CC O N S T R U C T I B L E H Y P E R S H E A V E SV I A E X I T P A T H S damien lejay
Abstract
The goal of this article is to extend a theorem of Lurie Sh A ( X ) = Fun ( Exit A ( X ) , S )representing constructible sheaves with values in S , the ∞ -category ofspaces, on a stratified space X with poset of strata A , as functors fromthe exit paths ∞ -category Exit A ( X ) to S . Lurie’s representation theoremworks provided A satisfy the ascending chain condition. This typicallyrules out infinite dimensional examples of stratified space.Building on it and with the help of a stratified homotopy invariancetheorem from Haine, we show that when X is a nice enough A -stratifiedspace and when A is itself stratified A ⩽ ⊂ A ⩽ ⊂ · · · ⊂ A by posets satisfy-ing the ascending chain condition, Hyp A ( X ) = Fun ( Exit A ( X ) , S )the ∞ -category of A -constructible hypersheaves on X is represented byfunctors from the exit paths ∞ -category of X .There are two types of nice stratified spaces on which this extendedrepresentation theorem applies: conically stratified spaces and spacesthat are sequential colimits of conically stratified spaces. Examplesof application include the metric and the topological exponentials ofa Fréchet manifold, locally countable simplicial complexes and moregenerally, locally countable cylindrically normal CW-complexes. c o n t e n t s ff erent notions of constructibility 41.1 Stratified spaces 41.2 Recollections on hypersheaves 51.3 Locally (hyper)constant (hyper)sheaves 61.4 (Hyper)constructible (hyper)sheaves 91.5 Constructible hypersheaves 102 Coincidences 112.1 Stratum case 112.2 Conical case 122.3 D -spaces 142.4 Colimit and conical D ω -spaces 15 This work was supported by
IBS-R003-D1 . a r X i v : . [ m a t h . A T ] F e b Representation via exit paths 193.1 Exit paths ∞ -category 193.2 Representation theorem 203.3 Homotopy invariance of exit paths 254 Examples of application 264.1 Simplicial complexes 264.2 CW-complexes 274.3 The exponentials 28When a topological space is nice enough, its category of locally constantsheaves of sets is equivalent to the category of representations of its funda-mental groupoid. The fundamental groupoid X has objects the points of X and arrows, the homotopy classes of continuous paths between two pointsin X up to homotopy. Going further, Lurie has shown that the ∞ -category oflocally constant sheaves of spaces is equivalent to the ∞ -category of represent-ations of Sing( X ), the simplicial set of maps ∆ n → X , which is a model for thefundamental ∞ -groupoid of X [1, A.2.15].This representation theorem can be further extended to stratified spaces.When X is a stratified space with poset of strata A , one can consider A -con-structible sheaves on X : sheaves whose restriction to each stratum X a is locallyconstant. In order to represent those constructible sheaves, the simplicialset Sing( X ) needs to be adapted to take into account the stratification of X .Following an idea of Treumann [2], Lurie considered the simplicial subset Exit A ( X ) ⊂ Sing( X ) where paths are only allowed to immediately escape adeeper stratum and never return. When the stratification is conical this sim-plicial subset is an ∞ -category, the exit paths ∞ -category of X . With such asetup, the representation theorem for A -constructible sheaves on X holds provided A satisfies the ascending chain condition . Definition (Ascending chain condition) . A poset A is said to satisfy the as-cending chain condition if A does not admit a chain a < a < · · · of infinitelength.This condition typically excludes stratified spaces of infinite dimension.For example, one may think of infinite dimensional simplicial complexeswhich are not locally finite or of any space X with a filtration by dimension X ⩽ ⊂ X ⩽ ⊂ · · · ⊂ X ⩽ n ⊂ · · · ⊂ X which happens every time X is for example the colimit lim −−→ n< ω X ⩽ n . Our goalis to extend the representation theorem to a large class of posets A that donot satisfy the ascending chain condition and which includes in particular theposet ω = { < < < · · · } .There are two obstacles to a generalisation of the representation theorem.The first one has to do with hypercompleteness of sheaves, a phenomenonthat starts appearing only in the ∞ -world. We shall dedicate a section tothe di ff erences between sheaves and hypersheaves [§ 1.2]. Having built acontinuous map Fun ( Exit A ( X ) , S ) −→ Sh A ( X )because every functor in Fun ( Exit A ( X ) , S ) is the limit of its truncation tower,it follows that its image must be a hypersheaf. There is thus no hope of epresenting all A -constructible sheaves in general and we shall instead focuson the full subcategory of A -constructible hypersheaves . Notice that when A satisfies the ascending chain condition, all A -constructible sheaves are alreadyhypersheaves.Constructible hypersheaves have already been used by Lurie to describethe equivalence between locally constant factorisation algebras on a finitedimensional manifold M and E M -algebras. A key tool in the proof of Lurie isthe use of the metric exponential of M , the metric space of finite subsets of M ,which is naturally stratified by the cardinality of the subsets. Lurie notes thathe had to add an hypercompletion hypothesis on the constructible sheaveson the exponential because the stratifying poset did not satisfy the ascendingchain condition [1, 3.3.12]. On the other side of the equation, Cepek hasshown that the combinatoric of the exit paths ∞ -category of the exponentialof R n is also related to the one of E n -algebras [3]. The study of the exponentialof a manifold is a major motivation for extending the representation theoremand we shall give more details about this particular example at the end of thearticle [§ 4.3].The second obstacle has to do with the proof of the representation theoremitself. It uses an induction on the depth of the stratification and a poset A admits a depth function if and only if it satisfies the ascending chaincondition. To circumvent this issue, we shall make use of the functoriality ofthe equivalence of ∞ -categories in the representation theorem. We shall thenwork with posets A which are themselves stratified A ⩽ ⊂ A ⩽ ⊂ · · · ⊂ A ⩽ n ⊂ · · · ⊂ A by posets satisfying the ascending chain condition. In such a case, the stratifiedspace X inherits a filtration X ⩽ ⊂ X ⩽ ⊂ · · · ⊂ X ⩽ n ⊂ · · · ⊂ X by closed stratified subspaces. We are thus considering posets that are strictind-objects in the category of posets with the ascending chain condition. Inanother perspective, Barwick, Glasman and Haine have considered spaces(and ∞ -toposes) stratified over profinite posets [4].The functoriality of the exit paths ∞ -category is easily adressed as it com-mutes with filtered colimits, since simplicies in Exit A ( X ) are only allowed tovisit a finite number of strata. Undertaking the functoriality of the ∞ -categoryof constructible hypersheaves is the real task here. We shall show that thecanonical dévissage map Hyp A ( X ) −→ lim ←−− n< ω Sh A ⩽ n ( X ⩽ n )is an equivalence in two special cases: when X is conically A -stratified andwhen the topology on X coincides with the colimit topology lim −−→ n< ω X ⩽ n . Inparticular, one can replace the topology of a conically stratified X with thecolimit topology and keep the same ∞ -category of constructible hypersheaves.This is coherent with the fact that the ∞ -category of exit paths does not seethe global topology of X , as every map ∆ n → X in Exit A ( X ) is required tovisit only a finite number of strata. These two sets of conditions are usuallyincompatible as explained in an impossibility theorem [2.14]. We then obtaintwo versions of the representation theorem. heorem [3.12] (conical case) . Let X be a paracompact conically A -stratifiedspace such that each stratum of X be locally of singular shape and A be ω -stratifiedwith A ⩽ n satisfying the ascending chain condition, for each n < ω . Then the ∞ -category of exit paths Exit A ( X ) represents Fun ( Exit A ( X ) , S ) = Hyp A ( X ) the ∞ -category of A -constructible hypersheaves on X . Theorem [3.13] (colimit case) . Let X be an A -stratified space, colimit of a se-quence of closed stratified embeddings of paracompact conically stratified spacesover posets satisfying the ascending chain condition and whose strata