n deﬁned b y α ∗ (({ X i } 0 ≤ i ≤ m , { f I })) = ({ X α ( j ) } 0 ≤ j ≤ n , { g J }), 1 Remember that morphisms in a dg-category are gr aded DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 5 where g J =      f α ( J ) if α | J is injective id X i if J = { j 0 > j 1 } with α ( j 0 ) = i = α ( j 1 ) 0 otherwise. The dg ner ve construction can be generalized to str ict unital A ∞ -categories in a simil ar way , adding higher differen tial in the cocycle condition (3.1). W e will denote N A ∞ ( C ) this A ∞ - nerve (which is an ∞ -categor y) and refer to [F ao17] for more details . W e consider coefﬁcients in some commutative ring k (we take k = Z /2 Z when dealing with Fukaya categories). W e denote by Ch ∞ ( k ) the dg-category of cochain complexes on k and b y Ch ∞ ( k ) = N d g (Ch( k )) the ∞ -categor y obtained via the dg nerve functor 2 . W e let D( k ) = Ch ∞ ( k )[ W − 1 ] be its localization along the class W of edges giv en by quasi- isomorphisms of complexes. A model for D( k ) is N d g (D d g ( k )) the dg nerve of the dg-derived category of k . F or a manifold N we set Sh( N ) = Sh( N , D( k )) as the ∞ -categor y of (homotop y)-sheaves valued in D( k ) (see [Lur18, Deﬁnition 1.1.2.1]). I n the case of manifold, we have that Sh( N ) is equivalent to the unbounded ∞ -derived categor y of A belian sheaves D(S h( N , Mod k )) (see for example [Sch25, Proposition 7.1]). Filter ed categor ies. A convenient way to deal with the ﬁltration on the categories is to in- troduce the categor y of ﬁltered complexes. W e let ← − R be the ∞ -category associated with the poset ( R , ≥ ) via the ner ve functor . W e set Ch ﬁl ( k ) = Fun( ← − R , Ch ∞ ( k )) and D ﬁl ( k ) = Fun( ← − R , D( k )) where Fun will always denote ∞ -functors. W e have D ﬁl ( k ) ≃ Ch ﬁl ( k )[ W − 1 ← − R ], where W ← − R is the set of ﬁltr ation-preserving quasi-isomorphisms, i.e ., for each t , f t : M t → N t is a quasi- isomorphism. Let R ≤ be the topological space R whose open subsets ar e the inter vals ( −∞ , a ). This is a special case of the γ -topology V γ for a cone γ in some vector space V introduced in [KS90]. W e remark that the poset ( R , ≥ ) is fully faithfully embedded (as a categor y) into the categor y Op( R ≤ ) op ∼ = ( R ∪ { ±∞ }, ≥ ). H ence there ar e restriction functors PSh( R ≤ , Ch ∞ ( k )) → Ch ﬁl ( k ), PSh( R ≤ , D( k )) → D ﬁl ( k ), F 7→ M ( F ) = [ t 7→ M ( F ) t = F (( −∞ , t ))]. out of the categor y of pr esheaves on R ≤ valued r espectively in Ch ∞ ( k ), D( k ). It is explained in [KZ25, Section 4.2] 3 that the functor PSh( R ≤ , D( k )) → D ﬁl ( k ) restricts to a fully faithful functor Sh( R ≤ ) , → D ﬁl ( k ) whose image satisﬁes the semi-continuous condition for persistence mod- ules, which is also a monoidal functor r espects certain monoidal str uctur es. T aking the left adjoint and sheaﬁﬁcation, we obtain functors (3.2) Ch ﬁl ( k ) − → D ﬁl ( k ) − → Sh( R ≤ ) , → Sh( R ). 2 Remember that an ∞ -category is a simplicial set satisfying the weak Kan extension: ever y inner horn has a ﬁller . 3 In [KZ25], the authors use the poset ( R , ≤ ), which causes a difference on sign and cohomology degree shifting. DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 6 W e r ecall the convenient equivalence of sheaf categories, in the ∞ -category setting, Sh( N × R ) ≃ Sh( N , Sh( R )) (see [V ol25, Cor . 2.24 and Prop . 2.30]). In particular , using (3.2) and sheaﬁ- fying on N , we obtain functors (3.3) Fun(Op ∞ ( N ) op , D ﬁl ( k )) − → Sh( N , D ﬁl ( k )) − → Sh( N × R ). where O p ∞ ( N ) is the ner ve of the poset Op( N ) of open subsets in N . Categories r elated to the microsupport . F or a manifold M and F ∈ S h( M ) we denote by SS( F ) ⊂ T ∗ M the microsupport of F deﬁned by K ashiwara-Schapira [KS90]. F or a conic sub- set W ⊂ T ∗ M we let Sh W ( M ) be the subcategor y of Sh( M ) for med by the sheaves F with SS( F ) ⊂ 0 M ∪ W . I n the case M = N × R we set for short { τ ≥ 0} = {( x , p , t , τ ) ∈ T ∗ ( N × R ) | τ ≥ 0} and deﬁne { τ R 0} for R ∈ { ≤ , > , < } in the same way . The micr olocal cut-off lemma of Kashiwara-Scha pira (see [KS90], Prop . 5.2.3(i), together with P rop . 3.5.3(iii)) sho ws that the essential image 4 of Sh( R ≤ ) , → Sh( R ) is identiﬁed with Sh τ ≥ 0 ( R ). The T amarkin categor y T ( N ) is deﬁned as the left or thogonal of Sh τ ≤ 0 ( N × R ), i.e. the full subcategor y spanned b y sheaves F such that Hom( F , G ) = 0 for all G ∈ Sh τ ≤ 0 . W e have T ( N ) , → Sh τ ≥ 0 ( N × R ), and a sufﬁcient (but not necessary) characterization is F | N × ( −∞ , A ) = 0 for some A ∈ R . F or W ⊂ T ∗ N we also let T W ( N ) be the subcategor y of T ( N ) formed by the F with SS( F ) ⊂ 0 N × R ∪ ρ − 1 ( W ), with ρ : T ∗ N × { τ > 0} − → T ∗ N , ( x , p , t , τ ) 7→ ( x , p / τ ). W e denote by RS( F ) : = ρ (SS( F )) the reduced microsupport. Inter leaving distances. Both F ( T ∗ N ) and T ( N ) come with“t ranslation functors ” T a , a ∈ R , inducing an interleaving distance on the set of objects. A “ translation functor” for a categor y C is a family of functors T a : C − → C for all a ∈ R and morphisms of functors τ a b : T a − → T b for a ≤ b satisfying some natura l compatibilities (higher coher ently). A good way to express these compatibilities is to ask that the data of T and τ gives a monoidal functor ( T , τ ) : ( R , ≤ ) − → End( C ), where ( R , ≤ ) is seen as a categor y through the order with a monoidal str uctur e given b y the addition (see [PS23, Def. 1.3 .4] or [BCZ, Def. 2.15]). This condition implies in par ticular that τ a b ≃ T a ( τ 0 b − a ) and we will only consider the family τ c = τ 0 c , c ≥ 0. For X , Y ∈ C we then set 5 γ ( X , Y ) = inf n a + b | a , b ≥ 0, ∃ u : X − → T a ( Y ), v : Y − → T b ( X ), T a ( v ) ◦ u : X − → T a + b ( X ), T b ( u ) ◦ v : Y − → T a + b ( Y ), are the maps τ a + b o Then γ is a pseudo-distance 6 on the set of (isomorphisms classes of ) objects of C (it is not a distance since we may have γ ( X , Y ) = 0 for non-isomorphic X , Y ). If C is an ∞ -categor y , we replace ( R , ≤ ) by − → R , the ∞ -categor y associated with the poset ( R , ≤ ) via the ner ve functor as befor e (we refer to the higher coher ent version [KZ25]). 4 This is the subcategory with objects equ ivalent to objects in the image. See [Kerodon, Def 4.6.2.12] for the ∞ -category case. 5 In the case wher e C is an ∞ -category , equality of morphisms is r eplaced by homotop y of 1-morphisms. 6 Several v ariations are possible, for example we could ask that a = b . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 7 Remar k 3.1 . (1) L et C , C ′ be two ∞ -categor ies endo wed with translation functors ( T , τ ) : − → R − → End( C ), ( T ′ , τ ′ ) : − → R − → End( C ′ ) and let γ , γ ′ be the associated pseudo-distances. Let F : C − → C ′ be a functor commuting with the − → R -actions. Then F is 1-Lipschitz, that is, γ ′ ( F ( X ), F ( Y )) ≤ γ ( X , Y ). If, moreo ver , F is fully faithful, then F preserves the distances . (2) When the categories are k -linear , we will assume the related functors are k -linear as well. W e will also consider the categor y of ﬁlter ed modules of C for k -linear C , M od ﬁl ( C ) = Fun( C o p , Ch ﬁl ( k )), which admits a translation functor induced by Ch ﬁl ( k ). If C is endo wed with translation functors ( T , τ ), then M od ﬁl ( C ) inher its another translation functor . M or e precisely , for X ∗ ∈ Mod ﬁl ( C ) we deﬁne T ∗ c ( X ∗ ) by T ∗ c ( X ∗ )( X ) = X ∗ ( T − c ( X )), where we intro- duce the minus sign so that the morphism of functors τ : id − → T c , c ≥ 0, induces a morphism τ ∗ : id − → T ∗ c , c ≥ 0. Then ( T ∗ , τ ∗ ) : − → R − → End(M od ﬁl ( C )) is a monoidal functor . In the article, we assume that the category C satisﬁes that the two translation functors on M od ﬁl ( C ) deﬁned abo ve ar e equivalent. This is the case for the ﬁltered Fukaya category of exact symplectic manifolds and exact Lagrangian br anes (see [BCZ, Remark 3.8]). W e then have pseudo-distances γ on C and γ ∗ on M od( C ). The Y oneda functor Y C : C − → M od( C ) commutes with T c , T ∗ c and sends τ to τ ∗ and is fully faith ful. B y (1) it follo ws that γ ∗ ( Y C ( X ), Y C ( Y )) = γ ( X , Y ) for any X , Y ∈ C . (3) I n the situation of (2), w e deﬁne D ﬁl ( C ) = Fun( C o p , D ﬁl ( k )) and call it the ﬁlter ed derived category of C . The ﬁlter ed derived category is identiﬁed with the localization of Mod ﬁl ( C ) with respect t o ﬁltration-pr eserving quasi-isomorphisms. Ther efore , it induces the tr ansla- tion functor from Mod ﬁl ( C ) and the localization functor Mod ﬁl ( C ) → D ﬁl ( C ) induces an isom- etry because the interleaving distance in Ch ﬁl ( k ) does not change under ﬁltration-preserving quasi-isomorphisms. 4. Q UA N T I Z AT I O N In [V it19] the second-named author associates a sheaf with any closed exac t Lagr angian. This gives a functor fr om th e Donaldson-F ukaya categor y of closed exact Lagrangian to the category of sheaves. In this section we see that this functor can be enhanced to a functor from a ﬁltered Fukaya categor y to the category of sheaves. The construction we give her e is actually the same as in [Vit19] but we ar e careful to deﬁne the functors at the level of th e A ∞ -category or ∞ -category and not only on the associated homotopy category . In this article, we use the construction of ﬁltered Fukaya categor ies from [Amb25] 7 . Even though the author constructs ﬁltered F ukaya categories for cer tain non-exact Lagrangian branes of general symplectic manifolds, here we focus on closed exact Lagrangian of the cotangent bundle T ∗ N , and denote it by F = F ( T ∗ N ). W e also r emark that the follo wing con- struction is basically independent of the deﬁnition of the ﬁltered Fukaya categories provided it is s tr ict unital (for example, such a construction based on categorical localization appr oach 7 T o be compatible with the ﬁltration convention of [V it19], our ﬁltration convention is opposite to loc. cit., i.e., F C ∗ ( L ′ , L ; t ) is the set of Floer chains of action gr eater than t . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 8 will appear in [KPS26]). The coefﬁcient ring could be an arbitrary commutative ring if moduli spaces are w ell-oriented. Remar k 4.1 . Her e, we did not emphasize the role of perturbation data in the deﬁnition of Fukaya categories and the follo wing construction of continuation morphisms, as w ell as sheaf quantization functors. As explained in [ABC26], w e may also organize F ukaya categories for different perturbation data as a system of ﬁltered A ∞ -categories with compar ison functors . In principle, those com- parison functors should be compatible with continuation morphisms and sheaf quantization functors we constructed below , but w e will not discuss that compatibility . W e will see later that those constructions work uniformly for (but depend on) different per turbation data, and the ﬁnal F ukaya density theor em does not involve perturbation data, si nce our sheafy result is perturbation-independent. Therefore , we do not need to pay much attention to perturbation data here . 4.1. Continuation morphisms. W e take C ≤ = ( C ∞ ( N ), ≤ ) to be the poset of smooth functions, and take k [ C ≤ ] the fr eely generated k -linear category b y setting H om( f , g ) = k e f g if and only if f ≤ g , and treat it as an A ∞ -category with trivial µ k for k = 2, and µ 2 ( e g h , e f g ) = e f h if and only if f ≤ g ≤ h . W e deﬁne a ﬁltration on H om( f , g ) = k b y setting Hom ≤ t ( f , g ) = k e f g if and only if t ≥ 0. W e deﬁne a coher ent choice of monotone continuation morphisms as a strict unital ( F , k [ C ≤ ]) ( A ∞ -)bimodule B with B ( L , f ) = F C ∗ ( L , Γ d f ) ∈ Ch ﬁl ( k ) together with bimodule maps (see [LM08]) 8 µ k | 1 | n : F ( L k , L k − 1 ) ⊗ · · · ⊗ F ( L 1 , L 0 ) ⊗ B ( L 0 , f 0 ) → B ( L k , f n )[1 − k − n ] that satisﬁes the usual bimodule S tasheff identity and preserves ﬁltration. H ere , we explain the existence of a coherent choice of monotone continuation morphisms . Review of constructi on of the right k [ C ≤ ] -module. The right k [ C ≤ ]-module str ucture is con- structed based on family Floer theory as explained in [Vit19, Prop . 