are locally ofsingular shape. The ∞ -category of exit paths Exit A ( X ) represents Fun ( Exit A ( X ) , S ) = Hyp A ( X ) = Sh A ( X ) the ∞ -category of A -constructible sheaves and all A -constructible sheaves on X arehypersheaves. The dévissage theorem in the conical case relies heavily on a stratifiedhomotopy invariance theorem from Haine [5, 2.3]. However, it is neitherconstructible sheaves nor constructible hypersheaves that are invariant but hyperconstructible hypersheaves . This other notion of constructibility stemsfrom the di ff erence in functoriality between sheaves and hypersheaves.For this reason, we shall dedicate the first section to the definitions of allthe types of sheaves and constructibilities that we have mentioned so far. Inthe second section, we shall see that for general types of spaces constructiblesheaves, constructible hypersheaves and hyperconstructible hypersheaves docoincide. The third section is dedicated to the exit paths ∞ -category andthe proof of the extended representation theorem. The last section shallpresent some examples that this new representation theorem allows us toconsider: the metric and the topological exponentials of a Fréchet manifold,locally countable simplicial complexes and more generally locally countablecylindrically normal CW-complexes. We start by presenting the di ff erent characters at play: stratified spaces, hy-perconstructible hypersheaves, constructible hypersheaves and constructiblesheaves. There exists many di ff erent non-equivalent notions of stratified spaces intopology. Here we shall use a very general one, following Lurie [1, A.5.1]. Definition 1.1 (Stratified space) . A stratified topological space is the data ofcontinuous map f : X → A where A is a poset viewed as a topological space bydefining U ⊂ A to be open if and only if it is closed upwards.A morphism of stratified spaces is a commutative square X YA B here the top map is continuous and the bottom map is a poset map.For each a ∈ A , we shall let X a denote the a -stratum, preimage f − ( a ). Weshall denote by s a , the embedding X a ⊂ X . We shall also let X ⩽ a denote thefibre product X × A ⩽ a A . Remark 1.2. If A is itself stratified by a poset Λ , we thus get two maps X → A → Λ and for every λ ∈ Λ , X ⩽ λ is naturally stratified over A ⩽ λ .We shall mainly be interested in stratified maps over an identity morphism A = B . Indeed we shall essentially focus on stratified homotopy equivalences.Also, if A is a subposet of B , then one can see X as being B -stratified withoutloss of generality. Definition 1.3 (Stratified homotopy equivalence) . Let X → A and Y → A betwo A -stratified spaces. An A -stratified homotopy between two A -stratifiedmap f , g : X → Y is an A -stratified map h : [0 , × X → Y such that h (0 , − ) = f and h (1 , − ) = g .We shall say that an A -stratified map f : X → Y is an A -stratified homotopyequivalence if there exists an A -stratified map g : Y → X and A -stratifiedhomotopies between f ◦ g and id Y on one hand, and between g ◦ f and id X onthe other hand. In the classical theory of sheaves of sets, it is well known that isomorph-isms can be detected on stalks. This is no longer true for sheaves in the ∞ -categorical world and this leads to a separation into two equally interestingobjects: sheaves and hypersheaves. The main di ff erence here to which we shallpay special attention is the di ff erence in functoriality: for sheaves one usespullbacks but for hypersheaves one needs to use hyperpullbacks. There areseveral equivalent definitions of hypersheaves [1, A.1.9]. Since we are onlyinterested in the case of hypersheaves on topological spaces, we shall choosethe most convenient definition. Definition 1.4 (Hypersheaf) . Given a topological space X , we shall denoteby Sh ( X ) the ∞ -category of sheaves (of spaces) on X and by Hyp ( X ) the fullsubcategory of hypersheaves: sheaves that are local with respect to maps F → G inducing an equivalence F x → G x on stalks, for every point x ∈ X .The inclusion Hyp ( X ) ⊂ Sh ( X ) admits a left exact reflector, which sends asheaf F to its hypercompletion (cid:98) F . It is such that the canonical map F x → (cid:98) F x is an equivalence for every point x ∈ X .Categories of sheaves admit the following functoriality: if f : X → Y is acontinuous map, it induces an adjunction Sh ( X ) Sh ( Y ) f ∗ f ∗ between the ∞ -categories of sheaves on X and sheaves on Y . The right adjoint f ∗ preserves hypersheaves but not the left adjoint f ∗ . Definition 1.5 (Hyperpullback) . Let f : X → Y be a continuous map and let Sh ( Y ) Hyp ( X ) (cid:98) f ∗ enote the hyperpullback along f , obtained by first using f ∗ and then hyper-completing.By construction, we obtain an adjunction Hyp ( X ) Hyp ( Y ) f ∗ (cid:98) f ∗ similar to the sheaf case. Remark 1.6.
Given two continuous maps f : X → Y and g : Y → Z and ahypersheaf H on Z , the canonical map (cid:128) ( gf ) ∗ H −→ (cid:98) f ∗ (cid:98) g ∗ H is an equivalence. This stem from the fact that ( gf ) ∗ = g ∗ f ∗ .In addition, there is a useful case where hyperpullbacks and pullbackscoincide for hypersheaves. Lemma 1.7 [1, A.3.6] . Let f : X → Y be a continuous map. Assume f ∗ admits aleft adjoint f ! , then for every hypersheaf H the canonical map f ∗ H −→ (cid:98) f ∗ H is an equivalence.This happens for example, when f is an open embedding. Because sheaves and hypersheaves do not share the same functoriality, thereare two possible candidates to extend the traditional notion of locally constantsheaves of sets to the ∞ -category world: locally constant sheaves and locallyhyperconstant hypersheaves.Let X be a topological space and let π : X → ∗ denote the projection to thepoint. Definition 1.8 (Constant sheaf) . A constant sheaf on X is a sheaf of the form π ∗ K for some K ∈ S , where S is the ∞ -category of spaces. Definition 1.9 (locally constant sheaf) . A sheaf F is said to be locally constantif there exists an open covering { j i : U i ⊂ X } i ∈ I of X such that j ∗ i F is constantfor each i ∈ I .We shall denote by Sh loc ( X ) ⊂ Sh ( X ) the full subcategory of locally con-stant sheaves on X Definition 1.10 (Hyperconstant hypersheaf) . A hyperconstant hypersheaf on X is a hypersheaf of the form (cid:98) π ∗ K for some K ∈ S . Definition 1.11 (Locally hyperconstant hypersheaf) . We shall say that anhypersheaf H is locally hyperconstant if there exists an open covering { j i : U i ⊂ X } i ∈ I of X such that the restriction j ∗ i H is a hyperconstant hypersheaf for each i ∈ I .We shall denote by Hyp loc − hyp ( X ) ⊂ Hyp ( X ) the full subcategory of locallyhyperconstant hypersheaves on X . arning 1.12. Even though, one has an inclusion
Hyp ( X ) ⊂ Sh ( X ), one doesnot have Hyp loc − hyp ( X ) ⊂ Sh loc ( X ).The two main results that we shall use in this article emanate from Lurieand Haine. Both authors use a di ff erent (but equivalent) definition for locallyconstant sheaves [5, 1.4] and locally hyperconstant hypersheaves [1, A.2.12]:the one of locally constant sheaves on an ∞ -topos. In what follows, we spendsome time explaining why their definition agrees with the one we have justgiven. Definition 1.13 (Locally constant sheaves on an ∞ -topos) . For an ∞ -topos X ,let π : X → ∗ be a final map. A constant sheaf on X , is a sheaf of the form π ∗ K for some K ∈ S .A sheaf F on X is locally constant if there is a small family of étale maps { j i : U i → X } i ∈ I such that ⨿ i ∈ I U i → X be an e ff ective epimorphism and j ∗ i F bea constant sheaf on U i for every i ∈ I . Notation 1.14.
Let us denote by O ( X ) the (nerve of the) frame of open subsets U ⊂ X of a topological space X and let us denote by O ( X ), the frame ofopen subtoposes U ⊂ X of an ∞ -topos X . We shall also denote by E ( X ) the ∞ -category of étale maps over X . Lemma 1.15.
The frame of open subtoposes of an ∞ -topos X O ( X ) E ( X ) is a left exact and reflexive localisation of E ( X ) .Proof. Open subtoposes correspond to the ( − E ( X ),it then is a reflexive subcategory of E ( X ) [6, 5.5.6.18]. Moreover, a morph-ism f in E ( X ) becomes invertible in O ( X ) if and only if f is an e ff ectiveepimorphism [6, 6.2.3.5(1)]. E ff ective epimorphisms are stable under pull-backs [6, 6.2.3.15] and thus, the localisation functor preserves finite lim-its [6, 6.2.1.1]. Lemma 1.16.