4.6]. W e shall now review its construction. F or a ﬁnite collection of functions ( f 0 ≤ · · · ≤ f n ), we can construct a ∆ n - family (here ∆ n is the topological standard n -simplex) of σ ( λ 1 , . . . , λ n ) = P n j = 0 ( λ j + 1 − λ j ) f n − j where λ 0 = 0 ≤ λ 1 ≤ · · · ≤ λ n ≤ λ n + 1 = 1 (notice that the set of ( λ 1 , . . . , λ n ) deﬁnes a topolog- ical n -simplex), and then consider a parameterized family of trajectories of the mixed Floer equation ∂ s u σ ( λ ( s )) ( s , t ) + J ∂ t u σ ( λ ( s )) ( s , t ) = 0, ∂ s λ ( s ) = τζ g ( λ ( s )), where ζ g ( λ ) = P n j = 1 λ j (1 − λ j ) ∂ λ j , subject to the boundar y condition u λ (0) ∈ L , u λ (1) ∈ Γ d σ ( λ ) , lim s →±∞ u λ ( s ) ( s , t ) ∈ L ∩ L σ ( λ ) , lim s →−∞ λ ( s ) = (0, . . . , 0), lim s →∞ λ ( s ) = (1, . . . , 1). Note that τ is a small parameter introduced because one needs “ slo w homotopies ” and in the abov e chain 8 In principle the map should be F ( L k , L k − 1 ) ⊗ · · · ⊗ F ( L 1 , L 0 ) ⊗ B ( L 0 , f 0 ) ⊗ Hom( f 1 , f 0 ) ⊗ · · · ⊗ Hom( f n , f n − 1 ) → B ( L k , f n )[1 − k − n ] but all the H om( f , g ) are k e f , g or 0 so we omit them from the notation. DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 9 map - which in principle depends on τ - w e take the limit as τ goes to 0. T o solve the equation, we transform the mo ving boundar y condition problem to a usual Floer equation b y introduc- ing a H amiltonian perturbation H as constructed in [V it19, Prop . 3.3, 4.6]. F or y 0 ∈ F C ∗ ( L , Γ d f 0 ) and y 1 ∈ F C ∗ ( L , Γ d f n ), we denote b y M ( y 0 , y 1 ) the moduli space of rigid solutions of the corr esponding Floer equation, and then we deﬁne the right k [ C ≤ ] mod- ule map B ( L 0 , f 0 ) = F C ∗ ( L , Γ d f 0 ) → B ( L 0 , f n ) = F C ∗ ( L , Γ d f n ), y 0 7→ X y 1 # M ( y 0 , y 1 ) y 1 , where the Stasheff identity corresponds exactly to Equation (3.1), which means that all those right module maps associated with ( f 0 ≤ · · · ≤ f n ) and its subsets form a n -simplex in Ch ﬁl ( k ). Construction of the ( F , k [ C ≤ ]) -bimodule. N ow , we sho w the compatibility with left multipli- cation of F . The construction follows from the idea of [GPS20, S ec. 5], but simpler . Proposition 4.2. There exists a coher ent choice of monotone continuation morphisms for F , whose right module r estriction is compatible with family Floer construction in [Vit19, Prop . 4.6]. Proof . W e start from the case that all Lagr angians ( L k , . . . , L 0 ) intersect transversely . W e consider moduli spaces of holomorphic strips with marked points: let R B k ; n be the com- pactiﬁed moduli space of strips R × [0, 1] with k marked points z 1 , ..., z k on R × {0} boundar y marked points and n -tuple of reals a 1 ≥ · · · ≥ a n deﬁned up to translation and view them as marked points on R × {1}, see F igure 1. W e r efer to [GPS20, Sec. 5.3] for details of a similar compactiﬁcation. W e set the universal cur ve C B k ; n = {( D , z ) | D ∈ R B k ; n , z ∈ D } together with sections z i , i = 1, . . . , k and a j , j = 1, . . . , n . As described in [GPS20, S ec. 4.1], R B 0; n describe a space of M orse ﬂo w lines on ∆ n , we could regar d a conﬁguration a 1 ≥ · · · ≥ a n as a universal map from the R × {1} part of C B k ; n to ∆ n b y associating a j to the edge ( j − 1, j ) and associating the inter val ( a j + 1 , a j ) to the vertex j ( a 0 = ∞ , a n + 1 = −∞ ). The Lagrangian labeling on R × {0} associates z i − 1 z i to L i for i = 0, . . . , k + 1 ( z − 1 = −∞ , z k + 1 = +∞ ). The ident iﬁcation of R × {1} with ∆ n compose with the n -dimension family of σ : ∆ n → C ∞ ( N ) give the Lagrangian labeling on R × {1} that associates a j + 1 a j to Γ d f j . Those data form the domain data of our counting. Conformally , we may regar d D ∈ R B k ; n as a disk with k + 2 marked points and n marked points located in the interior of the arc z k + 1 z 0 where z 0 = −∞ and z k + 1 = ∞ . N ext, we give the Floer datum: W e pick standard str ip-like ends ϵ i near z i and H amiltonian 1-form K , and almost complex structures J : C B k ; n → J ( T ∗ N ). W e take a Hamiltonian family H : C B k ; n × T ∗ N → R that gives the Hamiltonian perturbation introduced b y [Vit19, Prop . 3.3] near R × {1} and tr ivial other wise. M oreo ver , the choices of Floer data should be compatible as described in [GPS20, Sec. 5.4]. W e also assume that we choose a sequence of functions f 0 ≤ · · · ≤ f n such that f j − f i are M orse for i = j and, in addition, that Γ f 0 ⋔ L 0 and Γ f n ⋔ L k , then we take x i ∈ L i − 1 ∩ L i , i = DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 10 H z 1 z 2 z k a n a n − 1 a 1 Γ d f n Γ d f n − 1 Γ d f n − 2 Γ d f 0 L 0 L 1 L 2 · · · L k s = −∞ s = ∞ F I G U R E 1 . Lagrangian labeling and F loer data for the bimodule B 1, . . . , n , and y 0 ∈ L 0 ∩ Γ d f 0 , y 1 ∈ L k ∩ Γ d f n . W e consider the moduli space M ( x k , . . . , x 1 , y 0 ; y 1 ) of solutions of ( d u − X H ( u ) ⊗ K ) 0,1 J = 0 with the given boundary condition. Since all Lagrangians are closed exact and intersect trans- versely , then compactness and transversality can be achieved b y regular Floer datum. Then we deﬁne the bimodule map b y the rigid counting µ k | 1 | n B : F ( L k , L k − 1 ) ⊗ · · · ⊗ F ( L 1 , L 0 ) ⊗ B ( L 0 , f 0 ) → B ( L k , f n )[1 − k − n ] x k ⊗ · · · ⊗ x 1 ⊗ y 0 7→ # M ( x k , . . . , x 1 , y 0 ; y 1 ) y 1 . F or the Stasheff identity , we see that the Hamiltonian H is trivial far away from R × {1}, so homomorphic polygons split as the usual A ∞ relation near R × {0}; and n ear R × {1}, by our choice of H and J , the splitting o f holomorphic strips at s = ±∞ are the same as [Vit19, Pr op . 4.6] or [GPS20, Lemma 4.33] described in a simplicial way . T o ﬁnish the existence , we discuss th e non-transverse intersection case . In this case , we r e- ally need to follow [Amb25] to count holomorphic clusters rather than holomorphic disks. In case all intersections ar e transverse, the counting of holomorphic clusters reduces to counting of disks (or strips), so our previous discussion works directly in this case (and actually gives the same bimodule maps). The most important thing here is the existence of r egular Floer data such that all oper ations preserve ﬁltration. This is the main contribution of [Amb25] in case there is no Hamiltonian term H . In our case the Hamiltonian term H is associated to a “ monotonic ” family of exact graphs, hence it preserves the ﬁltration. The existence of regular ϵ i , K , J then follo ws from [Amb25]. I t follows that we can also construct the bimodule maps in the non-transverse intersec- tion case (but we need to be mor e careful when deﬁning the modu li space of the domain) b y replacing holomorphic strips with holomorphic clusters . T o check compatibility of this bimodule structur e in the case k = 0 with that from [Vit19, Prop . 4.6] we use the fact that R B 0; n is identiﬁed with a space of M orse ﬂo w lines on ∆ n we described befor e. N ote that the pseudo-gradient in [GPS20, R emark 4.1] is differ ent from the one in [Vit19], but a standard Floer homotop y argument shows that our perturbed counting is equal to the one from [Vit19]. □ DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 11 4.2. Quantization functor . F or a coherent choice of monotone continuation maps B (given b y Proposition 4.2) we may use [ LM08, Proposition 5.3, P age 569] to deﬁne a strict uni- tal A ∞ -functor k [ C ≤ ] → M od − F . In fact, the essential image consists of Y oneda modules Y Γ d f , so the A ∞ -functor factors through an A ∞ -functor b : k [ C ≤ ] → F . The morphism e f g ∈ H om k [ C ≤ ] ( f , g ) is mapped to a degree 0 and non-negative action element F C 0 ( Γ d f , Γ d g ; 0) (in particular , f = f is mapped to the unit of Γ d f in F ), so the functor b : k [ C ≤ ] → F preserves the ﬁltration. I t is clear that there exists a functor C ≤ → k [ C ≤ ] (between 1 -categories): it identiﬁes objects on boths sides and sends ∗ ∈ Hom C ≤ ( f , g ) to e f g . By using the ner ve constr uction of 1-category or A ∞ -categories we turn these categor ies into ∞ -categories. In particular we set C ∞ ≤ : = N ( C ≤ ), F ∞ : = N A ∞ ( F ). N otice that N ( k [ C ≤ ]) = N A ∞ ( k [ C ≤ ]). W e then deduce from b a functor (4.1) Gr : C ∞ ≤ − → F ∞ , f 7→ Γ d f . Let us ﬁrst recall the idea of the construction in [Vit19]. Let L be a closed exact Lagrangian. W e deﬁne a sheaf F L on N × R b y its sections on U × ( −∞ , c ) for any c ; the space of these sections is H om( k U × ( −∞ , c ) , F L ). There is no object in the Fukaya categor y of closed exact La- grangians corr esponding to k U × ( −∞ , c ) . Ho wever we hav e a ﬁber sequence k U × ( −∞ , c ) − → k N × R − → k ( N × R )\( U × ( −∞ , c )) + 1 − − → and k ( N × R )\( U × ( −∞ , c )) ≃ colim f > χ k { t ≥ f } where the colimit runs over smooth bounded functions f and χ = c on U , χ = −∞ outside U 9 . N o w k { t ≥ f } corresponds to Γ d f and we can deﬁne a presheaf F p r e L b y its sections Γ ( U × ( −∞ , c ); F p r e L ) : = colim f H om( Γ d f , L ). The sheaﬁﬁcation of F p r e L will give F L up to the constant sheaf with stalk H ∗ ( N , k ). As we said it is already proved in [V it19] that the associated sheaf F L has the expected properties; in particular i ts (reduced) microsupport is L . The problem is to check that L 7→ F L is a functor deﬁned on the F ukaya category F ∞ . N ow the language of ∞ -categories allows us to reformulate the construction dir ectly as a functor . The quantization functor Q : F ∞ − → Sh( N × R ) will be given b y the following composition F ∞ − → Fun(( F ∞ ) op , Ch ﬁl ( k )) (4.2) − → Fun(( C ∞ ≤ ) op , D ﬁl ( k )) (4.3) − → Fun(Op ∞ ( N ) op , D ﬁl ( k )) (4.4) − → Sh( N × R ), (4.5) 9 The sheaf k U × ( −∞ , c ) corresponds to, in the inﬁnitesimal wrapped Fukaya category , the external conormal T ∗ ∂ U , + N of ∂ U , when U has a smooth boundar y . The sheaf result indicates that T ∗ ∂ U , + N should be the colimit colim f > χ Γ d f in the inﬁnitesimal wrapped Fukaya category . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 12 that we explain no w . The map (4.2) is the Y oneda embedding. T o deﬁne (4.3) we com- pose with Gr (4.1) and the localization map Ch ﬁl ( k ) − → D ﬁl ( k ). The last map (4.5) is ex- plained in (3.3). It remains to explain (4.4). The functor C ≤ − → Op( N ), f 7→ { f > 0}, induces j : ( C ∞ ≤ ) op − → (Op ∞ ( N )) op . Since D ﬁl ( k ) is presentable , we can consider its left Kan extension ([K erodon, Corollary 7.3.5.2]) Lan j : Fun(( C ∞ ≤ ) op , D ﬁl ( k )) − → Fun((O p ∞ ( N )) op , D ﬁl ( k )). This is our map (4.4). T o make the link with [V it19] we describe how Q acts on objects . W e r ecall that the left Kan extension is given by a colimit as follo ws: Lan( j )( φ )( U ) = colim f φ ( f ), where f ∈ ( C ∞ ≤ ) op runs o ver the functions such that U ⊂ { f > 0} (equiv alently f | U > 0). S witch C ∞ ≤ and ( C ∞ ≤ ) op , we also have Lan( j )( φ )( U ) = lim f φ ( f ), where f ∈ C ∞ ≤ and f | U > 0. In the case where φ = Y L is Y oneda module of a Lagrangian L , we wr ite Q p r e ( L ) = Lan( j )( Y L ) where Y L ( f ) = F ( Γ d f , L ) and we r ecover th e deﬁnition of [Vit19] . Lastly , we notice that by the microlocal cut-off lemma and the compactness of L (see Sec- tion 3), the functor Q factors thr ough Q : F ∞ − → T ( N ) , → Sh( N × R ). Remar k 4.3 . In fac t, the composition of e Q : M od ﬁl ( F ∞ ) = Fun(( F ∞ ) op , Ch ﬁl ( k )) → Sh( N × R ) could also be understood as a sheaf quantization functor , and then Q ( L ) = e Q ( Y L ). A natural question is: for M ∈ M od ﬁl ( F ∞ ), what is the reduced microsupport of e Q ( M )? F or example , when M comes from any kind of Floer theory for singular Lagrangians. 