Let X be a topological space, with associated ∞ -topos X and hyper-complete subtopos (cid:98) X ⊂ X .Then, the maps sending an open U ⊂ X to the open subtoposes U ⊂ X and (cid:98)
U ⊂ (cid:98) X induce equivalences O ( X ) = O ( X ) = O ( (cid:98) X ) between the frames of open subsets of X , open subtoposes of X and open subtoposesof (cid:98) X .Proof. For every ∞ -topos X , open subtoposes U ⊂ X correspond to subsheaves U of the terminal sheaf X .To every open U ⊂ X corresponds the characteristic sheaf U ⊂ X on X where U ( V ) is punctual whenever V ⊂ U and is empty otherwise. Conversely,let F ⊂ X be a subsheaf of the terminal sheaf. Because F is a sheaf, if F ( U )and F ( V ) are both non-empty, the value F ( U ∪ V ) must also be non-empty.Thus there is a biggest open subset U ⊂ X for which F ( U ) is non-empty. Bydirect inspection F = U . We thus have O ( X ) = O ( X ).Since hypercompletion Sh ( X ) → Hyp ( X ) is a left exact reflexive localisationfunctor, it induces a left exact reflexive localisation functor O ( X ) → O ( (cid:98) X ).Lastly, since the terminal sheaf X is truncated, every subsheaf is also trun-cated and thus hypercomplete [6, 6.5.1.14]. So, O ( X ) = O ( (cid:98) X ). emma 1.17. Let X be a topological space with associated ∞ -topos X . Then forevery étale map V → X , there exists an e ff ective epimorphism ⨿ i ∈ I U i −→ V over X where I is a small set and each U i is an open subtopos of X .Proof. By construction of X , the open subtoposes U ⊂ X form a dense sub-category of the ∞ -category of étale maps over X . Thus given any étale map V → X , the canonical map (cid:97)
U⊂X (cid:97)
U→V over X U −→ V is an e ff ective epimorphism. Indeed, this can be checked using injectivity ofthe pullback of subobjects [6, 6.2.3.10]. Since subobjects of coproducts canbe identified with products of subobjects [6, 6.2.3.9], we may start by lettinglet Y , Z ⊂ V be two open subtoposes of V such that Y × V U = Z × V U for every U → V over X . Then Y = Y × V V = Y × V lim −−→ U→V over X U (by density)= lim −−→ U→V over X Y × V U (by universality of colimits)= lim −−→ U→V over X Z × V U (by assumption)= Z (by symmetry)one gets Y = Z . Proposition 1.18 (Translation) . Let X be a topological space, with associated ∞ -topos X and hypercomplete subtopos (cid:98) X ⊂ X .Then, Sh loc ( X ) = Sh loc ( X ) locally constant sheaves on X are the locally constant sheaves on X .Likewise, Hyp loc − hyp ( X ) = Sh loc ( (cid:98) X ) locally hyperconstant hypersheaves on X are the locally constant sheaves on (cid:98) X .Proof. Since open embeddings are étale maps, it is clear that locally constantsheaves on X are locally constant on X .Let us look at the reverse direction. Let F be a locally constant sheaf on X and let ⨿ i ∈ I V i ↠ X be an étale e ff ective epimorphism such that the pullbackof F to each V i is constant.By the previous lemma, for every i ∈ I there exists a covering ⨿ j ∈ J i U ij ↠ V i over X , by open subtoposes of X . As small coproducts of e ff ective epimorph-isms are again e ff ective epimorphisms [6, 6.2.3.11] and the composition oftwo e ff ective epimorphisms is again an epimorphism [6, 6.2.3.12], we get ane ff ective epimorphism ⨿ i ∈ I ⨿ j ∈ J i U ij −→ ⨿ i ∈ I V i −→ X overing X . As the pullback of a constant sheaf is again a constant sheaf, the re-striction of F to each U ij is constant. Finally, since e ff ective epimorphisms arepreserved by left exact functors, this one is sent to an e ff ective epimorphismin O ( X ) [1.15] which is just O ( X ) [1.16], that is ∪ i ∈ I ∪ j ∈ J i U ij = X .Now for (cid:98) X . Since it is a subtopos of X , every étale map with codomain (cid:98) X is of the form (cid:98) V → (cid:98) X with V → X an étale map. If ⨿ i ∈ I (cid:98) V i → (cid:98) X is an e ff ect-ive epimorphism, then as hypercompletion is left exact, we get an e ff ectiveepimorphism ⨿ i ∈ I ⨿ j ∈ J i (cid:99) U ij −→ ⨿ i ∈ I (cid:98) V i −→ (cid:98) X by the same argument as above. In addition to the previous arguments, we addthat for any open U ⊂ X , constant sheaves on (cid:98) U correspond to hyperconstantsheaves on U . Remark 1.19 (Terminology) . Since locally hyperconstant hypersheaves on X are the locally constant sheaves on (cid:98) X , it has naturally led Haine to callthem ‘locally constant hypersheaves’. We had to change this terminology inorder to distinguish between hypersheaves that are locally hyperconstant andhypersheaves that are locally constant (as sheaves). Continuing with the two possible functorialities, we shall obtain constructiblesheaves and hyperconstructible hypersheaves.
Definition 1.20 (Constructible sheaves) . A sheaf of spaces F on an A -strati-fied space X is said to be A -constructible if its restriction s ∗ a F to each stratum X a is locally constant for every a ∈ A .We shall denote by Sh A ( X ) ⊂ Sh ( X ) the full subcategory of A -constructiblesheaves on X Remark 1.21.
By construction, given a stratified map f : X → Y between an A -stratified space and a B -stratified space, the pullback functor Sh ( Y ) Sh ( X ) Sh B ( Y ) Sh A ( X ) f ∗ f ∗ preserves constructible sheaves. Definition 1.22 (Hyperconstructible hypersheaves) . A sheaf F on X shall becalled A -hyperconstructible if the hyperrestriction (cid:98) s ∗ a F is locally hypercon-stant for every a ∈ A .We shall denote by Hyp A -hyp ( X ) ⊂ Hyp ( X ) the full subcategory of A -hyper-constructible hypersheaves. Remark 1.23 (Terminology) . Following a previous remark [1.19] for theterminology about locally constant hypersheaves, what we have chosen tocall hyperconstructible hypersheaves are the constructible hypersheaves ofHaine. We had to change the terminology in order to distinguish betweenhypersheaves that are hyperconstructible and those that are constructible (assheaves). arning 1.24. Here again
Hyp A -hyp ( X ) (cid:49) Sh A ( X ), a priori. Remark 1.25.
By construction, given a stratified map f : X → Y between an A -stratified space and a B -stratified space, the hyperpullback functor Hyp ( Y ) Hyp ( X ) Hyp B -hyp ( Y ) Hyp A -hyp ( X ) (cid:98) f ∗ (cid:98) f ∗ preserves hyperconstructible hypersheaves.The main distinctive feature of hyperconstructible hypersheaves is their in-variance under stratified homotopy equivalences. It makes hyperconstructiblehypersheaves the natural ∞ -analogue of the usual theory of locally constantsheaves and constructible sheaves with values in sets. Theorem 1.26 [5, 2.3] . An A -stratified homotopy equivalence f : X → Y betweentwo A -stratified spaces induces an equivalence Hyp A - hyp ( Y ) Hyp A - hyp ( X ) (cid:98) f ∗ between their ∞ -categories of A -hyperconstructible hypersheaves. We have warned the reader that one does not have an inclusion
Hyp A -hyp ( X ) ⊂ Sh A ( X ) due to the di ff erent functorialities between constructibility and hyper-constructibility.They are in fact related by a correspondence Hyp A ( X ) Sh A ( X ) Hyp A -hyp ( X )via the ∞ -category of A -constructible hypersheaves. This third ∞ -categoryshall become the may object of study in this article. The left inclusion isobvious, the right one requires a lemma. Lemma 1.27.
Let X be a topological space. Then Hyp loc ( X ) ⊂ Hyp loc − hyp ( X ) locally constant hypersheaves are locally hyperconstant.More generally, if X is A -stratified, then Hyp A ( X ) ⊂ Hyp A - hyp ( X ) A -constructible hypersheaves on X are A -hyperconstructible. roof. Assume H be a locally constant constant hypersheaf. Then there existsan open covering j i : U i ⊂ X such that j ∗ i H be a constant sheaf. But since j i is an open embedding, j ∗ i H is a hypersheaf. A sheaf which is both constantand a hypersheaf is hyperconstant. constant if and only if By construction,the hypercompletion of a constant sheaf is a constant hypersheaf, so (cid:98) j ∗ i F is aconstant hypersheaf and F is hyperlocally constant.The case of an A -constructible sheaf now follows from the functoriality ofhypercompleted pullbacks [1.6]: for every a ∈ A , by assumption s ∗ a H is locallyconstant, so let j a,i : U i ⊂ X a be an open covering so that j ∗ a,i s ∗ a F is constant.Then (cid:130) ( s a j a,i ) ∗ H = (cid:99) j ∗ a,i (cid:98) s ∗ a H is a constant hypersheaf. Remark 1.28.
There is another correspondence relating constructible sheavesand hyperconstructible hypersheaves, Sh A ( X ) Hyp A -hyp ( X ) Sh A -hyp ( X )it is the ∞ -category of hyperconstructible sheaves. But we shall not use it. In this section we shall show that in some general cases, the ∞ -categories of hy-perconstructible hypersheaves, constructible hypersheaves and constructiblesheaves, actually coincide. We start with the case of a single stratum. Lurie has introduced the notion oftopological space ‘locally of singular shape’ [1, A.4.15]. These are spaces X forwhich the counit map | Sing( U ) | → U is a shape equivalence for every open U ⊂ X . Letting π X : X → ∗ denote the projection to the point, if X is locally ofsingular shape then the singular sheaf π ∗ X Sing( X ) admits a canonical globalsection X → π ∗ X Sing( X ). This canonical section allows the definition of afunctor S / Sing( X ) Sh ( X ) K π ∗ X K × π ∗ X Sing( X ) X ψ X which is fully faithful and whose image is equivalent to the subcategory oflocally constant sheaves on X [1, A.2.15].When X is locally of singular shape, the pullback functor π ∗ X admits a leftadjoint ( π X ) ! [1, A.2.8] and so does ψ X . In particular, ψ X preserves all smalllimits; it also preserves all small colimits. Proposition 2.1 (Coincidence, stratum case) . Let X be a space which is locallyof singular shape. Then Hyp loc − hyp ( X ) = Hyp loc ( X ) = Sh loc ( X ) locally hyperconstant hypersheaves are locally constant and locally constant sheavesare hypersheaves. roof. Since truncation towers converge in S / Sing( X ) and ψ X commutes withlimits, all locally constant sheaves on X are hypersheaves [1, A.2.17]. Sinceevery open U ⊂ X is again locally of constant shape, the notions of constantsheaves and hyperconstant sheaves on U coincide. By ripple e ff ect, locallyhyperconstant hypersheaves on X are locally constant.Being locally of singular shape also gives locally constant sheaves limitsand colimits, they are computed as in the ∞ -category of sheaves. Lemma 2.2.
Let X be locally of singular shape. The ∞ -category Sh loc ( X ) admitsall small limits and colimits. In addition, the inclusion Sh loc ( X ) ⊂ Sh ( X ) preservessmall limits and small colimits.Proof. The slice ∞ -category S / Sing( X ) has all small limits and colimits and ψ X preserves small limits and small colimits. Let X be an A -stratified space. The first thing one can ask of X to have somecoincidence theorem is that each of X be locally of singular shape. This isenough, for example, to guarantee that one can compute finite limits andsmall colimits of A -constructible sheaves on X . Lemma 2.3.
Let X be an A -stratified space. Assume that the stratum X a ⊂ X be locally of singular shape for each a ∈ A . The ∞ -category of A -constructiblesheaves on X admits finite limits and small colimits. Moreover, the inclusion Sh A ( X ) ⊂ Sh ( X ) preserves finite limits and small colimits.Proof. For each a ∈ A , the functor s ∗ a : Sh ( X ) → Sh ( X a ) preserves small colimitsand finite limits. Since Sh loc ( X a ) ⊂ Sh ( X a ) preserves all small limits andcolimits, it follows that a finite limit or a small colimit of A -constructiblesheaves on X is again A -constructible.But it is not enough to guarantee any coincidence between the di ff erentnotions of constructibility. Indeed, one needs to add a gluing assumption ofthe strata together.The one we shall use here is the conicality introduced by Lurie [1, A.5.5]; itis a less demanding condition than most other stratification hypothesis used intopology. A space is conically stratified when each point admits a coordinatedecomposition with on one side, a local coordinate dependant on the stratumand on the other side a radial coordinate describing a neighbourhood of thepoint around the stratum. Definition 2.4 (Open cone) . For a topological space X , the open cone of X isthe set C( X ) (cid:66) { } ⨿ ( R ∗ + × X )with topology defined as follows: A subset U ⊂ C( X ) is open if and only if U ∩ ( R ∗ + × X ) is open, and if 0 ∈ U , then (0 , ε ) × X ⊂ U for some positive realnumber ε .If X is stratified over a poset A , then C( X ) is naturally stratified over theposet A ◁ obtained from A by adding a new element smaller than every otherelement of A . arning 2.5. One should not confuse the cone on X with the collapsed rectangle defined as the quotient R + × X/ { } × X . When X is compact and separated,the cone on X and the collapsed rectangle on X are homeomorphic. This isno longer true in the general case: the cone on the open interval (0 ,
1) can beembedded in R , whereas the collapsed rectangle on (0 ,
1) is not metrizable.If (
X, d ) is a metric space, the topology of the cone C( X ) is metrizable byletting d (( λ, x ) , ( µ, y ) = max( | λ − µ | , d ( x, y )) and by adding d (0 , ( λ, x )) = λ . Definition 2.6 (Conically stratified space) . Let f : X → A be a stratified to-pological space. Let a ∈ A and let U ⊂ X a be an open subset. We shall saythat an open V ⊂ X is a conical extention of U if there exists a stratified space L over A a< such that V is homeomorphic to U × C( L ) over the poset map A ◁a< = A a ⩽ ⊂ A . We shall say that X is conically A -stratified if for every a ∈ A every point x ∈ X a admits an open neighbourhood in X a that can be conicallyextended to X . Figure 1:
A submanifold N ⊂ M with a tubular neighbourhood is an example of aconically stratified space with two strata. Argument 2.7 (Reduction) . Let us gather some properties of conically strati-fied spaces that shall be used as a core reduction argument in the proofs. • If a conically A -stratified space X is paracompact, then each stratum X a can be covered with opens U admitting a paracompact conical exten-sion [1, A.5.16]. As a consequence, for any local problem on paracompact X , one can assume that X = X a × C( L ); • In a paracompact space X a × C( L ), the closed subspace X a ⊂ X a × C( L ) isalso paracompact, thus F σ open subsets W ⊂ X a form a basis of paracom-pact open subsets, stable under intersection [6, 7.1.1.1]. Moreover, if W ⊂ X a is an open F σ , then W × C( L ) ⊂ X a × C( L ) is again an open F σ of X a × C( L ) and is again paracompact; • For a paracompact W , the closed subspace W ⊂ W × C( L ) admits a basisof open neighbourhoods W ⊂ V ⊂ W × C( L ) all of which are homeo-morphic to W × C( L ) as stratified spaces [1, A.5.12]; • Combining the above arguments, every point x ∈ X admits a basis ofconical open neighbourhoods V x (cid:27) U x × C( L ). Proposition 2.8 (Coincidence, conical case) . Let X be a paracompact A -stratifiedspace, such that each stratum X a be locally of singular shape, for a ∈ A . Then, for very A -hyperconstructible sheaf H on X , the canonical map s ∗ a H → (cid:98) s ∗ a H is anequivalence on X a , for each a ∈ A and thus Hyp A - hyp ( X ) = Hyp A ( X ) A -hyperconstructible hypersheaves on X are A -constructible.Proof. Constructible hypersheaves are hyperconstructible [1.27]. Also, sinceeach stratum X a is locally of singular shape, for a ∈ A , locally hyperconstanthypersheaves on X a coincide with locally constant sheaves on X a [2.1]. It shallthen be enough to show that for every A -hyperconstructible hypersheaf H thecanonical map α : s ∗ a H → (cid:98) s ∗ a H is an equivalence on X a .This question is local on X a , so by the reduction arguments for conicallystratified space [2.7], we can reduce to the case were X = X a × C( L ). Continuingthe reduction, it is enough to show that α ( W ) is an equivalence on each F σ open subset W ⊂ X a since this is a basis stable under intersection. For eachsuch W , W × C( L ) is again paracompact and we can thus reduce to show that α ( W ) is an equivalence in the case where X = W × C( L ).Now because X is paracompact, s ∗ a H ( W ) = lim −−→ W ⊂ V H ( V ) [6, 7.1.5.6]. Be-cause W is paracompact, the neighbourhoods V can be taken homeomorphicto W × C( L ) as stratified spaces [2.7]. In such a case, by homotopy invari-ance [1.26], the restriction map H ( V ) → (cid:98) s ∗ a H ( W ) is an equivalence, fromwhich we finally get that α ( W ) is an equivalence. D -spaces We have seen that on conically A -stratified spaces, hyperconstructibility coin-cides with constructibility for hypersheaves. In order to add coincidence withconstructible sheaves, as in the stratum case, one needs to add an assumptionon the poset allowing induction on depth. Namely, one needs to assume thatthe poset satisfy the ascending chain condition.Adding the ascending chain condition to the poset A , we get a type ofspaces that shall become the building brick of the next construction; we thusgive it a name. Definition 2.9 ( D -space) . A D -space (for ‘good’ depth stratified space) is an A -stratified space X such that • X is paracompact; • the stratum X a ⊂ X is locally of singular shape, for each a ∈ A ; • X is conically A -stratified; • A satisfies the ascending chain condition. Remark 2.10.
Any C -stratified space in the sense of Ayala-Francis-Tannakais a D -space [7, 2.1.15]. Proposition 2.11 (Coincidence, D -space case) . Let X → A be a D -space, then Hyp A - hyp ( X ) = Hyp A ( X ) = Sh A ( X ) A -constructible sheaves on X are hypersheaves and A -hyperconstructible hyper-sheaves are A -constructible.Proof. The first equality follows from the conical case [2.8]. The second canbe proven by induction on the depth of A [1, A.5.9]. .4 Colimit and conical D ω -spaces We now turn to the main object of study: a class of stratified spaces onwhich to extend the representation theorem. The idea is to consider to thoseposets A which do not satisfy the ascending chain condition but which can beobtained as a countable union of closed subposets satisfying the ascendingchain condition. Definition 2.12 ( D ω -space) . A D ω -space is a stratified space X → A such that • X is paracompact; • A is ω -stratified; • X ⩽ n → A ⩽ n is a D -space for every n < ω .We shall say that X → A → ω is a conical D ω -space if X is conically A -stratified; colimit D ω -space if X coincides with the colimit X < ω (cid:66) lim −−→ n< ω X ⩽ n .One may legitimately ask: why divide D ω -spaces into two categories?This is because the very topology of the cone is often incompatible with acolimit topology. For example, if L is an ω -stratified, then there is a continuousbijection lim −−→ n< ω C( L ⩽ n ) −→ C( L )which is not a homeomorphism in the general case. This leads to an impossib-ility theorem, where the two conditions become mutually exclusive. Remark 2.13.
Let X (cid:44) → · · · (cid:44) → X p (cid:44) → · · · be a sequence of closed embeddingsbetween T topological spaces and let X denote its colimit. Then every morph-ism K → X with K compact factors through one X p ⊂ X [8, 2.4.2].As a consequence, a sequence ( x n ) n ∈ N in X , converges only if it is bounded. Theorem 2.14 (Impossibility) . Assume X be an A -stratified space such that • X be T and not empty; • X ⩽ a ⊂ X have empty interior for each a ∈ A ; • X = lim −−→ a ∈ A X ⩽ a ; • A contain an ascending chain,then X is not conically stratified.Proof. One can assume that A = ω and that X is not empty without loss ofgenerality. Let x ∈ X ⩽ , if X is conically stratified, then there exists Z and Y such that Z × C( Y ) is stratifiedly homeomorphic to an open neighbourhood U x of x . Since each X ⩽ n has empty interior, U x is not contained in any of them.One can thus find a sequence in U x whose image in ω is strictly increasing. Let( y n ) be the corresponding coordinate in Y of this sequence. Then for any z ∈ Z ,the sequence ( z, ( λ n , y n )) converges to ( z,
0) in Z × C( Y ) for any sequence λ n → X since it is not bounded in the stratification [2.13].Nevertheless, conical and colimit D ω -spaces are very close; one can alwayschange the global topology of a conical D ω -space to make it become a colimit D ω -space. emma 2.15. Let X → A → ω be a D ω -space. Then X < ω → A → ω is a colimit D ω -space. Moreover, if Y → X is a continuous map making Y → X → A → ω intoa D ω -space, then one gets a continuous map Y < ω → X ω over A .Proof. The only non-trivial thing to check is the paracompactness. But sinceby hypothesis each X ⩽ n is paracompact, the space X < ω is built as a sequentialcolimit of closed embeddings of paracompact spaces and is thus paracom-pact [9, 8.2]. Lemma 2.16.