4.3. Properties of sheaf quantization functor. H ere , w e recall some properties of the sheaf quantization functor that is pro ven in [V it19], which follo w verbatim from arguments therein after adapting our constructions here. N otice that below , we map hom complex of the A ∞ - category F , say F ( L 1 , L 2 ) ∈ Ch ﬁl ( k ), to an object in D ﬁl ( k ) via (3.2). N otice that the general Floer theory tells that the action ﬁltration of F ( L 1 , L 2 ) satisﬁes the semi-continuity condition. I t follows from [KZ25, Theorem A-(2)] that under (3.2), we may also regar d F ( L 1 , L 2 ) ∈ D ﬁl ( k ) as an object of Sh τ ≥ 0 ( R ) , → D ﬁl ( k ). By the identiﬁcation Sh τ ≥ 0 ( N × R ) = Sh( N ; Sh τ ≥ 0 ( R )), we can deﬁne Γ : Sh τ ≥ 0 ( N × R ) → Sh τ ≥ 0 ( R ) as the Sh τ ≥ 0 ( R )-valued global section functor . M ore precisely , Γ is identiﬁed to the Sh τ ≥ 0 ( R )-linear direct image to a point ( a N ) Sh τ ≥ 0 ( R ) ∗ for a N : N − → pt, and Γ is also equivalent to the k -linear direct image functor ( a N × id R ) k ∗ : S h τ ≥ 0 ( N × R ) → Sh τ ≥ 0 ( R ). Proposition 4.4 ( [V it19, Prop . 8.1]) . The natural morphism induced by counit of sheaﬁﬁcation Q p r e ( L ) → Q ( L ) is an equivalence in PSh( N × R ) , i.e. Q p r e ( L ) i s alr eady a sheaf. I n par ticular , for any U ⊂ N open, F C ∗ ( v ∗ U , L ) : = colim f F C ∗ ( Γ d f , L ) ≃ − → Γ ( U , Q ( L )) is an equivalence in Sh τ ≥ 0 ( R ) . The follo wing is a slight generalization of [Vit19, Coro . 8.2] DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 13 Corollary 4.5. Let Z be a closed submanifold in N . Then for a closed exact Lagrangian L , we have F ( v ∗ Z , L ) ≃ − → Γ ( Z , Q ( L )) is an equivalence in Sh τ ≥ 0 ( R ) that is functorial with r espect to L . Proof . In the transverse case, we apply Proposition 4.4 to a δ -tubular neighborhood U δ of Z , and then take the colimit o ver δ > 0. This is exactly [Vit19, Coro . 8.2]. In the non-transv erse case, we use a similar argument to P roposition 4.8 belo w b y perturb- ing v ∗ Z b y C 2 -small H amiltonians. □ Remar k 4.6 . W e consider the particular case Z = { x } for some point x ∈ N . Then Proposi- tion 4.5 implies that Γ ( Z , Q ( L )) = i − 1 x Q ( L ) ∈ Sh τ ≥ 0 ( R ) (with i x : R → N × R ) is constructible for generic x since F ( v ∗ Z , L ) = F C ∗ ( T ∗ x N , L ), and is a γ τ -limit of constructible sheaves for all x since F ( T ∗ x n N , L ) converges to F ( T ∗ x N , L ) when x n tends to x . The sheaf convolution  and its adjoint H om  deﬁne closed symmetr ic monoidal str uc- tures on Sh τ ≥ 0 ( N × R ) , → Sh( N × R ) that is Sh τ ≥ 0 ( R )-linear b y the identiﬁcation S h τ ≥ 0 ( N × R ) ≃ Sh( N ; Sh τ ≥ 0 ( R )). Then [V it19, Pr op . 9.3, 9.8] and their proof imply the follo wing result. Proposition 4.7. For any closed exact Lagrangian L , the sheaf quantization Q ( L ) is dualizable with respect to  whose dual is identiﬁed with Q ( − L ) . In particular , for any F ∈ Sh ≥ ( N × R ) , we have equivalence Q ( − L )  F ≃ H om  ( Q ( L ), F ) that is functorial with respect to F . The follo wing is a slight generalization of [Vit19, Prop . 9.9] Proposition 4.8. The sheaf quantization functor Q induces functorial equivalences F ( L 1 , L 2 ) ≃ − → Γ ( N , H om  ( Q ( L 1 ), Q ( L 2 ))) in Sh τ ≥ 0 ( R ) for all pairs ( L 1 , L 2 ) . In particular , the sheaf quantization functor induces a fully faithful functor D ﬁl ( F ∞ ) 0 → Sh τ ≥ 0 ( N × R ) where D ﬁl ( F ∞ ) 0 is the full subcategory of Y oneda modules in D ﬁl ( F ∞ ). Remar k 4. 9 . The homotop y categor y of D ﬁl ( F ∞ ) 0 is exactly the ﬁltered version of Donaldson- Fukaya categor y since over ﬁelds two A ∞ modules ar e quasi-isomorphic if and only they ar e homotop y equivalent. Proof . W e ﬁrst explain the equiv alences for each pair ( L 1 , L 2 ). The proposition [Vit19, P rop . 9.9] pro ve the statement for transversely intersected pairs ( L 1 , L 2 ). T o the reader’ s convenience , let us pro vide its proof here . By dualizability P roposition 4.7, the right-hand side is computed b y Γ ( N , Q ( − L 1 )  Q ( L 2 )) ≃ Γ ( ∆ N , Q ( − L 1 ) □ ∗ Q ( L 2 )). The uniqueness of sheaf quantization sho ws that Q ( − L 1 ) □ ∗ Q ( L 2 ) ≃ Q ( − L 1 × L 2 ), where the later is the quantization in T ∗ ( N × N ). Then we apply Cor ollar y 4.5 to Z = ∆ N , and identify F C ( ∆ N , − L 1 × L 2 ) = F C ( L 1 , L 2 ). F or general pairs ( L 1 , L 2 ), we pick a seq uence C 2 -small H amiltonians H n for ϕ n = ϕ H n such that ϕ n ( L 1 ) are tr ansversely intersected with L 2 . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 14 On the sheaf side, we take K n the sheaf qu antization of H n (at time 1). By [Ike19, Appendix B] we have Q ( ϕ n ( L 1 )) ≃ K n ( Q ( L 1 )), and then Γ ( N , H om  ( Q ( ϕ n ( L 1 )), Q ( L 2 ))) ≃ Γ ( N , H om  ( K n ( Q ( L 1 )), Q ( L 2 ))) is convergent to Γ ( N , H om  ( Q ( L 1 ), Q ( L 2 ))) with r espect to the interleaving distance by [AI20, Prop . 4.11, Thm. 4.16]. Similarly , on the Fukaya side , we have F ( ϕ n ( L 1 ), L 2 ) is convergent to F ( L 1 , L 2 ) b y the same Hofer distance estimation via applying [BCZ, Theo . 3.-4-(i)] to F . Then the equivalence for the general pairs ( L 1 , L 2 ) follo ws from the transversely intersected pairs and the limit argument. T o see that Q induces a fully faithful functor D ﬁl ( F ∞ ) 0 → Sh τ ≥ 0 ( N × R ), we shall check that Q induces homotopy equivalence between the (Sh τ ≥ 0 ( R )-enriched) hom objects of D ﬁl ( F ∞ ) 0 and of Sh τ ≥ 0 ( N × R ). No w , we explain the ingredients and the proof for the claim. By the enriched Y oneda lemma, the hom objects of D ﬁl ( F ∞ ) 0 is F ( L 1 , L 2 ) ∈ Sh ≥ ( R ) , → D ﬁl ( k ). And the hom object of Sh τ ≥ 0 ( N × R ) is Γ ( N , H om  ( F 1 , F 2 )). At this moment, we already kno w Q is a functor , which induces the natural map F ( L 1 , L 2 ) → Γ ( N , H om  ( Q ( L 1 ), Q ( L 2 ))). I t remains to check it is an equivalence on cohomology . W e describe the natural map on cohomology: F or a morphism x : L 1 → L 2 , we have a mor- phism Y L 1 → Y L 2 of Y oneda modules induced by the pair of pants product, and induce a morphism Q ( L 1 ) → Q ( L 2 ) since Q are sheaﬁﬁcation of Y oneda modules (r estricted to exact graphs). H ow ever , it is shown that in [V it19, S ection 10] the equivalences w e desc r ibed in the ﬁrst part are exactly induced b y the pair of pants product on cohomology . □ 5. S T R AT E G Y O F P R O O F W e no w explain ho w we wil l use the quantization functor Q to pro ve the main theor em. W e ﬁrst sum up the results r ecalled so far and add some notations . Then we explain that it will be useful to consider the projector from T ( N ) to T D T ∗ N ( N ) to obtain the result. Some re minder. W e recall that we consider F ∞ ( D T ∗ N ) as an ∞ -categor y . W e consider the ﬁltered derived category D ﬁl ( F ∞ ( D T ∗ N )) = Fun( F ∞ ( D T ∗ N ) o p , D ﬁl ( k )) and we let Y F : F ∞ ( D T ∗ N ) − → D ﬁl ( F ∞ ( D T ∗ N )), be the D ﬁl ( k )-enr iched Y oneda embedding. F or x ∈ N , a ∈ R we denote by V ( x , a ) the ﬁber at x with pr imitive a of λ | V x = 0 and we deﬁne V ( x , a ) ∈ M od ﬁl ( F ∞ ( D T ∗ N )) as the functor L ′ 7→ F C ∗ ( L ′ , V ( x , a ) ), and we use the same notation V ( x , a ) to repr esent the correspond ing object in D ﬁl ( F ∞ ( D T ∗ N )). W e have r ecalled in §4 the quantization functor Q : F ∞ ( D T ∗ N ) − → T ( N ). The functor Q commutes with the translation functors, T c on F ∞ ( D T ∗ N ) and T c ∗ on T ( N ). By construction ρ ( S S ( Q ( L )) is L , and Q actually takes values in T D T ∗ N ( N ). The composition with Q induces an exact functor Q ∗ : D ﬁl ( T D T ∗ N ( N )) − → D ﬁl ( F ∞ ( D T ∗ N )), E 7→ E ◦ Q . W e let i : T D T ∗ N ( N ) − → T ( N ) be the embedding and i ∗ be the functor induced on the ﬁl- tered derived categories. W e let Y s : T ( N ) − → D ﬁl ( T ( N )), Y s 1 : T D T ∗ N ( N ) − → D ﬁl ( T D T ∗ N ( N )) DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 15 be the D ﬁl ( k )-enr iched Y oneda functors. N ote that so far we have four Y oneda functors: Y val- ued in M od f i l ( F ∞ ( D T ∗ N )), Y F valued in D ﬁl ( F ∞ ( D T ∗ N )) and Y s , Y s 1 on the T amarkin side , which we r egard Y F , Y s 1 , Y s as D ﬁl ( k )-enr iched Y oneda functors. W e obtain the commutative diagram F ∞ ( D T ∗ N ) Q / / Y F   T D T ∗ N ( N ) i / / Y s 1   T ( N ) Y s   D ﬁl ( F ∞ ( D T ∗ N )) D ﬁl ( T D T ∗ N ( N )) Q ∗ o o D ﬁl ( T ( N )) i ∗ o o Proposition 5.1. W e have equivalences in D ﬁl ( F ∞ ( D T ∗ N )) : (5.1) Q ∗ i ∗ Y s ( k { x } × [ a , +∞ ) ) ≃ V ( x , a ) . Proof . By identifying F ∞ ( D T ∗ N ) with its opposite F ∞ ( D T ∗ N ) op via L 7→ − L , we can iden- tify the Y oneda modules V ( x , a ) : L ′ 7→ F C ∗ ( L ′ , V ( x , a ) ) with the (anti-)coY oneda modules, L ′ 7→ F C ∗ ( V ( x , a ) , − L ′ ). On the other hand, F C ∗ ( V ( x , a ) , − L ′ ) ≃ Γ ( T x N , T a Q ( − L ′ )) ≃ ( Q ( − L ′ )) ( x , a ) , by Corollary 4.5, where the right-hand side is repr esented b y k { x } × [ a , +∞ ) in T ( N ). W e notice that all equivalences ar e functorial in L ′ with respect to evident nat ural module morphisms. □ The categories F ∞ ( D T ∗ N ), T D T ∗ N ( N ) and T ( N ) come with translation functors , hence interleaving distances. By Remar k 3.1 their derived categories inherit translation func tors and distances. W e denote all these distances b y γ but when we want to be mor e precise we let γ F , γ s 1 , γ s be t he distances on F ∞ ( D T ∗ N ), T D T ∗ N ( N ), T ( N ) and we let ˆ γ F , ˆ γ s 1 , ˆ γ s be t he distances on the derived categories. The functors Y F , Y s 1 , Y s are fully faithful and commute with the translation functors, hence they preserve the distances. Ho wever Q ∗ and i ∗ are only 1-Lipschitz (although Q , i are isomet- ric embeddings): (5.2) ˆ γ F ( Q ∗ ( F ), Q ∗ ( G )) ≤ ˆ γ s 1 ( F , G ), ˆ γ s 1 ( i ∗ ( F ), i ∗ ( G )) ≤ ˆ γ s ( F , G ). Projector. W e recall that w e want to pr ove the following density statement: for a given L ∈ F ∞ ( D T ∗ N ) an d a giv en ε > 0 there exists ﬁnitely many ﬁbers V ( x 1 , a 1 ) , . . . , V ( x p , a p ) and an iter- ated cone C of their Y oneda modules V ( x 1 , a 1 ) , . . . , V ( x p , a p ) such that ˆ γ F ( Y F ( L ), C ) < ε . In view of the isomorphism (5.1) and the Lipschitz bounds (5.2) it would be enough to have such a density result in T ( N ), that is, “ there exists an iterated cone F of the k { x i } × [ a i , +∞ ) ’ s such that γ s ( i Q ( L ), F ) < ε ” . Ho wever this latter statement cannot be true because there is no non-zero morphism in T ( N ) between k { x i } × [ a i , +∞ ) and k { x j } × [ a j , +∞ ) as soon as x i = x j . The crucial point is that the composition i ∗ Y s factors through T D T ∗ N ( N ). Indeed the func- tor i has both a left and a right adjo int, by [K uo23] and [KSZ23]. M ore precisely [K uo23] prov es the existence of adjoints of the embedding Sh Z ( N ) − → Sh( N ) (denoted W + and W − in loc. cit. and constr ucted by a wrapping procedure) in the case where N is compact and Z is a closed conic subset of T ∗ N (Sh Z ( N ) denotes the categor y of sheaves with microsupport contained in Z ). These r esults are extended to our setting (the non compact man ifold N × R ) in [KSZ23]. Let us denote b y P ′ D T ∗ N the right adjoint to i . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 16 W e then have, for any F ∈ T D T ∗ N ( N ) and G ∈ T ( N ) (5.3) i ∗ Y s ( G )( F ) = H om  T ( N ) ( i ( F ), G ) ≃ H om  T D T ∗ N ( N ) ( F , P ′ D T ∗ N ( G )) which shows that i ∗ Y s ( G ) is repr esented in T D T ∗ N ( N ) by P ′ D T ∗ N ( G ). W e thus also have the commutative diagram F ∞ ( D T ∗ N ) Q / / Y F   T D T ∗ N ( N ) Y s 1   T ( N ) Y s   P ′ D T ∗ N o o D ﬁl ( F ∞ ( D T ∗ N )) D ﬁl ( T D T ∗ N ( N )) Q ∗ o o D ﬁl ( T ( N )) i ∗ o o Let us set W ( x , a ) = P ′ D T ∗ N ( k { x } × [ a , ∞ ) ). Then (5.1) translates into (5.4) Q ∗ ( W ( x , a ) ) ≃ V ( x , a ) and Theorem 1.1 follo ws from the follo wing density statement for sheaves . Theorem 5.2 (Density Theorem for sheaves) . Let F ∈ T D T ∗ N ( N × R ) such that for any x ∈ N the sheaf F ⊗ k { x } × R is a γ -limit of constructible sheaves on R . Then for any ε > 0 , there are points ( x i ) i ∈ {1,..., l } , real numbers ( a i ) i ∈ {1,..., l } and C ∈ 〈 W ( x 1 , a 1 ) , . . . , W ( x l , a l ) 〉 in T D T ∗ N ( N × R ) such that γ s ( F , C ) < ε . T o apply it to Theorem 1.1, we note that b y R emark 4.6, Q ( L ) ⊗ k { x } × R is a γ -limit of con- structible sheaves on R for each x . Remar k 5.3 . The hypothesis on the sheaf F in Theor em 5.2 is w eaker than “ F is a γ -limit of constr uctible sheaves ” . Indeed, since we assume that its microsupport is contained in ρ − 1 ( D T ∗ N ) (which can be seen as a quantitative version of “ non-characteristic with respect to the ﬁbers { x } × R ”), we can bound the distance on { x } × R b y the distance on N × R using Proposition 6.10 belo w . 6. D I S TA N C E S O N T H E C AT E G O R Y O F S H E AV E S In this section we consider a manifold M , which will be either N or N × R , with N a closed manifold. W e recall and extend some results about interleaving distances on Sh( M ). The interleaving distance for sheaves was ﬁrst introduced in [KS18] and some variations are con- sidered in [AI20] and [GVit22a]. Any non-negative homogeneous Hamiltonian function on T ∗ M \ 0 M induces a translation f unctor on Sh( M ). This was ﬁrst noticed by T amarkin [T am08] in the special case M = N × R with T c the direct image b y the translation in the R direction. 6.1. Sheaf distance for a continuous norm. Let h : T ∗ M \0 M − → R be a smooth function which is positively homogeneous in the sense that h ( x , λ p ) = λ h ( x , p ) for λ > 0. W e set for short Sh h ≥ 0 ( M ) = Sh {( x , p ) | h ( x , p ) ≥ 0} ( M ). T o such an h we associate a distance γ h on Sh h ≥ 0 ( M ) deﬁned in the following way . W e let ϕ h be the Ham iltonian ﬂow of h and K ϕ h ∈ Sh( M 2 × R ) the sheaf deﬁned by ϕ h in [GKS12]. V er y often we write for short K h = K ϕ h . W e recall that K ϕ h is deﬁned so that S S ( K a ϕ h ) is the DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 17 graph of ϕ − a h , where K a ϕ h is the restriction of K ϕ h to M 2 × { a } – more pr ecisely (use [GKS12, Lem. A.1], applied to an autonomous Hamiltonian), away from the zer o section, (6.1) S S ( K ϕ h ) = © ( ϕ h ( x , p ), x , − p , s , − h ( x , p )) | ( x , p ) ∈ T ∗ M , s ∈ R ª . H ere we point out that the order of c omposition of sheaves can be misleading: if f : X − → Y is a function, with graph Γ f ⊂ X × Y , and y ∈ Y , then k Γ f ◦ k { y } = k { f − 1 ( y )} . So , if we want that S S ( K s h ◦ F ) = φ s h ( S S ( F )), we have to be careful that S S ( K s h ) should be the graph of ( φ s h ) − 1 . F or a given a , the composition with K a ϕ h , also sometimes denoted K a ϕ h : Sh( M ) − → Sh( M ), F 7→ K a ϕ h ( F ) = K a ϕ h ◦ F , preserves the subcategory Sh h ≥ 0 ( M ). M oreo ver , if we consider the whole family par ametrized b y a , the functor K ϕ h : S h( M ) → Sh( M × R ), F 7→ K ϕ h ◦ F , restricts to a functor Sh h ≥ 0 ( M ) → Sh α ≤ 0 ( M × R ), wher e α is the dual 10 of a ∈ R . This follo ws from (6.1) since the variable α is given b y − h (see also [GKS12, Prop . 4.8]). The restrictions of K a ϕ h ( − ) to Sh h ≥ 0 ( M ) (for all a ) come with the natural T amarkin mor- phisms τ h a , b ( F ) : K a ϕ h ( F ) − → K b ϕ h ( F ) for a ≤ b . Let us r ecall brieﬂy their construction. On the categor y Sh α ≤ 0 ( M × R ), the restriction to M × { a } can be computed by F | a ≃ π M ! ( F ⊗ k ( −∞ , a ) [1]) as can be seen by applying the functor π M ! ( F ⊗ − ) to the tr iangle k ( −∞ , a ] − → k { a } − → k ( −∞ , a ) [1] + 1 − − → and using the fact that Γ c ( R ; G ) ≃ 0 for a sheaf G ∈ Sh α ≤ 0 ( R ) with support con- tained in ( −∞ , a ]. Then the open inclusion ( −∞ , a ) ⊂ ( −∞ , b ) induces a natural morphism τ a , b ( F ) : F | a → F | b for a ≤ b and τ h a , b ( F ) is constructed as K ϕ h ( F ) | a ≃   τ a , b ( K ϕ h ( F )) / / K ϕ h ( F ) | b ≃   K a ϕ h ( F ) τ h a , b ( F ) / / K a ϕ h ( F ) As noted in [GVit22a] , the data ( K a ϕ h , τ h 0, a ) a ∈ R deﬁnes an − → R -action on Sh h ≥ 0 ( M ) (since τ a , b and K a ϕ h do), and it deﬁnes an interleaving type pseudo-distance on Sh h ≥ 0 ( M ) that we denote b y γ h : for F , G ∈ Sh h ≥ 0 ( M ), (6.2) γ h ( F , G ) = inf n a + b | a , b ≥ 0, ∃ u : F − → K b ϕ h ( G ), v : G − → K a ϕ h ( F ), K a ϕ h ( u ) ◦ v : G − → K a + b ϕ h G , K b ϕ h ( v ) ◦ u : F − → K a + b ϕ h F are homotopic to τ h 0, a + b ( − ) o I t will be useful in this paper to extend the deﬁnition of γ h to the case where h is only continuous. F or example , for two homogeneous functions h : T ∗ M \ 0 M − → R , h ′ : T ∗ M ′ \ 0 N − → R we deﬁne the sum h + h ′ b y ( h + h ′ )( x , p , x ′ , p ′ ) = h ( x , p ) + h ′ ( x ′ , p ′ ) on ( T ∗ M \ 0 M ) × ( T ∗ M ′ \ 0 M ′ ) and extend it b y continuity to T ∗ ( M × M ′ ) \ 0 M × M ′ . In general h + h ′ is not a C 1 function. T o deﬁne the distance γ h it is enough to know the sheaf K ϕ h . W e claim that a similar sheaf can be naturally deﬁned for a continuous h , by appro ximating h with smooth functions. Let us introduce some notations. W e recall that we assume that M is closed, or M = N × R with 10 The symplectic form on T ∗ ( M × R ) is d p ∧ d q + d α ∧ d a DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 18 N closed. W e let C ∞ p h ( T ∗ M ) (resp . C 0 p h ( T ∗ M )) be the space of smooth (resp . continuous) positively homogeneous ( p h stands for positively homogeneous) functions on T ∗ M ; in the case M = N × R we ask that the functions h in C ∞ p h ( T ∗ M ), or C 0 p h ( T ∗ M ), do not depend on the variable t on R (so for τ = 0, h ( x , p , t , τ ) = τ h 0 ( x , p / τ ) for some function on T ∗ N ). Let us pick a metric on N . In case M = N × R we endo w M with the product metric. Lemma 6.1. W e consider the non-negative function g on T ∗ ( M 2 × R ) \ 0 M 2 × R deﬁned b y g ( x , p , x ′ , p ′ , a , α ) = ( p 1 + a 2 ∥ ( p , p ′ ) ∥ 2 + α 2 ) 1/2 (we replace ∥ ( p , p ′ ) ∥ by ∥ ( p , τ , p ′ , τ ′ ) ∥ in the case M = N × R ). Let h ∈ C 0 p h ( T ∗ M ) be given and let ( h n ) n be a sequence in C ∞ p h ( T ∗ M ) C 0 -converging to h . Then the sequence K ϕ h n is Cauchy with r espect to the distance γ g . The limit K h : = γ g − lim n K ϕ h n is well-deﬁned and independent of the sequence ( h n ) n . Proof . The ﬁrst asser tion follo ws for example fr om [Asa+25, Thm. 1.13] which says that, for the function g ′ = ∥ ( p , p ′ ) ∥ , γ g ′ ( k ∆ M , K 1 h n ) is bounded b y the H ofer norm of h n . W e note that in loc. cit. the au thors consider the compact case and they consider time 1 and not all times together as we do her e. No w their proof extends to our non compact situation which is M = N × R , taking into account the fact that the Hamitlonians are independent of the variable t : the proof is actually given in Lemma 3.1 of [Asa+25] and the reader can see that the compac tness hypothesis is only used to have ﬁnite bounds on h n / g . The result for time 1 also extends because we have r escalled the metric b y the factor p 1 + a 2 ∼ | a | ( | a | − → ∞ ). Indeed K a h n ≃ K 1 a h n , hence γ | a | g ′ ( k ∆ M , K a h n ) = (1/ | a | ) γ g ′ ( k ∆ M , K 1 a h n ) is bounded b y the Hofer norm of h n . The fact that Cauchy sequences have a limit follows from [GVit22a, Prop . 6.22] or [AI24]. It is clear that the distance between two different limits (possibly for different Cauchy sequences ( h n ) n ) is 0. The distance γ g is degenerate but, if we endow M with a real analytic structur e and choose a real analytic metric, then [GVit22a, P rop . 6.22] says that γ g is non-degenerate when we r estr ict to sheaves that are limit of constructible sheaves. No w we can assume that the functions h n are real analytic, which proves that one limit K h (hence all) is a limit of constructible sheaves, hence uniquely deﬁned. □ Lemma 6.1 says that K h is uniquely deﬁned up to equivalence. W e can also giv e a func- torial construction of K h on M 2 × [0, ∞ ) as follows , which will also pro ve that K h is inde- pendent of the choice of a metric. W e consider C ∞ p h ( T ∗ M ) and C 0 p h ( T ∗ M ) as posets for the usual order ≤ . If h ≤ h ′ are smooth positively homogeneous functions, then we hav e a natural morphism, for any a ≥ 0, K a ϕ h − → K a ϕ ′ h . W e even have a morphism K ϕ h | M 2 × [0, ∞ ) − → K ϕ ′ h | M 2 × [0, ∞ ) . By [K uo23] the morphisms K ϕ h | M 2 × [0, ∞ ) − → K ϕ ′ h | M 2 × [0, ∞ ) organize into a func- tor from the poset C ∞ p h ( T ∗ M ) to the categor y Sh( M 2 × [0, ∞ )). F or a given h 0 ∈ C 0 p h ( T ∗ M ) we let C ∞ p h , < h 0 ( T ∗ M ) be the subposet of C ∞ p h ( T ∗ M ) of functions h such that h ( z ) < h 0 ( z ) for any z . Any incr easing sequence ( h n ) n in C ∞ p h , < h 0 ( T ∗ M ) which conv erges to h 0 is coﬁnal in C ∞ p h , < h 0 ( T ∗ M ). By [GVit22a, Lem. 6.21] we also have K h 0 | M 2 × [0, ∞ ) ≃ colim n K ϕ h n | M 2 × [0, ∞ ) . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 19 H ence we obtain a functorial description of K h 0 | M 2 × [0, ∞ ) (6.3) K h 0 | M 2 × [0, ∞ ) = colim h K ϕ h | M 2 × [0, ∞ ) , where h runs o ver C ∞ p h , < h 0 ( T ∗ M ). From (6.3) we see that h 7→ K h | M 2 × [0, ∞ ) extends the functor h 7→ K ϕ h | M 2 × [0, ∞ ) , initially deﬁned on C ∞ p h ( T ∗ M ), to a functor from C 0 p h ( T ∗ M ) to Sh( M 2 × [0, ∞ )). W e remark that (6.3 ) still holds if we let h runs o ver the poset C 0 p h , < h 0 ( T ∗ M ) which we deﬁne like C ∞ p h , < h 0 ( T ∗ M ). F or negative times we have a similar formula: K h 0 | M 2 × ( −∞ ,0] = colim h K ϕ h | M 2 × ( −∞ ,0] , where no w h r uns o ver the poset C 0 p h , > h 0 ( T ∗ M ) of functions greater than h 0 . Since colimits of sheav es commute with inverse image, the formula (6.3) implies , for any a ∈ R , K a h 0 : = K h 0 | M 2 × { a } ≃ colim h K a ϕ h where h r uns o ver C ∞ p h , < h 0 ( T ∗ M ) (resp . C ∞ p h , > h 0 ( T ∗ M )) when a ≥ 0 (r esp . a ≤ 0). F or a = 0 we ﬁnd K 0 h 0 ≃ k ∆ M as expected. Lemma 6.2. Let h ∈ C 0 p h ( T ∗ M ) . W e deﬁne Γ h ⊂ T ∗ ( M × R ) by Γ h = {( x , − p , a , − h ( x , p )) | ( x , p ) ∈ T ∗ M , a ∈ R }. Then S S ( K h ) ⊂ T ∗ M × Γ h . In par ticular the functor K h : Sh( M ) → Sh( M × R ) , F 7→ K h ◦ F , r estricts to a functor Sh h ≥ 0 ( M ) → Sh α ≤ 0 ( M × R ) . Proof . W e pick a sequence ( h n ) n ∈ N in C ∞ p h , > h ( T ∗ M ) converging to h . W e have S S ( K h ) ⊂ lim inf S S ( K h n ) by [KS90, Ex. 5.7] (or [GVit22a, P rop . 6.26] for details), where lim inf j X j = © x | ∃ ( x j ) j ≥ 1 , x j ∈ X j , lim j x j = x ª . Since S S ( K h n ) is given b y (6 .1), we have the rough bound S S ( K h n ) ⊂ T ∗ M × Γ h n and we deduce the ﬁrst assertion. N ow we have a bound for S S ( K h ◦ F ) given by a “ set theoretic composition ” S S ( K h ) ◦ S S ( F ) (see [GKS12, (1.12)]). The bound is {( x , p , a , α ) | ∃ x ′ , p ′ , ( x , p , x ′ , − p ′ , a , α ) ∈ S S ( K h ), ( x ′ , p ′ ) ∈ S S ( F )}. The result follo ws. □ By Lemma 6.2 and the discussion in the beginning of this section, ther e exist natural mor- phisms τ h a ( F ) : F − → K a h ( F ) for F ∈ Sh h ≥ 0 ( M ) and a ≥ 0, which induce an − → R -action on Sh h ≥ 0 ( M ). W e can no w deﬁne a pseudo-distance as in the smooth case: Deﬁnition 6.3. Let h : T ∗ M − → R be a positively homogeneous continuous function and let K h be the sheaf deﬁned in Lemma 6.1. U sing this sheaf K h we deﬁne a pseudo-distance γ h on Sh h ≥ 0 ( M ) as in (6.2) . 6.2. Some properties of the distance. Lemma 6.4. Let h : T ∗ M \ 0 M − → R , h ′ : T ∗ M ′ \ 0 M ′ − → R be two continuous homogeneous func- tions. Then K h + h ′ ≃ K h ⊠ R K h ′ , where ⊠ R denotes the exterior tensor product r elative to the factor R . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 20 Proof . W e ﬁrst assume that h , h ′ are smooth on T ∗ M \ 0 M , T ∗ M ′ \ 0 M ′ . W e pick a sequence k n in C ∞ p h ( T ∗ ( M × M ′ )) converging to h + h ′ . W e can assume that k n = h + h ′ on U ( ε n ) = ( T ∗ M \ D ε n T ∗ M ) × ( T ∗ M ′ \ D ε n T ∗ M ′ ) for some sequence ( ε n ) n converging to 0 and that d k n is small on Z n = T ∗ ( M × M ′ ) \ U n . Let us denote by Γ ϕ h the graph of ϕ h as described in the right hand side of (6.1). Then Γ ϕ k n ∩ U ( ε n ) = ( Γ ϕ h × R Γ ϕ h ′ ) ∩ U ( ε n ) and, choosing d k n small enough on Z n , we ﬁnd lim inf n Γ ϕ k n = Γ ϕ h × R Γ ϕ h ′ . U sing the “limit property” of microsupports as in the proof of Lemma 6.2, we deduce that S S ( K h + h ′ ) ⊂ Γ ϕ h × R Γ ϕ h ′ . T o conclude it is enough to pro ve that there exists a unique K ∈ Sh(( M × M ′ ) 2 × R ) such that K | ( M × M ′ ) 2 × {0} ≃ k ∆ M × M ′ and S S ( K ) ⊂ Γ ϕ h × R Γ ϕ h ′ . W e follo w the proof of the uniqueness of the quantization [GKS12, Prop . 3.2]. The main point is that K ϕ h is invertible with respect to the composition of sheaves, with inverse K ⊗− 1 ϕ h ≃ K ϕ − h . W e consider the composition r elative to the factor R and set K ′ = K ◦ R ( K ⊗− 1 ϕ h ⊠ R K ⊗− 1 ϕ h ′ ). Then the bounds for the behaviour of the microsupport under sheaf oper ations in [KS90, §5.4] (also used in the proof of Lemma 6.2) give S S ( K ′ ) ⊂ T ∗ ∆ M × M ′ × R (( M × M ′ ) 2 × R ). S ince we also have K ′ | ( M × M ′ ) 2 × {0} ≃ k ∆ M × M ′ , this im- plies that K ′ ≃ k ∆ M × M ′ × R and then K ≃ K ϕ h ⊠ R K ϕ h ′ . This gives the r esult when h , h ′ are smooth. In gener al we pick increasing sequences ( h n ) n , ( h ′ n ) n in C ∞ p h , < h ( T ∗ M ) and C ∞ p h , < h ′ ( T ∗ M ′ ) converging to h , h ′ . Then the sequence ( h n + h ′ n ) n is coﬁnal in C 0 p h , < h + h ′ ( T ∗ ( M × M ′ )). H ence (6.3) (which also holds when the colimit i s taken over continuous functions, as we already noticed) gives the result over ( M × M ′ ) 2 × [0, ∞ ) because colimits of sheaves commute with ⊠ R . W e have in the same way the result for negative times and this concludes the proof. □ Lemma 6.5. Let h : T ∗ M \ 0 M − → R , h ′ : T ∗ M ′ \ 0 M ′ − → R be two continuous positively homoge- neous functions. Let F , G ∈ Sh h ≥ 0 ( M ) , F ′ , G ′ ∈ Sh h ′ ≥ 0 ( M ′ ) . Then F ⊠ F ′ ∈ Sh h + h ′ ≥ 0 ( M × M ′ ) and (6.4) γ h + h ′ ( F ⊠ F ′ , G ⊠ G ′ ) ≤ max{ γ h ( F , G ), γ h ′ ( F ′ , G ′ )}. Proof . By [KS90, Prop . 5.4.4], we have F ⊠ F ′ ∈ Sh h + h ′ ≥ 0 ( M × M ′ ). By Lemma 6.4 we also have K a ϕ h + h ′ ≃ K a ϕ h ⊠ K a ϕ h . W e deduce τ h + h ′ 0, a ( F ⊠ F ′ ) ≃ τ h 0, a ( F ) ⊠ τ h ′ 0, a ( F ′ ) for a ≥ 0. Let d > max{ γ h ( F , G ), γ h ′ ( F ′ , G ′ )} and let u , v (r esp . u ′ , v ′ ) be interleaving morphisms for F , G (resp . F ′ , G ′ ) for v alues a , b with a + b ≤ d . Then u ⊠ u ′ and v ⊠ v ′ are interleaving morphisms for F ⊠ F ′ , G ⊠ G ′ and the values a , b . The result follo ws. □ Let i : S − → M be the inclusion of a closed submanifold. W e denote the transpose derivative b y ( d i ) t : S × M T ∗ M − → T ∗ S . Lemma 6.6. Let h : T ∗ M \ 0 M − → R , h ′ : T ∗ S \ 0 S − → R be smooth positively homogeneous func- tions. W e assume that h | S × M T ∗ M = h ′ ◦ ( d i ) t . Then K h ′ ≃ K h | S 2 × R . Proof . W e ﬁrst check that the ﬂo w ϕ h preserves the coisotropic submanifold S × M T ∗ M , whose symplectic reduction is T ∗ S , and induces ϕ h ′ on T ∗ S . W e take coordinates ( s , s ′ ) on M (and ( s , s ′ , ξ , ξ ′ ) on T ∗ M ) such that S = { s ′ = 0}. The hypothesis says that h ( s , 0, ξ , ξ ′ ) = h ′ ( s , ξ ). Since DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 21 h is homogeneous, we have h ( s , s ′ , ξ , ξ ′ ) = P i ξ i ( ∂ h / ∂ξ i ) + P j ξ ′ j ( ∂ h / ∂ξ ′ j ). H ence the hypoth- esis implies ( ∂ h / ∂ξ ′ j )( s , 0, ξ , ξ ′ ) = 0, for any j , and the Hamiltonian vector ﬁeld of h is tan- gent to S × M T ∗ M . M ore pr ecisely X h | S × M T ∗ M = X h ′ + P j ( ∂ h / ∂ s ′ j ) ∂ ξ ′ j , and then, ϕ a h ( s , 0, ξ , ξ ′ ) = ( s 1 , 0, ξ 1 , ξ ′ 1 ) with ( s 1 , ξ 1 ) = ϕ a h ′ ( s , ξ ), as claimed. W e want to understand the inverse image of K h b y the inclusion j = i × i × id R : S 2 × R − → M 2 × R . W e r ecall that there exists an easy bound for the microsupport of the inverse image of a sheaf F b y a map f : X − → Y if f is non-characteristic for S S ( F ), which means, when f is an inclusion, that S S ( F ) \ 0 Y does not meet T ∗ X Y : in this case we have S S ( f − 1 ( F )) ⊂ ( d f ) t ( S S ( F ) ∩ X × Y T ∗ Y ). When F is a direct image b y f , F = f ∗ ( F ′ ) for some F ′ , then we are in general in a characteristic case , although f − 1 ( F ) ≃ F ′ is easy to understand. W e are in a similar case here and j is a prior i characteristic for S S ( K h ). W e decompose j = j 2 ◦ j 1 with j 1 : S 2 × R − → M × S × R , j 2 : M × S × R − → M 2 × R the inclusions. W e set C = ( M × S × R ) × M 2 × R T ∗ ( M 2 × R ) Γ = ( d j 2 ) t ( Γ ϕ h ∩ C ) C ′ = ( S 2 × R ) × M × S × R T ∗ ( M × S × R ). Since ϕ h preserves S × M T ∗ M and ϕ a h is a bijection, we obtain Γ ⊂ C ′ and Γ = (( d j 1 ) t ) − 1 ( Γ ϕ h ′ ). N ow Γ ϕ h is non-characteristic for j 2 and we hav e S S ( j − 1 2 ( K h )) ⊂ Γ by [KS90, Prop . 5.4.13]. H ence the microsupport of j − 1 2 ( K h ) is contained in the zero-sect ion away from S 2 × R and it follo ws that j − 1 2 ( K h ) is locally constant there . Since at time 0 K h ≃ k ∆ M we deduce that the support of j − 1 2 ( K h ) is contained in S 2 × R , hence j − 1 2 ( K h ) ≃ j 1, ∗ ( j − 1 ( K h )). By [KS9 0, Prop . 5.4.4] we deduce that S S ( j − 1 ( K h )) ⊂ Γ ϕ h ′ . H ence j − 1 ( K h ) satisﬁes the conditions which de- termine K h ′ , as requir ed. □ Lemma 6.7. Let h ∈ C 0 p h ( T ∗ M ) , h ′ ∈ C 0 p h ( T ∗ S ) be continuous positively homogeneous func- tions such that h | S × M T ∗ M = h ′ ◦ ( d i ) t . Then K h ′ ≃ K h | S 2 × R . Proof . It is enough to pro ve the result when we restrict to S 2 × I ± with I + = [0, ∞ ) and I − = ( −∞ , 0]. In both cases we can use the description of K h as a colimit explained in (6.3): K h | M 2 × I + = colim h K h 1 | M 2 × I + where h 1 runs o ver C ∞ p h , < h ( T ∗ M ) (and the similar for mula for I − ). W e let C ∞ p h , < h , < h ′ ( T ∗ M ) ⊂ C ∞ p h , < h ( T ∗ M ) be the subposet formed b y the h 1 such that h 1 | S × M T ∗ M = h ′ 1 ◦ ( d i ) t for some function h ′ 1 ∈ C ∞ p h , < h ′ ( T ∗ S ). W e can see that C ∞ p h , < h , < h ′ ( T ∗ M ) is coﬁnal in C ∞ p h , < h ( T ∗ M ) and the r esult follo ws from Lemma 6.6 and the fact that the inverse image of sheaves commutes with colimits . □ Lemma 6.8. Let h ∈ C 0 p h ( T ∗ M ) , h ′ ∈ C 0 p h ( T ∗ S ) be continuous positively homogeneous func- tions such that h | S × M T ∗ M = h ′ ◦ ( d i ) t . Then for any F , G ∈ Sh ( M ) we have γ h ′ ( F | S , G | S ) ≤ γ h ( F , G ) . Proof . The proof is the same as the pr oof of Lemma 6.5, using Lemma 6.7 instead of Lem- ma 6.4. □ DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 22 N ow we compare two distances γ h 1 , γ h 2 assuming h 1 ≤ h 2 on some given conic subset W . W e have to be careful that this inequality h 1 | W ≤ h 2 | W holds for any time and a natural hy- pothesis is to assume moreo ver that W is stable by ϕ h 1 and ϕ h 2 . H owev er , when h 1 , h 2 are only continuous, we cannot consider their ﬂows . Instead we assume that W is given by W = { f ≥ 0} for some smooth function f . For h smooth, ϕ h preserves W if X h is tangent to ∂ W , that is, 〈 X h , d f 〉| ∂ W = 0, or equ ivalently , 〈 X f , d h 〉| ∂ W = 0. The latter condition means tha t h is con- stant along the trajectories of X f contained in ∂ W , which is meaningful also when h is only continuous. M oreo ver if a continuous h satisﬁes this condition, it can be appro ximated b y smooth functions also satisfying the same condition. Lemma 6.9. Let W ⊂ T ∗ M \ 0 M , be a closed conic subset. W e assume that W = T k i = 1 W i with W i = { f i ≥ 0} for smooth functions f i on T ∗ M \ 0 M . Let h 1 , h 2 : T ∗ M \ 0 M − → R be two continuous homogeneous functions. W e assume that (1) h 1 | W ≤ h 2 | W , (2) h 1 and h 2 ar e constant along the orbits of the ﬂow ϕ f i contained in ∂ W , for i = 1, . . . , k . Let i W : S h W ( M ) − → Sh( M ) be the embedding. Then ther e exist morphisms of functors σ s : K s h 1 ◦ i W − → K s h 2 ◦ i W for s ≥ 0 such that τ h 2 s = σ s ◦ τ h 1 s . Proof . W e ﬁrst assume that h 1 and h 2 are smooth and h 1 | W < h 2 | W . W e deﬁne an isotopy ψ b y ψ s = ϕ − 1 h 1 , s ◦ ϕ h 2 , s . I ts H amiltonian function h is given by h s ( z ) = h 2 ( ϕ h 1 , s ( z )) − h 1 ( ϕ h 1 , s ( z )). Since ϕ h 1 , ϕ h 2 preserve W , we have h > 0 on W . W e can ﬁnd a conic neighborhood Ω of W × R in T ∗ M × R such that h > 0 on Ω . Let b : ( T ∗ M \ 0 M ) × R − → [0, 1] be a bump function such that b ( x , λ p , s ) = b ( x , p , s ) for any λ > 0 and b = 1 on W × R , b = 0 outside Ω . W e set h ′ = b h and let ψ ′ be its H amiltonian ﬂo w . Then ψ ′ is nonnegative and coincides with ψ on W × R . W e deduce morphisms of functors id − → K ψ ′ s for s ≥ 0, hence i W − → K ψ ′ s ◦ i W = K ψ s ◦ i W . The compatibility with the natural morphism τ follo ws from the more gener al fact (recalled abo ve) that the morphisms K ϕ h − → K ϕ ′ h organize into a functor from C ∞ p h ( T ∗ M ) to Sh( M 2 × [0, ∞ )). N ow , composing with K ϕ h 1 , s we obtain the lemma for smooth functions. If h 1 and h 2 are only continuous and satisfy (1) and (2), we can ﬁnd increasing sequences of smooth functions ( h 1, n ) n , ( h 2, n ) n converging to h 1 , h 2 and satisfying h 1, n | W < h 2, n | W and (2). W e then have morphisms K s ϕ h 1, n ◦ i W − → K s ϕ h 2, n ◦ i W . T aking the colimit over n we obtain the result b y (6.3). □ Proposition 6.10. W e use the notations of Lemma 6.9 and make the same hypotheses. M oreo ver we assume h 1 ≥ 0 on W . Then, for F , G ∈ Sh W ( M ) we have γ h 2 ( F , G ) ≤ γ h 1 ( F , G ) . Proof . Let d ≥ γ h 1 ( F , G ). There exist a , b ≥ 0 such that a + b ≤ d and morphisms u : F − → K a h 1 ( G ), v : G − → K b h 1 ( F ) such that u ◦ v and u ◦ v ar e homotopic to the morphisms τ h 1 . U sing DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 23 the morphisms σ s of Lemma 6.9 we set u ′ = σ a ◦ u , v ′ = σ b ◦ v . W e have the diagram F   u / / K a h 1 ( G )   v / / K a + b h 1 ( F )   u / / K 2 a + b h 1 ( G )   F u ′ / / K a h 2 ( G ) v ′ / / K a + b h 2 ( F ) u ′ / / K 2 a + b h 2 ( G ) where the horizontal compositions of two consecutive arro ws are the morphisms τ h 1 and τ h 2 . W e deduce that γ h 2 ( F , G ) ≤ d , as requir ed. □ Lemma 6.11. W e choose a Riemannian metric ∥ • ∥ on M and we let g : T ∗ M \ 0 M − → R be the norm function ( x , p ) 7→ ∥ p ∥ . Let x ∈ M and let Z ⊂ M be a closed contractible subset. W e assume that x ∈ Z and Z ⊂ B ε , where B ε is the open ball with center { x } and radius ε , for some ε < r i n j ( M )/3 . Then γ g ( k { x } , k Z ) ≤ 2 ε . Proof . (i) Let u s set F s = K s g ◦ k { x } , G s = K s g ◦ k Z . For 0 < r < r i n j ( M ) we have F − r ≃ k B r and F r ≃ k B r [ n ], where n = dim M . I ndeed we know that the microsupport of F r is the outer conormal of ∂ B r outside the zero section and that F r is supported in B r . It follo ws that F r must be of the form F r ≃ E B r for some E ∈ D( k ). Since the global sections ar e independent of r , we ﬁnd k ≃ Γ ( M ; E B r ) ≃ E [ − n ] and the r esult follows . The hypotheses x ∈ Z and Z ⊂ B ε imply the existence of natural r estri ction morphisms F − ε a − → G 0 b − → F 0 and the composition b ◦ a is the morphism τ ε ( F − ε ). W e set a ′ = K ε g ( a ) and a ′′ = τ ε ( G ε ) ◦ a ′ : F 0 − → G 2 ε . Then K 2 ε g ( b ) ◦ a ′′ = τ 2 ε ( F 0 ) and it remains to pro ve that a ′′ ◦ b = τ 2 ε ( G 0 ). (ii) Let us pro ve that Hom( G 0 , G 2 ε ) ≃ k . The ﬂow φ g ,2 ε sends any point ( x , p ) with x ∈ B ε to a point ( x ′ , p ′ ) such that x ′ is at distance 2 ε of x in the direction p . In particul ar x ′ ∈ B 3 ε \ B ε . I t follows that S S ( G 2 ε | B ε ) is contained in the zero section, hence G 2 ε is constant on B ε , say G 2 ε | ε ≃ E B ε for some E ∈ D( k ). N ow k ≃ Hom( F − ε , G 0 ) ≃ Hom( F ε , G 2 ε ) ≃ Hom( k B ε [ n ], E B ε ) and we deduce E ≃ k [ n ]. Finally , H om( G 0 , G 2 ε ) ≃ H om( k Z , k B ε [ n ]) ≃ k , as claimed, because supp( G 0 ) = Z is contained in B ε . Since Γ ( M ; G 0 ) ≃ Γ ( M ; G 2 ε ) ≃ k , we deduce that the global section morphism u 7→ Γ ( M ; u ), from H om( G 0 , G 2 ε ) to H om( Γ ( M ; G 0 ), Γ ( M ; G 2 ε )), is an isomorphism. In particular u is deter- mined b y Γ ( M ; u ). (iii) W e recall that Γ ( M ; G s ) is independent of s and moreo ver , for c ≤ d , the natural isomor- phism Γ ( M ; G c ) ∼ − → Γ ( M ; G d ) is induced b y τ c , d ( G ) (see [GKS12, Prop . 