Let X → A → ω be a D ω -space. For each n < ω , let us denote by j ( n ) the closed stratified embedding X ⩽ n ⊂ X . Then for each A ⩽ n -constructiblesheaf F on X ⩽ n , F ∈ Sh A ⩽ n ( X ⩽ n ) = ⇒ j ( n ) ∗ F ∈
Hyp A ( X ) its pushforward j ( n ) ∗ F is an A -constructible hypersheaf on X .Proof. Let a ∈ A . Either X a ∩ X ⩽ n = ∅ , in which case s ∗ a j ( n ) ∗ F is the initialsheaf by proper base change [6, 7.3.2.13], or X a ⊂ X ⩽ n , in which case onehas s ∗ a j ( n ) ∗ F = s ∗ a F again by proper base change and is locally constant byhypothesis.Since by assumption X ⩽ n is a D -space, F is a hypersheaf [2.11] and aspushforwards preserve hypersheaves, j ( n ) ∗ F is also a hypersheaf. Theorem 2.17 (Dévissage, colimit case) . Let X → A → ω be a colimit D ω -spaceThe inclusion maps j ( n ) : X ⩽ n ⊂ X induce an adjunction Sh ( X ) lim ←−− n< ω Sh ( X ⩽ n ) j ∗ j ∗ which is an equivalence of ∞ -categories and which reduces to an equivalence Sh A ( X ) = lim ←−− n< ω Sh A ⩽ n ( X ⩽ n ) between the ∞ -category of A -constructible sheaves on X and the inverse limit ofthe ∞ -categories of constructible sheaves on each X ⩽ n . Moreover, Hyp A ( X ) = Sh A ( X ) A -constructible sheaves on X are hypersheaves.Proof. The fact that j ∗ ⊣ j ∗ is an equivalence of ∞ -categories follows from thefact that X is the colimit of a sequence of closed embeddings of paracompactspaces [6, 7.1.5.8]. Then the restriction to the subcategories of contructiblesheaves follow directly. Finally, since since X ⩽ n is a D -space for every n < ω ,all A ⩽ n -constructible sheaves on X ⩽ n are hypersheaves [2.11] and as limits ofhypersheaves are again hypersheaves, we get that all A -constructible sheaveson X are hypersheaves.This theorem for colimit D ω -spaces, together with the coincidence pro-position for conically stratified spaces shall let us see that constructible hy-persheaves (which are a priori not functorial) inherit the functoriality ofhyperconstructible hypersheaves. In particular it shall also inherit its homo-topy invariance. orollary 2.18 (Functoriality of constructible hypersheaves) . Let X → A → ω be a colimit or a conical D ω -space and let Y → B be a stratified space. Let f : X → Y be a stratified map. Then for any B -constructible hypersheaf HH ∈
Hyp B ( Y ) = ⇒ (cid:98) f ∗ H ∈
Hyp A ( X ) its hyperpullback (cid:98) f ∗ H is A -constructible.In particular, if f is a stratified homotopy equivalence between two conical orcolimit D ω -spaces, then (cid:98) f ∗ induces an equivalence between the ∞ -categories ofconstructible hypersheaves.Proof. When X is a conical D ω -space: then since H is a B -constructible hy-persheaf, it is B -hyperconstructible [1.27]. The hyperpullback (cid:98) f ∗ H is then A -hyperconstructible. But since X is conically stratified, it is also A -construct-ible by coincidence [2.8].When X is a colimit D ω -space, then f ∗ H is A -constructible because H is B -constructible. By the previous theorem f ∗ H is then a hypersheaf and so, thecanonical map f ∗ H → (cid:98) f ∗ H is an equivalence of sheaves on X .As a consequence, when restricted to colimit and conical D ω -spaces, con-structible hypersheaves form a subfunctor of the functor of hyperconstructiblehypersheaves [2.8]. It is thus invariant under stratified homotopy equival-ences [1.26]. Theorem 2.19 (Dévissage, conical case) . Let X → A → ω be a D ω -space andassume that X is either a conical or a colimit D ω -space. The inclusion maps j ( n ) : X ⩽ n ⊂ X induce an adjunction Sh ( X ) lim ←−− n< ω Sh ( X ⩽ n ) j ∗ j ∗ which reduces to an equivalence Hyp A ( X ) = lim ←−− n< ω Sh A ⩽ n ( X ⩽ n ) between the ∞ -category of A -constructible hypersheaves on X and the inverse limitof the ∞ -categories of constructible sheaves on each X ⩽ n .Proof. Using the dévissage theorem for the colimit case, we shall identify theright hand side of the equivalence with the ∞ -category of A -constructiblesheaves on X < ω and see the adjunction j ∗ ⊣ j ∗ as steming from the canonicalmap j : X < ω → X . For a sheaf F on X < ω , we shall denote by F ⩽ n its restrictionto X ⩽ n for each n < ω and by F a its restriction to X a for each a ∈ A .We shall start by showing that for every A -constructible sheaf F on X < ω the map ψ F : s ∗ a j ∗ F → F a is an equivalence for each a ∈ A . This is of localnature, thus by the reduction arguments [2.7], one can assume that X = X a × C( L ). Continuing the reduction, it shall then su ffi ce to show that ψ ( W )is an equivalence for every F σ open subset W ⊂ X a , since this is a basis stableunder intersection. For each such W , W × C( L ) is again paracompact and wethus reduce to show that ψ ( X a ) is an equivalence.We shall prove by induction on k < ω that for every F the map ψ F ( X a ) is k -connective. For the case k = 0, the canonical map π j ∗ F ( X ) = π (cid:16) lim ←−− n< ω j ( n ) ∗ F ⩽ n ( X ) (cid:17) −→ lim ←−− n< ω π j ( n ) ∗ F ⩽ n ( X ) s surjective. As each j ( n ) ∗ F ⩽ n is a constructible hypersheaf [2.16], by homo-topy invariance [1.26] the stratified deformation retract X a × C( L ) → X a givesus j ( n ) ∗ F ⩽ n ( X ) = F a ( X a ). We deduce that π j ∗ F ( X ) → π F a ( X a ) is surjective,and thus that π ψ F ( X a ), through which the surjection factors, is surjective.Assume that we have shown that ψ F ( X a ) is k -connective for every A -con-structible F , for some k < ω . To show that ψ F ( X a ) is ( k + 1)-connective, it isequivalent to show that for every section η ∈ s ∗ a j ∗ F ( X a ), the induced map ∗ × s ∗ a j ∗ F ( X a ) ∗ ∗ × F a ( X a ) ∗ ψ ′ is k -connective. Because X is paracompact and X a ⊂ X is a closed embedding,there exists an open neighbourhood V of X a in X on which η can be extendto a section η [6, 7.1.5.5]. Since X a is paracompact, shrinking V if necessary,one can assume that V is homeomorphic to X a × C( L ) as a stratified space [2.7].We may thus reduce to the case where η is a global section of j ∗ F on X . Theinduced map X < ω → F allows us to define G (cid:66) × F . The sheaf G is again A -constructible since it is a finite limit of A -constructible sheaves [2.3]. By leftexactness, one has ψ ′ = ψ G ( X a ) which is then k -connective by the inductionhypothesis.We deduce that j ∗ F is A -constructible. Since hypersheaves are stable underpushforwards and all A -constructible sheaves on X < ω are hypersheaves, j ∗ F isan A -constructible hypersheaf.So, j ∗ j ∗ F is also A -constructible and thus, a hypersheaf. We also deducethat for every point x ∈ X < ω , the counit map ( j ∗ j ∗ F ) x → F x is an equivalenceand thus that j ∗ j ∗ F → F is an equivalence.We now show that for every A -constructible hypersheaf H on X , the unitmap υ : H → j ∗ j ∗ H is an equivalence. Since H is a hypersheaf, one needs toshow that υ x : H x → ( j ∗ j ∗ H ) x is an equivalence for every x ∈ X . For this it isenough to show that υ ( V x ) : H ( V x ) → j ∗ j ∗ H ( V x ) is an equivalence for a basisof open subsets x ∈ V x ⊂ X . By the conical nature of X , it is possible to select V x (cid:27) U x × C( L ) [2.7]. We then reduce to showing that υ ( X ) is an equivalence inthe case where X = X a × C( L ).Let p be the unique natural number such that X a ⊂ X p . Since j ∗ j ∗ H ( X a × C( L )) = lim ←−− p ⩽ n< ω H ⩽ n ( X a × C( L ⩽ n ))it shall be enough to see that H ( X a × C( L )) → H ⩽ n ( X a × C( L ⩽ n )) is an equivalencefor every n ⩾ p . One has a commutative diagram of restriction maps H ( X a × C( L )) H ⩽ n ( X a × C( L ⩽ n )) s ∗ a H ( X a ) = (cid:98) s ∗ a H ( X a ) [2.8]for which both of the vertical arrows are equivalences by homotopy invari-ance [1.26], We conclude using the two-out-of-three property of equival-ences. Remark 2.20.
Combining the two dévissage theorems with the coincidencetheorem in the conical case [2.8], we see that for a conically A -stratified D ω -space, one has the following coincidences Hyp A -hyp ( X ) = Hyp A ( X ) = Hyp A ( X < ω ) = Sh A ( X < ω ) hich can be extended with Hyp A -hyp ( X < ω ) = Hyp A ( X < ω )a coincidence between A -hyperconstructible hypersheaves on X < ω and A -con-structible hypersheaves on X . Contrarily to the conical case, this last equalitymight not happen systematically for colimit D ω -spaces. However, one canshow that this is the case whenever the colimit D ω -space arises from a conical D ω -space. In this section, we extend the representation theorem for constructible sheaveson D -spaces given by Lurie, to a representation theorem for constructiblehypersheaves on colimit or conical D ω -spaces. ∞ -category Definition 3.1.