4.8]). I t follo ws that Γ ( M ; τ ε ( G ε )) = id k . The same holds for a , hence a ′ , and for b . Finally Γ ( M ; a ′′ ◦ b ) = id k . By (ii) we conclude that a ′′ ◦ b = τ 2 ε ( G 0 ) and b y (i) this ﬁnishes the proof. □ 6.3. Main example: S mall wrapping of the diagonal. W e consider the case M = N × R and put a metric on N . W e denote by ( x , p ) 7→ ∥ p ∥ the induced norm on T ∗ N . W e pick some radius r and consider the function h : T ∗ ( N × R ) \ 0 N × R − → R which is the homogenization of max{0, ∥ p ∥ − r }. W e want to compute the sheaf K h ∈ Sh(( N × R ) 2 × R ) associated with h . Since we work mainly in the T amarkin category we only need to consider its composition w ith the DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 24 projector to the T amarkin category , that is, K − s h ◦ k ∆ N × { t ′ ≤ t } . For this it i s enough to know h on { τ ≥ 0}. Lemma 6.12. Let f : R − → R be a smooth fu nction. W e assume that ther e exist 0 < a < b and r ∈ R such that f ( u ) = 0 on [0, a ] , f ( u ) = u − r on [ b , ∞ ) , and f ′ gives a bijection from [ a , b ] to [0, 1] . W e let h : T ∗ ( N × R ) \ 0 N × R − → R be the homogenization of ( x , p ) 7→ f ( ∥ p ∥ ) , that is, h ( x , t , p , τ ) = τ f ( ∥ p / τ ∥ ) for τ > 0 , extended smoothly to T ∗ ( N × R ) \ 0 N × R . Let s > 0 be less than the injectivity radius of N . Then K − s h ◦ k ∆ N × { t ′ ≤ t } ≃ k C s , wher e C s = {( x , x ′ , t , t ′ ) | d ( x , x ′ ) ≤ s , t ≥ t ′ − s g ( d ( x , x ′ )/ s )} and g ( v ) = f (( f ′ ) − 1 ( v )) − v ( f ′ ) − 1 ( v ) . Proof . W e choose the follo wing convenient way to extend h smoothly to all of T ∗ ( N × R ): we set h ( x , t , p , τ ) = ∥ p ∥ − r τ near { τ = 0} and h ( x , t , p , τ ) = − τ f ( ∥ p / τ ∥ ) − 2 r τ for τ < 0. (If we extend h symmetr ically b y h ( x , t , p , τ ) = | τ | f ( ∥ p / τ ∥ ), we ﬁnd h ( x , t , p , τ ) = ∥ p ∥ − r | τ | near { τ = 0}, which is continuous but not smooth). Let n be the norm function on T ∗ N \0 N , n ( x , p ) = ∥ p ∥ . Let X n be its Hamiltonian vector ﬁeld and ϕ n its ﬂo w (the normalized geodesic ﬂo w on T ∗ N \ 0 N ). For τ > 0 we have h = τ f ( ∥ p / τ ∥ ), hence X h = f ′ ( ∥ p / τ ∥ ) X n + f 1 ( ∥ p / τ ∥ ) ∂ t , with f 1 ( u ) = f ( u ) − u f ′ ( u ). Since ∥ p ∥ is preserved b y ϕ n and τ b y the ﬂo w of ∂ t , we see that the coefﬁcients of X n and ∂ t are constant along the tr a- jectories of X h . W e deduce ϕ h ( x , t , p , τ , − s ) = ( x ′ , t ′ , p ′ , τ ) with ( x ′ , p ′ ) = ϕ n ( x , p , − s f ′ ( ∥ p / τ ∥ )) and t ′ = t − s f 1 ( ∥ p / τ ∥ ). I n particular d ( x , x ′ ) = s f ′ ( ∥ p / τ ∥ ). F or d ( x , x ′ ) ∈ (0, s ) we thus obtain ∥ p / τ ∥ = ( f ′ ) − 1 ( d ( x , x ′ )/ s ), hence t ′ = t − s g ( d ( x , x ′ )/ s ). F or τ < 0 the computation is the same (up to changing some signs and adding X − 2 r τ = − 2 r ∂ t ), which yields ϕ h ( x , t , p , τ , − s ) = ( x ′ , t ′ , p ′ , τ ) with ( x ′ , p ′ ) = ϕ n ( x , p , − s f ′ ( ∥ p / τ ∥ )) and t ′ = t + s (2 r + g ( d ( x , x ′ )/ s )). W e deduce that the front projection of the graph of ϕ − s h is Γ − ∪ Γ + with Γ − = {( x , x ′ , t , t ′ ) | d ( x , x ′ ) ≤ s , t ′ = t − s g ( d ( x , x ′ )/ s )}, Γ + = {( x , x ′ , t , t ′ ) | d ( x , x ′ ) ≤ s , t ′ = t + s (2 r + g ( d ( x , x ′ )/ s ))}. Since K h is uniquely deter mined by the graph of ϕ h and the condition K h | { s = 0} = k ∆ N , and taking into account the fact that S S ( K s h ) is the graph of ϕ − s h , we deduce K − s h ≃ k Γ 0 with Γ 0 = {( x , x ′ , t , t ′ ) | d ( x , x ′ ) ≤ s , t ′ − s g ( d ( x , x ′ )/ s ) ≤ t ≤ t ′ + s (2 r + g ( d ( x , x ′ )/ s ))}. N ow composing with k ∆ N × { t ′ ≤ t } yields the result. □ Lemma 6.13. Let r ≥ 0 be given and let f be the funct ion f ( u ) = max{0, u − r } . W e set h ( x , t , p , τ ) = | τ | f ( ∥ p / τ ∥ ) for τ = 0 and extend by continuity , so h ( x , t , p , τ ) = max{0, ∥ p ∥ − r | τ | } . Let s > 0 be less than the injectivity radius of N . Then K − s h ◦ k ∆ N × { t ′ ≤ t } ≃ k C s with C s = {( x , x ′ , t , t ′ ) | d ( x , x ′ ) ≤ s , t ≥ t ′ + r d ( x , x ′ )} . Proof . W e choose a s equence ( f n ) n of differ entiable functions C 0 converging to f . W e assume that the f n ’ s satisfy the hypotheses of Lemma 6.12, that is, f n coincides with f on the set DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 25 R ≥ 0 \ I n , where I n = [ r − 1 n , r + 1 n ], and f ′ n is increasing on I n . W e set h n ( x , t , p , τ ) = τ f n ( ∥ p / τ ∥ ) for τ > 0. W e extend h n smoothly to T ∗ ( N × R ) \ 0 N × R . By Lemma 6.12 we h ave K − s h n ◦ k ∆ N × { t ′ ≤ t } ≃ k C n , s with C n , s = {( x , x ′ , t , t ′ ) | d ( x , x ′ ) ≤ s , t ≥ t ′ − s g n ( d ( x , x ′ )/ s )} and, for v ∈ [0, 1], g n ( v ) = f n (( f ′ n ) − 1 ( v )) − v ( f ′ n ) − 1 ( v ) (recall that f ′ n identiﬁes I n and [0, 1]). Since ( f ′ n ) − 1 ( v ) ∈ I n , we have 0 ≤ f n (( f ′ n ) − 1 ( v )) ≤ 1/ n and g n tends to the function g ( v ) = − r v . H ence C n , s tends to the set C s of the lemma. □ 7. P R O O F O F T H E M A I N R E S U L T In this section we pro ve Theor em 5.2. Since it deals with the obj ects W ( x , a ) , images of k { x } × [ a , ∞ ) b y the projector P ′ D T ∗ N , we ﬁrst r ecall some results on P ′ D T ∗ N . 7.1. Some r esults on the projector . W e recall the f ollowing description of the projector taken from [KSZ23]. Recall that for any closed subset Z ⊂ T ∗ N , ther e exists a projector P ′ Z , right adjoint to the em bedding i Z of T Z ( N ) in T ( N ) (see the discussion befor e formula 5.3). M ore- o ver P ′ Z is given b y a convolution functor described as follows . Proposition 7.1 (Prop 6.5 of [KSZ23]) . Let Z ⊂ T ∗ N be a closed set, and let H n , n ∈ N , be any incr easing sequence of compactly supported Hamiltonians supported on T ∗ N \ Z such that H n ( u ) − → ∞ for al l u ∈ Z . Let K n = K − 1 H n be their sheaf quantizations, which form an inverse system along cont inuation maps K n − → k ∆ M × [0, ∞ ) . Then we have P ′ Z ( F ) = lim n K n ( F ) and the counit P ′ Z ⇒ id is intertwined with the limit of con tinuation maps. Remar k 7.2 . In the proof of [ KSZ23, Pr op 6.5], the compactly support condition was made for two purposes: 1) T o guarantee the existence of sheaf q uantization K n in the gr eat generality . 2) T o make sur e the H amiltonian sequence is coﬁnal under the assumption H n ( u ) − → ∞ for all u ∈ Z . So , for any coﬁnal Hamiltonian sequence H n (that may not to be compactly supported) such that H n ( u ) − → ∞ for all u ∈ Z and whose sheaf quantizations K n exist, we can conclude the same formula as [KSZ23, Prop 6.5]. In par ticular , w e can construct the projector P ′ Z for Z = D T ∗ N in the follo wing way . W e t ake the continuous function H ( x , p ) = max{0, ∥ p ∥ − 1} and h is the homogenization of H , i.e. the function h in Lemma 6.13. Then for H n = α n H where α n ↗ ∞ , we take K n : = K − α n h ◦ k ∆ N × { t ′ ≤ t } as their sheaf quantization. Then we have P ′ Z ( F ) = lim n K n ( F ) as well. W e remark that the char acterization of P ′ Z as a right adjoint of i Z implies that it is uniquely determined by Z . Lemma 7.3. Let Z ⊂ T ∗ N be a closed subset. Let φ : T ∗ N − → T ∗ N be a Hamiltonian isotopy which admits a contact lift e φ on J 1 ( N ) (or equivalently a lift to T ∗ ( N × R ) \ 0 N × R as a homoge- neous Hamiltonian isotopy). Let K e φ be the sheaf associated with e φ . W e assume that φ ( Z ) = Z and e φ | ρ − 1 ( Z ) = id ρ − 1 ( Z ) . Then P ′ Z ≃ P ′ Z ◦ K e φ . Proof . The hypothesis implies that i Z ≃ K − 1 e φ ◦ i Z and the result follo ws b y adjunction. □ DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 26 Lemma 7.4. Giv en a metric on N and pick s > 0 to be smaller than the injectivity radius. F or ( x 0 , a 0 ) ∈ N × R we de ﬁne a closed subset of N × R , C ( x 0 , a 0 , s ) = {( x , t ) | d ( x , x 0 ) ≤ s , t ≥ a 0 + d ( x , x 0 )} . Then (7.1) W ( x 0 , a 0 ) ≃ P ′ D T ∗ N ( k C ( x 0 , a 0 , s ) ). x t s ( x 0 , a 0 ) x 0 a 0 a 0 + s d ( x , x 0 ) ≤ t − a 0 d ( x , x 0 ) ≤ s F I G U R E 2 . The graph of k C ( x 0 , a 0 , s ) . Proof . Let r > 1. W e use the notations o f Lemma 6.13: f and h are deﬁned b y f ( u ) = max{0, u − r } and h ( x , t , p , τ ) = τ f ( ∥ p / τ ∥ ). Let ( f n ) n be a sequence of differentiable functions C 0 converg- ing to f and set h n ( x , t , p , τ ) = τ f n ( ∥ p ∥ / τ ). W e assume that f n | [0,1] = 0. By Lemma 7.3 we have P ′ Z ≃ P ′ Z ◦ K s h n for all n and s , hence P ′ Z ≃ P ′ Z ◦ K s h . Lemma 6.13 says that K − s h ◦ k ∆ N × [0, ∞ ) ≃ k C s for s > 0 and for some “ conic ” set C s . No w , k C s ◦ k { x 0 } × [ a 0 , ∞ ) ≃ k C ( x 0 , a 0 , s , r ) where C ( x 0 , a 0 , s , r ) = {( x , t ) | d ( x , x 0 ) ≤ s , t ≥ a 0 + r d ( x , x 0 )}. Hence W ( x 0 , a 0 ) ≃ P ′ D T ∗ N ( k C ( x 0 , a 0 , s , r ) ). The sheaves k C ( x 0 , a 0 , s , r ) , r > 1, organize into an inverse system and taking the limit as r goes to 1 gives the result. □ A consequence of Lemma 7.3 is the following distance comparison. W e use again the notations of Lemma 6.13 (with r = 1): f and h are deﬁned by f ( u ) = max{0, u − 1} and h ( x , t , p , τ ) = | τ | f ( ∥ p / τ ∥ ). W e also set h + ( x , t , p , τ ) = h ( x , t , p , τ ) + | τ | . Hence h + is the ho- mogenization of the continuous H amiltonian ( x , p ) 7→ max{1, ∥ p ∥ }. Proposition 7.5. Let F , G ∈ T ( N ) . Then γ τ ( P ′ D T ∗ N ( F ), P ′ D T ∗ N ( G )) ≤ γ h + ( F , G ). Proof . As in the pr oof of Lemma 7.4 we appro ximate f by a sequence of smooth functions ( f n ) n and let ( h n ) n be the corresponding Hamiltonians . Since h + n = h n + | τ | and h n and τ commute, we hav e φ h + n , s = φ h n , s ◦ φ τ , s = φ τ , s ◦ φ h n , s . As a r esult the same relation holds for their associated sheaves . W e can also see that P ′ D T ∗ N commutes with K τ , s . Hence , if there exists a morphism u : F − → K h + n , a ◦ G , applying P ′ D T ∗ N and using Lemma 7.3, we obtain P ′ D T ∗ N ( u ) : P ′ D T ∗ N ( F ) − → K τ , a ( P ′ D T ∗ N ( G )). S ince the deﬁnition of the distance only involves morphisms of the kind F − → K h + n , a ◦ G and G − → K h + n , b ◦ F , the result follo ws. □ DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 27 Remar k 7.6 . It is clear that Proposition 7.5 is true for all closed Z ⊂ T ∗ N and homogeneous H amiltonians h + : T ∗ ( N × R ) \ 0 N × R → R that r estrict to h + | ρ − 1 ( Z ) = τ and F , G ∈ T ( N ) ∩ Sh h + ≥ 0 ( N × R ), where we only take Z = D T ∗ N and a particular h + here . Thanks to Proposition 7.5, w e shall estimate γ h + for related sheav es. Lemma 7.7. Let x ∈ N , ε > 0 and let Z ⊂ N be a closed contr actible subset contained in the ball with cent er x and radius ε < r i n j ( N )/3 . Then, for any F ∈ Sh( N × R ) such that S S ( F ) ⊂ { τ ≥ ∥ p ∥ } , we have γ h + ( F ⊗ k { x } × R , F ⊗ k Z × R ) ≤ 4 ε . Proof . W e r ecall that h + ( x , p , t , τ ) = max{ τ , ∥ p ∥ } for τ ≥ 0 . Since we work on Sh τ ≥ 0 ( N × R ) we can as well