Let p < ω , we shall view the topological simplex | ∆ p | = (cid:110) ( t , . . . , t p ) ∈ [0 , p +1 : t + · · · + t p = 1 (cid:111) as a stratified space over the poset { < · · · < p } with | ∆ p | ⩽ i being the set oftuples ( t , . . . , t i , , . . . ,
0) for every i ⩽ p .Given a stratified space X → A , we let Exit A ( X ) denote the simplicial setwhose p -simplicies are the stratified maps | ∆ p | → X , for every p < ω . Theorem 3.2 [1, A.6.4] . Let X be a conically A -stratified space. Then Exit A ( X ) isan ∞ -category. Proposition 3.3.
Let X → A be a stratified space. Assume A be also stratifiedover a filtered poset Λ and assume that X ⩽ λ be conically A ⩽ λ -stratified space foreach λ ∈ Λ . Then, the simplicial set Exit A ( X ) is an ∞ -category and the canonicalisomorphism of simplicial sets lim −−→ λ ∈ Λ Exit A ⩽ λ ( X ⩽ λ ) = Exit A ( X ) exhibits Exit A ( X ) as a colimit of the diagram { Exit A ⩽ λ ( X ⩽ λ ) } λ ∈ Λ in the ∞ -categoryof ∞ -categories.Proof. We notice that
Exit A ⩽ λ ( X ⩽ λ ) → Exit A ( X ) is a monomorphism for every λ ∈ Λ , thus the canonical map lim −−→ λ ∈ Λ Exit A ⩽ λ ( X ⩽ λ ) → Exit A ( X ) is also a mono-morphism. It is an isomorphism because, as Λ is filtered, any map of posets { < · · · < n } → A must factor trough A ⩽ λ for some λ ∈ Λ and by definition,an n -simplex of Exit A ( X ) is a stratified map | ∆ n | → X , which must then factorthrough X ⩽ λ for some λ ∈ Λ .To show that Exit A ( X ) is an ∞ -category, we need to check that it has theright lifting property Λ ni Exit A ( X ) = lim −−→ λ ∈ Λ Exit A ⩽ λ ( X ⩽ λ ) ∆ n ∗ ith respect to inner horns inclusions. Because inner horns Λ ni are finite sim-plicial set for every n < ω and every 0 < i < n , and Λ is filtered, any map Λ ni → Exit A ( X ) must factor through Exit A ⩽ λ ( X ⩽ λ ) for some λ ∈ Λ . By assumption X ⩽ λ is conically A ⩽ λ -stratified and a lift to a map ∆ n → Exit A ⩽ λ ( X ⩽ λ ) ⊂ Exit A ( X )exists by the previous theorem.Since marked simplicial sets form a simplicial model category [6, 3.1.4.4],in order to show that Exit A ( X ) is a colimit of the diagram { Exit A ⩽ n ( X ⩽ λ ) } λ ∈ Λ inthe ∞ -category of ∞ -categories, it will be enough to show that Exit A ( X ) ♮ is ahomotopy colimit of the diagram { Exit A ⩽ λ ( X ⩽ λ ) ♮ } λ ∈ Λ [6, 3.1.4.1 & 4.2.4.1]. Thisis the case since the model category of marked simplicial sets admits cofibrantgenerators with ω -small domains and codomains (the marked simplicial maps,with underlying simplicial map (cid:128) ∆ n ⊂ ∆ n ), and filtered colimits in such casescoincide with homotopy colimits [10, 7.3]. Corollary 3.4.
For any D ω -space X , one has an equality Exit A ( X ) = Exit A ( X < ω ) between the exit paths ∞ -categories of X and X < ω . In other words, the exit paths ∞ -category does not depend on the globaltopology of the space. Let us recall how the ∞ -category Fun ( Exit A ( X ) , S ) represents A -constructiblesheaves in the case where X is a D -space [1, A.10]. First the ∞ -category offunctors Fun ( Exit A ( X ) , S ) can be replaced by N( A ◦ X ), the ∞ -category associatedto the simplicially enriched category of fibrant-cofibrant objects of the category Set ∆ / Exit A ( X ) endowed with the covariant model structure, via Fun ( Exit A ( X ) , S ) N (cid:18)(cid:18) Set C [ Exit A ( X )] ∆ (cid:19) ◦ (cid:19) N (cid:16) A ◦ X (cid:17) a chain of equivalences [6, 2.2.1.2 & 4.2.4.4], where the left functor is aforgetful functor and the right functor is the unstraightening functor. Onecan then define a functor O ( X ) op × A ◦ X → Set ◦ ∆ ( U , Y ) (cid:55)−→ Fun
Exit A ( X ) ( Exit A ( U ) , Y )which will induce a functorN (cid:16) A ◦ X (cid:17) PSh ( X ) Ψ X with values in the ∞ -category of presheaves on X . Theorem 3.5 [1, A.10.5, A.10.10 & A.10.3] . Let X → A be a D -space. Thefunctor Ψ X : N( A ◦ X ) → PSh ( X ) is fully faithful and its image is equivalent to thesubcategory of A -constructible sheaves on X . We shall now extend this theorem to the case where X is a D ω -space. Lemma 3.6.
Let X be an A -stratified space, then N( A ◦ X ) is a presentable ∞ -cat-egory and the functor Ψ X is a right adjoint. roof. The simplicial model category A X is combinatorial [6, 2.1.4.6], so itsassociated ∞ -category is presentable [6, A.3.7.6].In view of the adjoint functor theorem [6, 5.5.2.9], it shall be enough toshow that Ψ X is both continuous and accessible.For every open subset U ⊂ X , the functor Y (cid:55)→ Fun
Exit A ( X ) ( Exit A ( U ) , Y ) from A ◦ X to Set ◦ ∆ preserves homotopy limits since A X is a simplicial model categoryand Exit A ( U ) is cofibrant (as all objects are). As a consequence the functor Y (cid:55)→ Ψ X ( Y )( U ) preserves all small limits [6, 4.2.4.1 & 4.2.3.14]. It follows that Ψ X preserves all small limits because limits in presheaves ∞ -categories arecomputed pointwise [6, 5.1.2.3].Using the same arguments, to prove that Ψ X is accessible, it shall beenough to find a cardinal κ such that the functors Y (cid:55)→ Ψ X ( Y )( U ) commutewith κ -filtered homotopy colimits for every U ⊂ X . Since A X is combinat-orial, there exists a cardinal κ such that all κ -filtered colimits be homotopycolimits [10, 7.3]. Enlarging κ if necessary, we can also demand that Exit A ( U )be κ -small for every open subset U ⊂ X . As a consequence, Y (cid:55)→ Ψ X ( Y )( U )commutes with all (homotopy) κ -filtered colimits for every open U ⊂ X . Lemma 3.7.
Let X be a D ω -space. For each n < ω , denote by j ( n ) the inclusion of X ⩽ n into X . One has a bireflexive simplicial model localisation A X A X ⩽ n j ( n ) ∗ j ( n ) ! j ( n ) ∗ where j ( n ) ! is the forgetful functor, where j ( n ) ∗ ( Y ) (cid:66) Exit A ⩽ n ( X ⩽ n ) × Exit A ( X ) Y and where both categories are endowed with the covariant model structure.Proof. The existence of the right adjoint to j ( n ) ∗ can be obtained by the adjointfunctor theorem, using the presentability of A X and the universality of colim-its in A X ⩽ n . The obvious fully faithfulness of the left adjoint j ( n ) ! implies thefully faithfulness of the right adjoint j ( n ) ∗ .The forgetful functor j ( n ) ! has an obvious simplicial enrichment. Moreover,one has evident isomorphisms j ( n ) ! ( ∆ p × Y ) = ∆ p × j ( n ) ! ( Y ) natural in Y ∈ A X ⩽ n for every p < ω . This is enough to endow j ( n ) ∗ and the adjunction j ( n ) ! ⊣ j ( n ) ∗ with a simplicial enrichment [11, 3.7.10]. Likewise one has isomorphisms j ( n ) ∗ ( ∆ p × Y ) = ∆ p × j ( n ) ∗ ( Y ) natural in Y ∈ A X for every p < ω , giving j ( n ) ∗ andthe adjunction j ( n ) ∗ ⊣ j ( n ) ∗ a simplicial enrichment.In this kind of setup, the pair j ( n ) ! ⊣ j ( n ) ∗ is always a model adjunc-tion [6, 2.1.4.10]. The embedding Exit A ⩽ n ( X ⩽ n ) ⊂ Exit A ( X ) is a right fibration.This follows from the fact that if a stratified path ends in X ⩽ n , then all thepath must be in X ⩽ n . As a consequence, the adjunction j ( n ) ∗ ⊣ j ( n ) ∗ is a modeladjunction [12, 11.2].As a consequence, both functors j ( n ) ∗ and j ( n ) ∗ preserve fibrant objects bythe previous lemma and induce a reflexive localisationN (cid:16) A ◦ X (cid:17) N (cid:16) A ◦ X ⩽ n (cid:17) j ( n ) ∗ j ( n ) ∗ etween the associated ∞ -categories [6, 5.2.4.5]. These adjunctions induce anadjunction N (cid:16) A ◦ X (cid:17) lim ←−− n< ω N (cid:16) A ◦ X ⩽ n (cid:17) j ∗ j ∗ where j ∗ is given by the family of the functors j ( n ) ∗ and where j ∗ ( { Y ⩽ n } ) (cid:66) lim ←−− n< ω j ( n ) ∗ Y ⩽ n for any { Y ⩽ n } ∈ lim ←−− n< ω N( A ◦ X ⩽ n ). Lemma 3.8.
Let C be an ∞ -category and consider two towers of monomorphisms X (cid:44) → X (cid:44) → . . . (cid:44) → X n (cid:44) → . . . (cid:44) → X and Y (cid:44) → Y (cid:44) → . . . (cid:44) → Y n (cid:44) → . . . (cid:44) → Y in C .Assume that we have also given a morphism X → Y and equivalences X n ≃ Y n together with natural transformations making the square X n Y n X Y ≃ commute, for every n < ω . This data is enough to induce a commutative square lim −−→ n< ω X n lim −−→ n< ω Y n X Y ≃ whose top map is an equivalence and whose vertical maps are the canonical projec-tions.Proof. By composition with X → Y , we get a commutative trianglelim −−→ n< ω X n X Y letting us reduce to the case where X = Y .Up to an equivalence, we can assume that the ∞ -category of subobjectsof Y is a 0-category [6, 2.3.4.18]. We thus reduce to the case where the maps X n (cid:44) → Y and Y n (cid:44) → Y are equal for every n < ω and the result is trivial. Remark 3.9.
We shall use this lemma in the special case where C is the ∞ -cat-egory of presentable ∞ -categories and right adjoints. In this case fully faithfulright adjoint functors are monomorphisms [13, 5.7]. As a consequence, reflex-ive localisation functors are epimorphisms in the ∞ -category of presentable ∞ -categories and left adjoint functors [6, 5.5.3.4]. Proposition 3.10 (Dévissage) . Let X be a D ω -space. The adjunction N (cid:16) A ◦ X (cid:17) lim ←−− n< ω N (cid:16) A ◦ X ⩽ n (cid:17) j ∗ j ∗ induced by the pairs j ( n ) ∗ ⊣ j ( n ) ∗ is an equivalence of ∞ -categories. roof. Let
Set C [ Exit A ( X )] ∆ denote the category of simplicially enriched functorsfrom C [ Exit A ( X )] to Set ∆ endowed with the projective model structure. Thediagram of right model adjoints Set C [ Exit A ( X )] ∆ A X Set C [ Exit A ( X ⩽ n )] ∆ A X ⩽ n unstraightening j ( n ) ∗ j ( n ) ∗ unstraightening is commutative [6, 2.2.1.1].Moreover, all functors are simplicially enriched [6, 2.2.2.12] and the com-mutation holds at the enriched level. Let us see why. The enrichment for theunstraightening functor stems from the following sequence: for any two sim-plicial functors F and G and any simplicial set K , any natural transformation F ⊠ K → G induces a natural transformation α ( F, G, K )Un( F ) × K −→ Un( F ) × Sing Q • ( K ) = Un( F ⊠ K ) −→ Un( G )while the enrichment for both pullback functors comes straight from com-mutation with tensoring with simplicial sets. Using this fact, one gets that j ( n ) ∗ α ( F, G, K ) is isomorphic to α ( j ( n ) ∗ F, j ( n ) ∗ G, K ).Because
Exit A ⩽ n ( X ⩽ n ) ⊂ Exit A ( X ) is a monomorphism, the associated sim-plicial functor C [ Exit A ⩽ n ( X ⩽ n )] → C [ Exit A ( X )] is a cofibration [6, 2.2.5.1], asa consequence the associated pullback functor j ( n ) ∗ preserves cofibrant ob-jects [6, A.3.3.9]. Then, all functors in this square preserve fibrant-cofibrantobjects and one gets N (cid:18)(cid:18) Set C [ Exit A ( X )] ∆ (cid:19) ◦ (cid:19) N (cid:16) A ◦ X (cid:17) N (cid:18)(cid:18) Set C [ Exit A ( X ⩽ n )] ∆ (cid:19) ◦ (cid:19) N (cid:16) A ◦ X ⩽ n (cid:17) j ( n ) ∗ j ( n ) ∗ an equivalence between the two induced pullback functors at the ∞ -categorylevel [6, 2.2.3.11]. One also has a commutative diagramN (cid:18)(cid:18) Set C [ Exit A ( X )] ∆ (cid:19) ◦ (cid:19) Fun ( Exit A ( X ) , S )N (cid:18)(cid:18) Set C [ Exit A ( X ⩽ n )] ∆ (cid:19) ◦ (cid:19) Fun (cid:16)
Exit A ⩽ n ( X ⩽ n ) , S (cid:17) j ( n ) ∗ j ( n ) ∗ letting us identify j ( n ) ∗ with the pullback at the level of ∞ -categories offunctors [6, 4.2.4.4].We then reduce to show that the canonical map Fun ( Exit A ( X ) , S ) lim ←−− n< ω Fun (cid:16)
Exit A ⩽ n ( X ⩽ n ) , S (cid:17) j ∗ is an equivalence [3.8, 3.9], which follows from the fact that Exit A ( X ) is acolimit of the diagram { Exit A ⩽ n ( X ⩽ n ) } n< ω in the ∞ -category of ∞ -categories [3.3]. heorem 3.11. Let X → A → ω be a D ω -space and assume that X be either aconical or a colimit D ω -space.The functor Ψ X is fully faithful and induces an equivalence of ∞ -categories N (cid:0) A ◦ X (cid:1) = Hyp A ( X ) between N( A ◦ X ) and the ∞ -category of A -constructible hypersheaves on X .Proof. For each n < ω and each open subset U ⊂ X one has j ( n ) ∗ Exit A ( U ) = Exit A ⩽ n ( U ∩ X ⩽ n ), implying that for each left fibration Y over Exit A ⩽ n ( X ⩽ n ), θ ( U , j ( n ) ∗ Y ) = θ ( U ∩ X ⩽ n , Y ) since j ( n ) ∗ ⊣ j ( n ) ∗ is a simplicially enriched ad-junction [3.7]. As a consequence, the squareN ( A X ) PSh ( X )N (cid:16) A X ⩽ n (cid:17) PSh ( X ⩽ n ) Ψ X j ( n ) ∗ Ψ X ⩽ n j ( n ) ∗ commutes up to a natural equivalence. In fact, because X ⩽ n is a D -space and j ( n ) ∗ preserves sheaves among presheaves and constructible sheaves amongsheaves [2.16], we have an inducedN ( A X ) PSh ( X )N (cid:16) A X ⩽ n (cid:17) Sh A ⩽ n ( X ⩽ n ) Ψ X j ( n ) ∗ Ψ X ⩽ n j ( n ) ∗ commutative square of right adjoints [3.6] by the representation theorem for D -spaces [3.5].One thus gets a commutative diagramN ( A X ) PSh ( X )lim −−→ n< ω N (cid:16) A X ⩽ n (cid:17) lim −−→ n< ω Sh A ⩽ n ( X ⩽ n ) Ψ X j ∗ lim −→ n< ω Ψ X ⩽ n j ∗ where the colimits are taken in the ∞ -category of presentable ∞ -categoriesand right adjoints [3.8, 3.9]. By the representation theorem for D -spaces [3.5],the bottom map is an equivalence of ∞ -categories. By dévissage [3.10], the leftarrow is an equivalence of ∞ -categories. Indeed, colimits in the ∞ -categoryof presentable ∞ -categories and right adjoints are canonically equivalent tothe associated limit in the ∞ -category of presentable ∞ -categories and leftadjoints [6, 5.5.3.4]. Using the same argument, one deduces that the rightarrow is fully faithful and its image is the subcategory of A -constructiblehypersheaves again by dévissage [2.17, 2.19]. Corollary 3.12 (Representation theorem, conical case) . Let X → A → ω be aconical D ω -space. Then the ∞ -category of exit paths Exit A ( X ) represents Fun ( Exit A ( X ) , S ) = Hyp A ( X ) the ∞ -category of A -constructible hypersheaves on X . orollary 3.13 (Representation theorem, colimit case) . Let X → A → ω be acolimit D ω -space. The ∞ -category of exit paths Exit A ( X ) Fun ( Exit A ( X ) , S ) = Hyp A ( X ) = Sh A ( X ) represents the ∞ -category of A -contructible sheaves. Remark 3.14.
It is natural to conjecture that the equivalences Ψ X are part ofa natural equivalence of functors. In particular, we should have commutativesquares Fun ( Exit A ( Y ) , S ) Fun ( Exit A ( X ) , S ) Hyp A ( Y ) Hyp A ( X ) ≃ Exit A ( f ) ∗ ≃ (cid:98) f ∗ whenever given a conical or colimit D ω -space Y → A → ω and a continuousmap f : X → Y letting X → Y → A → ω be a conical or colimit D ω -space.The existence of these commutative squares has been shown by Lurie inthe case where X and Y are D -spaces [1, A.10.16]. Using a similar proof, onecan show their existence in the case of conical or colimit D ω -spaces. Ayala, Francis and Rozenblyum have shown that the exit paths ∞ -categoryfunctor is fully faithful between the ∞ -category of smooth stratified spacesand the ∞ -category of ∞ -categories [14]. We prove here a modest extensionto D ω -spaces, showing that the exit paths functor is homotopy invariant. Thishomotopy invariance result invites us to think that the stratified homotopyhypothesis for smooth stratified spaces could be extended to stratified spaceswhose poset of strata does not satisfy the ascending chain condition. Lemma 3.15.