assume h + ( x , p , t , τ ) = max{ τ , ∥ p ∥ } for any τ . W e ﬁrst give a bound on N 2 × R 2 and then take the pull-back by the diagonal embedding δ : N × R − → ( N × R ) 2 . T o distinguish, we denote points of the second cop y of T ∗ ( N × R ) with prime, and then points of T ∗ ( N 2 × R 2 ) are denoted b y ( x , p , x ′ , p ′ , t , τ , t ′ , τ ′ ). W e will apply Lemma 6.8 and we look for a function h 1 on T ∗ ( N 2 × R 2 ) such that h 1 | ∆ 1 = h + ◦ ( d δ ) t , where ∆ 1 = ∆ N × R × N 2 × R 2 T ∗ ( N 2 × R 2 ). W e have ( d δ ) t ( x , p , x , p ′ , t , τ , t , τ ′ ) = ( x , t , p + p ′ , τ + τ ′ ) so we should extend the function ∥ p + p ′ ∥ outside the diagonal. W e let ∆ ( r ) be the neighbor hood of ∆ N formed by the pairs ( x , x ′ ) such that d ( x , x ′ ) < r . F or ( x , x ′ ) ∈ ∆ ( r i n j ( N )) we identify T ∗ x N and T ∗ x ′ N through the parallel transport along the shortest geodesic between x and x ′ ; we can then make sense of p + p ′ . W e choose a partition of unity α , β with α = 1 on ∆ ( r i n j ( N )/2) and β = 1 outside ∆ ( r i n j ( N )). W e set h 0 ( x , p , x , p ′ , t , τ , t , τ ′ ) = α ∥ p + p ′ ∥ + β ∥ ( p , p ′ ) ∥ . W e have the inequality ∥ p ∥ + h 0 ≥ ∥ p ′ ∥ because o ver ∆ ( r i n j ( N )) we hav e ∥ p ∥ + h 0 ≥ α ∥ p ∥ + α ∥ p + p ′ ∥ + β ∥ p ′ ∥ ≥ ( α + β ) ∥ p ′ ∥ = ∥ p ′ ∥ and outside ∆ ( r i n j ( N )) we even have h 0 ≥ ∥ p ′ ∥ . N ow we deﬁne h 1 on T ∗ ( N 2 × R 2 ) b y h 1 = max{ τ + τ ′ , h 0 }. W e bound the distance between F ⊠ k { x } × R and F ⊠ k Z × R for the distance γ h 1 . W e deﬁne h 2 on the second copy of T ∗ ( N × R ) b y h 2 = ∥ p ′ ∥ . W e r emark that Lemma 6 .5 makes sense in the degenerated case w her e h or h ′ is the zero function. W riting h 2 = 0 + 0 + ∥ p ′ ∥ + 0 w e deduce , b y Lemma 6.11, that γ h 2 ( F ⊠ k { x } × R , F ⊠ k Z × R ) ≤ 2 ε . F or G ∈ Sh τ ′ ≥ 0 ( N × R ) we have S S ( F ⊠ G ) ⊂ W with W = { τ ≥ ∥ p ∥ , τ ′ ≥ 0} = { τ 2 − ∥ p ∥ 2 ≥ 0, τ ≥ 0, τ ′ ≥ 0} ⊂ T ∗ ( N 2 × R 2 ). When we restrict to W , we have h 1 ≥ 1 2 ( τ + τ ′ ) + 1 2 h 0 ≥ 1 2 ∥ p ∥ + 1 2 h 0 ≥ 1 2 h 2 . By P roposition 6.10 we deduce γ h 1 ( F ⊠ k { x } × R , F ⊠ k Z × R ) ≤ 2 γ h 2 ( F ⊠ k { x } × R , F ⊠ k Z × R ) ≤ 4 ε and b y Lemma 6.8, we have γ h + ( F ⊗ k { x } × R , F ⊗ k Z × R ) ≤ γ h 1 ( F ⊠ k { x } × R , F ⊠ k Z × R ) ≤ 4 ε , which ﬁnishes the proof. □ DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 28 7.2. Some lemmata on iterated cones. Let C be a stable ∞ -category [Lur17, Def. 1.1.1.9] 11 and F , G 0 , . . . , G n be objects of C . W e say that F is an ( n steps) iterated cone on ( G 0 , . . . , G n ) if there exist objects F 0 = G 0 , . . . , F n = F and morphisms f i : F i − → G i + 1 such that F i + 1 is iso- morphic to the cone of f i . The next lemmas hold for any stable ∞ -category with an − → R -action, where γ denotes the corresponding interleaving distance (instead of γ h + and T s = K s h + ◦ − for sheaves). Lemma 7.8. Let u : F ′ − → F and f : F − → G be two morphisms. W e assume that ther e exists v : F − → T ε ( F ′ ) such that v ◦ u and T ε ( u ) ◦ v are the canonical morphisms τ . L et C ( f ) , C ( f ◦ u ) be the cones of f , f ◦ u . Then γ ( C ( f ), C ( f ◦ u )) < 4 ε . Proof . By the oct ahedral axiom (now a proposition in stable categor ies), we have a ﬁber se- quence C ( u ) − → C ( f ◦ u ) w − → C ( f ) + 1 − − → such that C ( w ) ≃ C ( u )[1]. B y [GVit22a, Lem. 6 .8 (i)] and the existence of v we have γ (0, C ( u )) < 2 ε , hence γ (0, C ( w )) < 2 ε , and then we apply [GVit22a, Lem. 6.8 (ii)] to w to deduce that γ ( C ( f ), C ( f ◦ u )) < 4 ε . □ Lemma 7.9. Let f : F − → G be a morphism and let F ′ , G ′ be two objects such that γ ( F , F ′ ) , γ ( G , G ′ ) < ε . Then ther e exist 0 ≤ a , b < ε and a morphism f ′ : T − a ( F ′ ) − → T b ( G ′ ) such that γ ( C , C ′ ) < 8 ε , where C , C ′ ar e the cones of f , f ′ . Proof . Since γ ( F , F ′ ) < ε , there exist 0 ≤ a , a ′ and morphisms u 1 : F − → T a ′ ( F ′ ), u 2 : F ′ − → T a ( F ) such that T a ′ ( u 2 ) ◦ u 1 and T a ( u 1 ) ◦ u 2 are homotopic to canonical morphisms τ a ′ + a and a ′ + a < ε . W e have similar morphisms v 1 : G − → T b ( G ′ ), v 2 : G ′ − → T b ′ ( G ). W e deﬁne f ′ = v 1 ◦ f ◦ T − a ( u 2 ) and also f 1 = f ◦ T − a ( u 2 ). By Lemma 7.8 (and its variant which deals with u ◦ f ) both γ ( C , C ( f 1 )) and γ ( C ( f 1 ), C ′ ) are less than 4 ε . The r esult follo ws from the triangle inequality . □ Lemma 7.10. Let F be an n steps iter ated cone on G 0 , . . . , G n . Let ε > 0 and G ′ 0 , . . . , G ′ n be such that γ ( G i , G ′ i ) < ε for all i . Then there exist a 0 , . . . , a n and an n steps iterated cone F ′ on T a 0 ( G ′ 0 ), . . . , T a n ( G ′ n ) such that γ ( F , F ′ ) < 8 n ε . Proof . W e pro ve this b y induction on n . F or n = 1 the r esult is given by Lemma 7.9. W e assume the result holds for n − 1 and we consider an n steps iterated cone. By deﬁnition there exist F 0 = G 0 , . . . , F n = F and morphisms f i : F i − → G i + 1 such that F i + 1 is isomorphic to the cone of f i . The induction step gives a 0 , . . . , a n − 1 , F ′ 0 = T a 0 ( G ′ 0 ), . . . , F ′ n − 1 , and morphisms f ′ i : F ′ i − → T a i G ′ i + 1 such that F ′ i + 1 is isomorphic to the cone o f f ′ i and γ ( F n − 1 , F ′ n − 1 ) < 8 n − 1 ε . W e apply Lemma 7.9 to f n − 1 : F n − 1 − → G n , F ′ n − 1 and G ′ n , with 8 n − 1 ε instead of ε . W e obtain a morphism f ′ n : T − a ( F ′ n − 1 ) − → T b ( G ′ n ), for some a , b , such that γ ( F ′ n , F n ) < 8 n ε , wher e F ′ n is the cone of f ′ n . W e shift a 0 , . . . , a n − 1 b y − a and set a n = b . This proves the r esult. □ Lemma 7.11. Let ε > 0 and let F i , G i , i ∈ I , be two families of objects such that γ ( F i , G i ) ≤ ε for any i ∈ I . W e assume that C admits I -indexed dir ect sums. Then γ ( L i ∈ I F i , L i ∈ I G i ) ≤ 2 ε . 11 Stable ∞ -categories are ∞ -categories that play the role of triangulated categories [Lur17, Theorem 1.1.2.14]. Sheaf categories Sh( M ) are stable . DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 29 Proof . The hypothesis imply in particular th at there exist ϕ i : F i − → T ε ( G i ), ψ i : G i − → T ε ( F i ) such that ϕ i ◦ ψ i and ψ i ◦ ϕ i are homotopic to canonical morphisms. W e deduce the lemma b y considering ϕ = L i ϕ i and ψ = L i ψ i . □ Fr om now on, we consider sheaves . U sing a ˇ Cech resolution we can express any sheaf F as an n steps iterated cone on sheaves of the for m G i = L j ∈ I i F ⊗ k Z j , i = 0, . . . , n (see Lemma 7.12). U sing L emma 7.7 and previous approximation Lemmas pro ved in this section we can replace the F ⊗ k Z j b y F ⊗ k { x j } × R , for some x j ∈ Z j , and obtain an iterated cone o n the L j ∈ I i F ⊗ k { x j } × R , which approximate F . No w F ⊗ k { x j } × R is appro ximated by the iter- ated cones of ﬁbers using Lemma 7 .13 and we will be able to conclude. W e give more details in §7.3. Lemma 7.12. Let M = S i ∈ I U i be a ﬁnite covering. W e set U c l J = T j ∈ J U j and we assume that U c l J = ; for | J | ≥ n + 2 and tha t all U c l J ar e contractible. Let k M ≃ C 0 − → C 1 − → · · · − → C n be the ˇ Cech resolution of the constant sheaf , where C i = L J ⊂ I , | J |= i + 1 k U c l J . Then for any sheaf F on M , we have F ∈ 〈 F ⊗ C 0 , . . . , F ⊗ C n 〉 Sh( M ) , and mor e precisely F is an n steps itera ted cone on F ⊗ C 0 [1] , F ⊗ C 1 , . . . , F ⊗ C n [1 − n ] . Proof . Let D i be the truncated complex D i = C 0 − → C 1 − → · · · − → C i . W e hav e ﬁber sequences D i − → D i − 1 − → C i [1 − i ] + 1 − − → . W e deduce by induction on i that F ⊗ D i is an i steps iterated cone on F ⊗ C 0 , . . . , F ⊗ C i [1 − i ]. For i = n we obtain the lemma. □ Lemma 7.13. Let k be ring such that any ﬁnite type module has a free r esolution of ﬁnite length. Then any constructible F ∈ T (pt) (with coefﬁcients k ) is an iterated cone of sheaves of the type k [ a , ∞ ) [ d ] . Proof . W e assume that F is constructible with respect to a stratiﬁcation given b y ﬁnitely many points a 1 < · · · < a N and the open inter vals they deﬁne. Then F ≃ E ( −∞ , a 1 ) on ( −∞ , a 1 ) for some k -module E and H om( F , k ( −∞ , a ] ) ≃ E for any a < a 1 . Since F ∈ T (pt), it is left orthogonal to k ( −∞ , a ] and we obtain E ≃ 0. W e deduce that F | I ≃ ( E 1 ) [ a 1 , ∞ ) | I for some neighborhood I of { a 1 } and some k -module ( E 1 ) (see for example [Gui23, Example 1.2.3- (v)]). Since S S ( F ) ⊂ { τ ≥ 0}, the microlocal Morse theorem implies that the isomorphism F | I ≃ ( E 1 ) [ a 1 , ∞ ) | I extends to a morphism u : F − → ( E 1 ) [ a 1 , ∞ ) . Then the cone of u is con- structible with respect to the stratiﬁcation induced by a 2 < · · · < a N . An induction on N sho ws that F is an iterated cone of sheaves of the for m ( E i ) [ a i , ∞ ) . No w the hypothesis on k implies that it is also an iterated cone of sheaves of the form k [ a i , ∞ ) [ d ]. □ 7.3. Proof of Theorem 5.2. Let 0 < ε < r i n j ( N )/3. W e pick a ﬁn ite co vering N = S i ∈ I U i be a ﬁnite cov ering as in Lemma 7.12 and assume moreo ver that all U i ’ s are contained in a ball of radius less than ε . W e set U c l J = T j ∈ J U j . By Lemma 7.12 F is an n steps iterated cone on F ⊗ C 0 [1], F ⊗ C 1 , . . . , F ⊗ C n [1 − n ], where C i = L J ⊂ I , | J |= i + 1 k U c l J × R . For each J ⊂ I we pick x J such that U c l J ⊂ B ε ( x J ). Lemma 7.7 says that γ h + ( F ⊗ k { x J } × R , F ⊗ k U c l J × R ) ≤ 4 ε . W e set C ′ i = L J ⊂ I , | J |= i + 1 k { x J } × R . By Lemma 7.11 we have γ h + ( F ⊗ C i , F ⊗ C ′ i ) ≤ 8 ε . N ow for any J , we set F J = F ⊗ k { x J } × R . By our hypotheses F J is a limit of constructible sheaves and, by Lemma 7.13, we deduce that there exists D J , which is an iterated cone of DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 30 sheaves of the type k { x J } × [ a , ∞ ) [ d ], such that γ τ ( F J , D J ) < ε . Because τ ≤ h + on { τ ≥ 0}, we have γ h + ( F J , D J ) ≤ γ τ ( F J , D J ) < ε by Lemma 6.10. Therefor e, for any i , the sheaf D i = L J ⊂ I , | J |= i + 1 D J satisﬁes γ h + ( F ⊗ C ′ i , D i ) < 2 ε , and then γ h + ( F ⊗ C i , D i ) < 10 ε by triangle inequality . I t follo ws from Lemma 7.10 that there exist a 0 , . . . , a n and an n steps iterated cone F ′ on T a 0 ( D 0 ), . . . , T a n ( D n ) such that γ h + ( F , F ′ ) < 10 × 8 n ε . N ow we apply the projector P ′ D T ∗ N to our iterated cones. Each G i = P ′ D T ∗ N ( T a i ( D i )) is a direct sum of sheav es of the type P ′ D T ∗ N ( T a i ( D J )), then an iterated cone of W ( x J , a i + b ) [ d ] = P ′ D T ∗ N ( k { x J } × [ a i + b , ∞ ) [ d ]). Then C = P ′ D T ∗ N ( F ′ ) is an iterated cone o n G 0 , . . . , G n , and then an iterated cone of W ( x J , a ) [ d ]. Since F ≃ P ′ D T ∗ N ( F ), P roposition 7.5 implies that γ τ ( F , C ) ≤ γ h + ( F , F ′ ) < 10 × 8 n ε , which pro ves the r esult. □ Remar k 7.14 . The last proof works with k = Z . If we work o ver a ﬁeld, we can r eplace the use of Lemma 7.13 by the fact that any constructible sheaf is a dir ect sum of sheaves of the type k { x J } × [ a , b ) [ d ] (by Gabriel’ s theorem) with b a real number or b = +∞ . The case b < +∞ is given as the cone of k { x J } × [ a , ∞ ) − → k { x J } × [ b , ∞ ) . Then the proof shows that it is enough to take dim( N ) + 1 cones to get an approximation of a sheaf by the sheaves W ( x , a ) (if we don ’ t count taking direct sums into th e total count of cones). A P P E N D I X A. I N T E R L E AV I N G R O U Q U I E R D I M E N S I O N The Remark 7.14 l eads to a generalization of Rouquier dimension for stable ∞ -categories with − → R -actions. F or a stable ∞ -categor y C equipped with an − → R -action T , we set its interleaving distance to be γ C . W e say a subset of objects G = { G i : i ∈ I } that is closed under the − → R -action of C is a set of solo-appro ximators if: For any i , j ∈ I , we have γ C ( G i , G j ) < +∞ , and mor eover , for any object X ∈ C