Let f : C → D be a functor between two small idempotent complete ∞ -categories. If the pullback functor Fun ( D , S ) Fun ( C , S ) f ∗ is an equivalence of ∞ -categories, then so is f .Proof. Let f ! denote the left adjoint to f ∗ . The following square Fun ( C , S ) Fun ( D , S ) C op D op f ! f op δ C δ D is commutative [6, 5.2.6.3]. By assumption, f ∗ is an equivalence of ∞ -categor-ies and so is its left adjoint f ! . Then, f ! induces an equivalence of ∞ -categoriesbetween the respective full subcategories of completely compact objects. Byidempotent completeness, these full subcategories are equivalent to the fullsubcategories of representable functors [6, 5.1.6.8]. As a consequence f op andthus f are equivalences of ∞ -categories. Lemma 3.16.
Let X be an A -stratified space, then Exit A ( X ) is an idempotentcomplete ∞ -category. roof. An idempotent stratified path in X must stay in the same stratum, inwhich case it is invertible and thus trivial. Theorem 3.17.
Let Λ be a filtered poset, A be a Λ -stratified poset, Y be an A -strat-ified space and let f : X → Y be a continuous map.Assume that for every λ ∈ Λ , both X ⩽ λ → A ⩽ λ and Y ⩽ λ → A ⩽ λ be D -spaces. If f is an A -stratified homotopy equivalence, then the functor Exit A ( X ) Exit A ( Y ) Exit A ( f ) is an equivalence of ∞ -categories.Proof. Since f is an A -stratified homotopy equivalence, it induces an A ⩽ λ -strat-ified homotopy equivalence f ⩽ λ : X ⩽ λ ≃ Y ⩽ λ for each λ ∈ Λ . By homotopyinvariance, the pullback functor ( f ⩽ λ ) ∗ : Sh A ⩽ λ ( Y ⩽ λ ) → Sh A ⩽ λ ( X ⩽ λ ) is an equi-valence of ∞ -categories [2.8, 2.11, 1.26]. As a consequence, the pullbackfunctor Exit A ⩽ λ ( f ⩽ λ ) ∗ from Fun ( Exit A ⩽ λ ( Y ⩽ λ ) , S ) to Fun ( Exit A ⩽ λ ( X ⩽ λ ) , S ) is also anequivalence of ∞ -categories [3.14]. Then by the previous lemmas, Exit A ⩽ λ ( f ⩽ λ )is an equivalence between the ∞ -category Exit A ⩽ λ ( X ⩽ λ ) and Exit A ⩽ λ ( Y ⩽ λ ). Tak-ing the colimit [3.3], we get that Exit A ( f ) is an equivalence of ∞ -categories. We shall give examples of applications of the representation theorem for con-structible hypersheaves in the case where the stratifying poset does not satisfythe ascending chain condition. Simplicial complexes and CW-complexesare natural examples of colimit D ω -spaces. We shall then describe a conicalexample of D ω -space, the metric exponential.One common point of these examples is that each stratum is locally homeo-morphic to a locally convex topological vector space and we shall need toknow that such spaces are locally of singular shape. Lemma 4.1.
Let X be a topological space locally homeomorphic to a locally convexreal topological vector space V . Then X is locally of singular shape.Proof. By assumption X is covered by spaces homeomorphic to V , so it shallbe enough to show that V is locally of singular shape [1, A.4.16]. Let U ⊂ V bean open subset of V and let A be the set of all convex open subsets of U . Since V is locally convex U = ∪ C ∈ A C . Moreover for any finite B ⊂ A , the intersection ∩ C ∈ B C is still convex. Lastly, any convex subset in a topological vector spaceis contractible and thus of singular shape [1, A.4.11]. This is enough to showthat U is also of singular shape [1, A.4.14]. Let X be a simplicial complex. Its constitutive simplices naturally give X astructure of stratified space X → S .Lurie has shown that when X is locally finite, then it is conically S -strat-ified [1, A.7.3]. The poset of simplices S also satisfies the ascending chaincondition. Moreover, Exit S ( X ) is always an ∞ -category [1, A.7.4] and thecanonical map Exit S ( X ) → N( S ) is an equivalence of ∞ -categories [1, A.7.5]. f X is a locally finite simplicial complex, it is then a D -space and therepresentation theorem applies. We shall extend this to the class of all locallycountable simplicial complexes. Definition 4.2.
We shall say that a simplicial complex X is locally countableif each vertex ν ∈ X belongs to at most a countable number of edges of X . Theorem 4.3.
Let X be a locally countable simplicial complex with poset of sim-plices S . One can represent S -constructible sheaves on X Fun (N( S ) , S ) = Sh S ( X ) as functors from N( S ) to S .Proof. For each vertex ν let E ν be the set of edges ϵ for which ν ∈ ϵ . Byassumption E ν is countable so one can find an exhaustion by finite subsetsE ν ⩽ ⊂ · · · ⊂ E ν ⩽ n ⊂ · · · ⊂ E ν for each vertex ν . For each n < ω , let X ⩽ n ⊂ X denote the biggest simplicialsubcomplex for which each edge ϵ ∈ X ⩽ n is such that ν ∈ ϵ = ⇒ ϵ ∈ E ν ⩽ n . Byconstruction the poset S ⩽ n of simplices of X ⩽ n is a downward closed subposetof S and X → S is exhausted by the subcomplexes X ⩽ n → S ⩽ n which are all D -spaces as discussed earlier.Every simplicial complex is paracompact and each simplex is locally ofsingular shape [4.1]. Thus X → S is a colimit D ω -space and the representationtheorem applies [3.13]. The case of CW-complexes resembles the one of simplicial complexes: theset of cells C of a CW-complex X can be made into a poset by letting d ⩽ c whenever d ⊂ c . However, this poset is generally odd with cells attached toother cells but isolated in the poset. It is then natural to restrict one’s attentionto normal CW-complexes. Definition 4.4.
A CW-complex X with set of cells C is said to be normal iffor every c ∈ C , its boundary is a again a union of cells of X .We shall say that a normal CW-complex X is locally finite whenever C ν< isfinite for every 0-cell ν , and we shall say that X is locally countable if instead C ν< is countable.The last remaining obstacle is the conicality of the stratification as locallyfinite normal CW-complexes might not be conically stratified. Tamaki hasintroduced the notion of cylindrically normal CW-complex [15, 4.1] and withTanaka, they have shown that for finite cylindrically normal CW-complexes,the cell stratification is conical [16, 1.7]. Since conicality is a local condition,this immediately extends to locally finite normal CW-complexes. RegularCW-complexes and, real or complex projective spaces are examples of cyl-indrically normal CW-complexes [15, 4.3]. Theorem 4.5.
Let X be a locally countable cylindrically normal CW -complex withposet of cells C . Then Exit C ( X ) is an ∞ -category Fun ( Exit C ( X ) , S ) = Sh C ( X ) representing C -constructible sheaves on X . roof. Every CW-complex is paracompact and each cell being homeomorphicto a locally convex topological vector space, it is locally of singular shape [4.1].By assumption if X is locally countable, for every 0-cell ν , one can find anexhaustion of C ν< C ν< ⊂ · · · C nν< ⊂ · · · C ν< by finite subsets. Let X ⩽ n ⊂ X denote the biggest subcomplex such that foreach cell e ∈ X ⩽ n , if ν ⩽ e , then e ∈ C nν< . By construction X ⩽ n is locally finiteand cylindrically normal, so it is conically stratified by the discussion above.In addition, because the border of every cell e ∈ X is a finite union of cellsof lower dimension, a direct induction shows that each cell e belongs to oneof the X ⩽ n . It follows that X is a colimit D ω -space and the representationtheorem applies [3.13]. Given a set X , the exponential on X is the set Exp( X ) of all finite subset S ⊂ X .If ( X, d ) is a metric space, the formula D ( S, T ) (cid:66) max max s ∈ S min t ∈ T d ( s, t )max t ∈ T min s ∈ S d ( s, t )canonically endows the exponential with a generalised metric. That is, ametric space in which two points can be at infinite distance from one another.Here this is the case of the empty configuration: D ( ∅ , S ) = + ∞ if S is not empty.The set Exp( X ) when endowed with its metric D is the metric exponentialExp M ( X ). The metric exponential is canonically stratified over the poset ω ∗ = { } ⨿ { < < · · · } , with a finite subset S ⊂ X being sent to its cardinality.Replacing the metric topology of Exp M ( X ) with the colimit one arising fromthis stratification, one obtainsExp T ( X ) (cid:66) lim −−→ n ∈ ω ∗ Exp M ⩽ n ( X )the topological exponential. When M is a Fréchet manifold, one can show thatthe metric exponential is conically stratified [17]. It follows that Exp M ( M )is a conical D ω -space and Exp T ( M ) is its associated colimit D ω -space. Inparticular Hyp ω ∗ (Exp M ( M )) = Sh ω ∗ (Exp T ( M )) = Fun ( Exit ω ∗ (Exp M ( X )) , S )both have the same ∞ -category of constructible hypersheaves and these arerepresentable using exit paths.Constructible cosheaves on the exponential form a key ingredient in thetheory of locally constant factorisation algebras. Constructible hypersheaveson the metric exponential are for example used by Lurie, to bridge locallyconstant factorisation algebras on a finite dimensional manifold M with thetheory of E M -algebras [1, 3.6.10]. Acknowledgements
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