and any ε > 0, there exists an iterated cone C out of some G i such that γ C ( X , C ) < ε . F or d ≥ − 1, we denote 〈 G i | i ∈ I 〉 d the smallest full thick (that is, closed under taking dir ect summand) subcategor y of C that contains all step d + 1 iterated cones out of direct sums L i ∈ I k G i , k = 0, . . . , d , with I k ⊂ I . Deﬁnition A.1. The interleaving Rouquier dimension of ( C , T ) , denoted by IRdim( C , T ) (or simply IRdim( C ) if T is clear), is deﬁned as the minimal d such that there exists a set of solo- appro ximators G = { G i | i ∈ I } such that for any X ∈ C and any ε > 0 , there exists C ∈ 〈 G i | i ∈ I 〉 d such that γ C ( X , C ) < ε . W e no w explain the r elation between the interleaving Rouquier dimension and the usual Rouquier dimension Rdim 12 . For any stable ∞ -categor y C equ ipped with an − → R -action, one can deﬁne a stable ∞ -categor y C ∞ as the V erdier quotient of C by T amarkin torsion objects of C (see [GS14, Deﬁnition 6.1], [KZ25, Deﬁnition 5.19]). W e have 12 The R ouquier dimension of C is deﬁned as the minimal d such that for some G we have C = 〈 G 〉 d . It is clear that Rdim only depends on th e homotopy category of h o ( C ) (similarly , IR dim only depends on h o ( C ) and h o ( T ).) DENSITY OF FIBERS FOR THE FIL TERED FUKA Y A CA TEGOR Y OF T ∗ N 31 Proposition A.2. If C has a set of solo-approximators G = { G i | i ∈ I } , then C ∞ is generated by a single G i (for any i ∈ I ), and moreo ver , Rdim( C ∞ ) ≤ IRdim( C ) . Proof . Any two objects X , Y in C with γ C ( X , Y ) < ∞ become isomorphic in C ∞ . Then the ﬁnite distance condition for solo-approximators tells that all G i , G j are isomorphic to each other . M or eover , the density condition sho ws that any X ∈ C is isomorphic i n C ∞ to an iter- ated cone out of G i . This pro ves the ﬁrst claim. M oreo ver , if IRdim( C ) = d , then any X ∈ C are isomorphic in C ∞ to a iterated cone of G i of step d + 1, which means Rdim( C ∞ ) ≤ d . □ N ow we relate Remark 7.14 and the interleaving Rouquier dimension. Let T D T ∗ N ( T ∗ N ) l c to be the full subcategory of T D T ∗ N ( T ∗ N ) consisting of objects F satisfying that for any x ∈ N the sheaf F ⊗ k { x } × R is a γ s -limit of constructible sheaves on R . W e hav e that T D T ∗ N ( T ∗ N ) l c is a stable ∞ -category that is invariant under the − → R -action of T D T ∗ N ( T ∗ N ): The − → R -invariance is clear; and we only need to verify the stability in the case N = pt because − ⊗ k { x } × R is an exact functor . The N = pt case follows from Lemma 7.9. Consider { W ( x , a ) | ( x , a ) ∈ N × R = I }, the density theorem almost tells that it forms a set of solo-approximator of T D T ∗ N ( T ∗ N ) l c , except that it is not clear whether the W ( x , a ) ’ s are indeed objects of T D T ∗ N ( T ∗ N ) l c (which should be, but we will not verify it her e). So , we take C ( D T ∗ N ) as the thick stable subcategory of T D T ∗ N ( T ∗ N ) spanned by { W ( x , a ) | ( x , a ) ∈ N × R = I } and T D T ∗ N ( T ∗ N ) l c , which is clearly − → R - invariant. Proposition A.3. Let C ( D T ∗ N ) be the st able ∞ -categor y deﬁned as abo ve equipped with the − → R -action of T D T ∗ N ( T ∗ N ) . Then { W ( x , a ) | ( x , a ) ∈ N × R = I } is a set of solo-approximators of C ( D T ∗ N ) , and if k is a ﬁeld, then we have IR dim( C ( D T ∗ N )) ≤ dim N . Proof . It remains to c heck that any two W ( x , a ) have a ﬁnite interleaving distance. In fact, b y Lemmata 7.4 and 6.11 as well as P roposition 7.5, we have that if d ( x , y ) < ε and | a − b | < ε then γ s ( W ( x , a ) , W ( y , b ) ) < 2 ε . No w , we pick a path between ( x , a ) and ( y , b ) in N × R , and the ﬁnite distance follo ws if we subdivide the path ﬁnely enough. □ Remar k A.4 . H ere , we consider the (unﬁltered) wrapped Fukaya categor y WF ( T ∗ N ) (and its perfect modules categor y): I t is proven in [BC23, Section 4] that Rdim(P erf( WF ( T ∗ N ))) ≤ dim N . On the other hand, we have R dim(( C ( D T ∗ N )) ∞ ) ≤ dim N by P ropositions A.2 and A.3. Those two results ar e related in the follo wing way: we claim that I dem(( C ( D T ∗ N )) ∞ ) ≃ Perf( WF ( T ∗ N )), where I dem means idempotent completion. Then two R dim bounds are the same one (notice that idempotent completion does not increase Rouquier dimension). Her e is the idea of the proof: By [Abo11], the latter is generated b y a single cotangent ﬁber T ∗ x N whose endomor - phism algebra is isomorphic to the chain over the based loop space. On the other hand, we already kno w that ( C ( D T ∗ N )) ∞ is generated b y a single W ( x , a ) (which also a wrapped version of a single cotangent ﬁber), so does I dem(( C ( D T ∗ N )) ∞ ), it remains to check that the endo- morphism algebra of W ( x , a ) (in ( C ( D T ∗ N )) ∞ ) is also the chain over the based loop space . This computation can be done b y [KZ25, Proposition 5.21] and a variant of [Zha23, Thm. 4.4]. REFERENCES 32 The proof of Rdim(P erf( WF ( T ∗ N ))) ≤ dim N we give her e seemingly differs fr om [BC23], where the authors use the sectorial descent of [GPS24]. H o wever , our proof for the density theory uses a ˇ Cech descent argument for sheaves on the base N , which should be regar ded as a ﬁltered analogy of the sectorial descent of [GPS24] (that is not pro ven yet.) W e then believe both proofs ar e essentially the same in a certain context. R E F E R E N C E S [Abo11] M. Abouzaid. A cotangent ﬁbre generates the Fukaya categor y . A dvances in Math- ematics 228.2 (2011), pp . 894–939. D O I : 10.1016/j.aim.2011.06.007 . [Amb25] G. Ambrosioni. Filtered F ukaya categories . J ournal of S ymplectic Geometry 23.2 (2025), pp . 423–509. D O I : 10.4310/jsg.250624013956 . [ABC26] G. Ambrosioni, P . Biran, an d O . Cornea. A pproximability for Lagr angian subman- ifolds . 2026. arXiv: 2601.12506 . [AI20] T . Asano and Y . Ike. P ersistence-like distance on T amarkin ’ s categor y and sym- plectic displacement energy . J. S ymp. Geometr y 18.3 (2020), pp . 613–649. D O I : 10.4310/JSG.2020.v18.n3.a1 . arXiv: 1712.06847 [math.SG] . [AI24] T . Asano and Y . Ike. Completeness of der ived interleaving distances and sheaf quantization of non-smooth objects . Math. Ann. 390.2 (2024), pp . 2991–3037. I S S N : 0025-5831. D O I : 10.1007/s00208- 024- 02815- x . [Asa+25] T . Asano et al. C 0 -rigidity of Lege ndrians and coisotropics via sheaf quantization . 2025. arXiv: 2510.01746 [math.SG] . [BC23] S. B ai and L. Côté. On the Rouquier dimension of wr apped Fukaya categories and a conjecture of Orlo v . Compositio Mathematica 159.3 (2023), pp . 437–487. D O I : 10.1112/S0010437X22007886 . [Bir24] P . Biran. Approximat ions in the Fukaya Category . T alk at the C onference “F rom H amiltonian Dynamics to Symplectic T opology and Beyond ”. J une 2024. [BCZ] P . Bir an, O . Cornea, and J. Zhang. T riangulation, Persistence, and F ukaya cate- gories . arXiv: 2304.01785 [math.SG] . [F ao17] G. F aonte. Simplicial ner ve of an A-inﬁnity categor y . Theor y and Applications of Categories 32 (2017), pp . 31–52. [Fuk21] K. Fukaya. Gromo v-H ausdorff distance between ﬁlter ed A ∞ - categories 1: L agr angian Floer theory . 2021. arXiv : 2106.06378 [math.SG] . [GPS20] S. G anatra, J. P ar don, and V . S hende. Co variantly functorial wr apped Floer theory on Liouville sectors . Publications mathématiques de l’IHÉS 1 31 (2020), pp . 73–200. D O I : 10.1007/s10240- 019- 00112- x . [GPS24] S. Ganatra, J. P ar don, and V . Shende. Sectorial descent for wrapped Fukaya cat- egories . J ournal of the American Mathematical Society 37.2 (2024), pp . 499–635. D O I : 10.1090/jams/1035 . [Gui23] S. Guillermou. Sheaves and symplectic geometr y of cotangent bundles . A stérisque 440 (2023). D O I : 10.24033/ast.1199 . REFERENCES 33 [GKS12] S. Guillermou, M. Kash iwara, and P . Schapira. Sheaf quantization of H amilton- ian isotopies and applications to non-displaceability problems . Duke Math. J. 161, no. 2 (2012), 201-245 161.2 (2012), pp. 201–245. D O I : 10 . 1215 / 00127094 - 1507367 . [GS14] S. Guillermou and P . Schapira. “Microlocal Theory of Sheaves and T amarkin ’ s N on Displaceability Theorem ”. In: Homological Mirror Symmetry and T ropical G eom- etry . Ed. by R. C astano-Bernard et al. V ol. 15. Lecture Notes of the U nione M atem- atica Italiana. Springer V erlag, 2014, pp. 43–85. D O I : 10 . 1007 / 978 - 3 - 319 - 06514- 4_3 . [GVit22a] S. Guillermou and C. Viterbo. The support of sheaves is γ -coisotropic . Geometric and Functional Analysis 34 (2024), pp . 1052–1113. D O I : 10 . 1007 / s00039 - 024 - 00682- x . arXiv : 2203.12977 [math.SG] . [Ike19] Y . Ike. Compact Exact Lagrangian I ntersections in Cotangent B undles via S heaf Quantization . P ubl. Res. Inst. Math. Sci. 55.4 (Oct. 2019), pp. 737–778. D O I : 10 . 4171/prims/55- 4- 3 . [KS90] M. K ashiwara and P . Schapira. S heaves on manifolds . V ol. 292. G rundlehren der Math. Wissenschaften. S pringer-V erlag, 1990. D O I : 10.1007/978- 3- 662- 02661- 8 . [KS18] M. Kashiwara and P . Schapira. Persistent homolog y and microlocal sheaf theory . 2018. D O I : 10.1007/s41468- 018- 0019- z . arXiv: 1705.00955 [math.AT] . [K uo23] C. K uo. W rapped Sheaves . Advances in Mathematics 415 (2023), p . 108882. D O I : 10.1016/j.aim.2023.108882 . [KSZ23] C. K uo, V . Shende, and B . Zhang. On the H ochschild cohomolog y of T amarkin cat- egories . 2023. arXiv: 2312.11447 [math.SG] . [KPS26] T . Kuwagaki, A. P etr, and V . Shende. C omplete Fukaya category . In preparation. 2026. [KZ25] T . K uwagaki and B . Zhang. Almost mathematics, P ersistence module, and Tamarkin category . 2025. arXiv : 2503.15933 . [Lur17] J. L urie. Higher Algebra . P reprint, av ailable at https : / / www . math . ias . edu / ~lurie/papers/HA.pdf . 2017. [Lur18] J. Lurie. Spectral Algebraic Geometry . Preprint, available at https :// www.math. ias.edu/~lurie/papers/SAG- rootfile.pdf . 2018. [K erodon] J. Lurie et al. An online resour ce for homotopy-coherent mathematics. 2025. U R L : https://kerodon.net . [LM08] V . L yubashenko and O . Manzyuk. “ A ∞ -bimodules and Serr e A ∞ -functors”. In: G e- ometry and Dynami cs of Groups and Spaces: I n Memory of A lexander Rezniko v . Basel: Birkhäuser Basel, 2008, pp . 565–645. I S B N : 978-3-7643-8608-5. D O I : 10 . 1007/978- 3- 7643- 8608- 5_14 . [PS23] F . Petit and P . Schapira. Thickening of the diagonal and interleaving distance . Se- lecta Mathematica 29.5 (Sept. 2023). D O I : 10.1007/s00029- 023- 00875- 6 . [Sch25] P . Scholze. S ix-F unctor F ormalisms . 2025. arXiv: 2510.26269 [math.AG] . REFERENCES 34 [T am08] D . T amarkin. Microlocal condition f or n on-displaceablility . Ed. by M. Hitrik et al. 2008. D O I : 10.1007/978- 3- 030- 01588- 6_3 . arXiv: 0809.1584 [math.SG] . [Vit19] C. Viterbo. S heaf Quantization of Lagr angians and Floer cohomolog y . 2019. arXiv : 1901.09440 [math.SG] . [V ol25] M. V olpe. The six operations in topology . J ournal of T opolog y 18.4 (2025), e70050. D O I : https://doi.org/10.1112/topo.70050 . [Zha23] B . Zhang. Idempotence of microlocal kernels and S 1 -equivariant Chiu- T amarkin invariant . 2023. arXiv: 2306.12316 . U M R C N R S 6 6 2 9 D U C N R S , L A B O R AT O I R E D E M AT H É M AT I Q U E S J E A N L E R AY , 2 C H E M I N D E L A H O U S S I N I È R E , B P 9 2 2 0 8 , F - 4 4 3 2 2 N A N T E S C E D E X 3 F R A N C E Email addr ess : Stephane.Guillermou@univ-nantes.fr U N I V E R S I T É P A R I S - S A C L AY , C N R S U M R 8 6 2 8 , L A B O R AT O I R E D E M AT H É M AT I Q U E S D ’ O R S AY , 9 1 4 0 5 , O R S AY , F R A N C E . Email addr ess : claude.viterbo@universite-paris-saclay.fr C E N T R E F O R Q UA N T U M M AT H E M AT I C S . U N I V E R S I T Y O F S O U T H E R N D E N M A R K . C A M P U S V E J 5 5 , 5 2 3 0 O D E N S E , D E N M A R K . Email addr ess : bingyuzhang@imada.sdu.dk

Density of fibers for the filtered Fukaya category of $T^*N$

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment