Introduction
Contents and future applications.
This paper is the first step in a larger project devoted to a systematic development of the theory of perverse schobers. The latter are categorical analogs of perverse sheaves, in which vector spaces are replaced by (enhanced) triangulated categories. The idea of perverse schobers was proposed in [Reference Kapranov and Schechtman31] based on the features of various ‘elementary’ descriptions of perverse sheaves in terms of quivers. Namely, these descriptions are often of such form that a natural categorical analog (quiver representations formed by categories instead of vector spaces) suggests itself readily. For example, for the classical description [Reference Beilinson2, Reference Galligo, Granger and Maisonobe24] of perverse sheaves on the disk in terms of diagrams
with $\operatorname {id}  ab$ and $\operatorname {id}  ba$ invertible, such a categorical analog is found in the concept of a spherical adjunction; see [Reference Kapranov and Schechtman31].
However, the quiver descriptions do not give satisfying definitions of the category of perverse sheaves since they depend on auxiliary choices. For example, in the above case, a choice of a direction at the origin is needed to define vanishing and nearby cycles. On the other hand, from the customary point of view, a perverse sheaf is an object of an abelian category that arises as the heart of a certain tstructure on the derived category of constructible sheaves on a stratified topological space. It is not clear whether such an approach can be categorified directly.
In this paper, we identify perverse sheaves (not yet schobers) on a stratified surface X with socalled Milnor sheaves (Theorem 3.1.13). Similarly to the description of constructible sheaves as representations of the exit path category (see [Reference Treumann45]), our result follows from an alternative parametrization in terms of a hybrid of the exit and entrance path categories, called the Milnor category of the surface. Its objects, Milnor disks, are given by disks in X together with a choice of a finite number of boundary intervals. These intervals determine the interaction with the stratification: A disk may move on the surface via isotopy such that the points in the zerodimensional stratum exit the disk through the chosen boundary intervals and enter the disk through their complement. In addition, the boundary intervals themselves can interact in a way familiar from Connes’ cyclic category (see below for more details). A Milnor sheaf is then defined as a representation of the Milnor category subject to certain natural gluing conditions that arise from cutting Milnor disks into pieces.
As a result, we obtain an intrinsic definition of perverse sheaves on Riemann surfaces that is internal to the framework of abelian categories, without reference to derived categories, and which can therefore serve as an alternative to the definition given in [Reference Beilinson, Bernstein and Deligne3]. Our main incentive is that the definition has a comparatively straightforward categorification offering a good framework for perverse schobers. This approach will be elaborated in sequels to this paper.
Even in the uncategorified context of perverse sheaves, Milnor sheaves provide a novel perspective on classical aspects of the theory. For example, one motivation for the introduction of perverse sheaves is the fact that, in contrast to constructible sheaves, they are preserved under Verdier duality. This phenomenon becomes almost selfevident in the Milnor sheaf model. Namely, it is a direct consequence of a canonical selfduality of the Milnor category obtained by swapping the boundary intervals with their complements (generalizing the wellknown selfdualities of the cyclic and paracyclic categories).
In higher complex dimensions, a possible generalization could involve mimicking more closely the topology related to forming perverse sheaves of vanishing cycles associated to holomorphic functions. When such a perverse sheaf is supported at a single point (the ‘isolated microlocal singularity’ case), it reduces to a single vector space so we have purity just like for Riemann surfaces. We hope to explore this approach in future work.
Details of the main result
Fundamental for us is the concept of a Milnor disk, a pair $(A,A')$ where $A \subset X$ is a closed disk, containing at most one point from the zerodimensional stratum N, and $A'\subset \partial A$ is a finite nonempty disjoint union of closed intervals. These Milnor disks will be depicted by the symbols
We call the points in the zerodimensional stratum N special and signify them via the symbol
. For example, a Milnor disk $(A,A')$ with one boundary interval containing a special point will be referred to as
leaving the embedding of A into the surface X implicit. Milnor disks form the objects of the Milnor category $M(X,N)$ where a morphism from $(A,A')$ to $(B,B')$ is given by an equivalence class of isotopies $H: I \times {\mathbb {D}} \to X$ with $H_0: {\mathbb {D}} \cong A$ and $H_1: {\mathbb {D}} \cong B$ , together with a choice of bordism $P \subset I \times S^1$ from $H_0^{1}(A')$ to $H_1^{1}(B')$ such that the inclusion $H_1^{1}(B') \subset P$ is a homotopy equivalence (see Figure 1). Here, roughly speaking, the trajectories $H^{1}(N)$ of the special points are required to enter the cylinder through $(I \times S^1) \setminus P$ and exit through P. This hybrid exit–entry behaviour puts the Milnor category ‘in between’ the exit and entrance path categories of $(X,N)$ . As will be explained in the main body of this work, this phenomenon can be regarded as a geometric manifestation of the fact that the perverse tstructure lies ‘in between’ the standard tstructure and its Verdier dual.
In particular, while the exit and entrance path categories are dual to one another, the Milnor category is selfdual: On objects, the duality is given by
on morphisms, it is obtained by replacing the bordism P by the closure of $(I \times S^1) \setminus P$ and reversing the direction of the isotopy H. For example, the action of the selfduality associates to the morphism
depicted in Figure 1, the morphism
Given an object of the derived constructible category and a morphism $(H,P): (A,A') \to (B,B')$ of Milnor disks, we obtain a correspondence on relative (hyper) cohomology
and hence a functor
We note that can be identified with , the sheaf of vanishing cycles for with respect to an appropriate holomorphic function f (possibly with a zero of arbitrary order), hence the name ‘Milnor disk’, modelled after ‘Milnor fibers’ in singularity theory. In particular, we may now express the local classification data (0.1) at a special point ϵ N in terms of our terminology:

1. The space of vanishing cycles:

2. The space of nearby cycles:

3. The variation map
$$\begin{align*}a = \operatorname{var}: \Phi \to \Psi \end{align*}$$is the value of on the morphism (0.2).

4. The canonical map
$$\begin{align*}b = \operatorname{can}: \Psi \to \Phi \end{align*}$$is the value of on the morphism (0.3).
See §4.2 for a discussion of how to recover the relations $T_{\Psi } = \operatorname {id}  ab$ and $T_{\Phi } = \operatorname {id}  ba$ , expressing the monodromy in terms of these data.
Our main result is based on the observations that

(1) Perverse sheaves can be characterized by the fact that their relative (hyper) cohomology on Milnor disks is concentrated in degree $0$ ,

(2) A perverse sheaf F is completely described by its values on Milnor disks.
Observation (1) immediately implies that, for a perverse sheaf , the functor from equation (0.5) takes values in the abelian category given by the heart of the standard tstructure. Observation (2) then leads to the main result of this work: Theorem 3.1.13 establishes that the association provides an equivalence between the abelian category of perverse sheaves on the stratified Riemann surface $(X,N)$ and the category of Milnor sheaves: valued presheaves on the Milnor category $M(X,N)$ that satisfy descent conditions with respect to the cutting and pasting Milnor disks.
Method of proof: ${\infty }$ categorical Kan extension
Although the statement of Theorem 3.1.13 is ‘purely abelian’, the proof utilizes the ambient derived category and relies on $\infty $ categorical techniques. That is, we establish a result (Corollary 3.1.12) identifiying constructible sheaves with values in a stable ${\infty }$ category , and appropriately defined Milnor sheaves valued in . When is the ${\infty }$ categorical enhancement of the derived category of a Grothendieck abelian category , then perverse sheaves are recovered among all constructible complexes via the observation (1) above.
The method of proof of Corollary 3.1.12 is as follows. In general, identifying two given ${\infty }$ categories is hard to achieve by hand due to the infinite amount of coherence data involved. The technique of Kan extensions allows for an efficient means of handling such data and ‘mediating’ it across parametrizing diagram categories (see Proposition A.3). Using this technique, we produce equivalences between representations of various subcategories of the larger paracyclic category $\Lambda (X,N)$ to mediate the subcategories of standard disks, Milnor disks, and bounded disks. In this framework, we provide an alternative construction of the presheaf on the Milnor category $M(X,N)$ as a Kan extension from the category of standard disks (cf. §3).
Corollary 3.1.12 and various technical tools developed for its proof provide not only a stepping stone for the more classicallooking Theorem 3.1.13 but also present a possible framework for the generalization to perverse schobers. In that generalization, a stable ${\infty }$ categorical enhancement of triangulated categories is important from the very beginning.
The role of paracyclic Segal objects
Our approach to perverse sheaves via Milnor sheaves naturally involves structures familiar in the theory of cyclic homology [Reference Connes11, Reference Elmendorf21, Reference Loday36]. One of them is the paracyclic category $\Lambda _{\infty }$ which can be regarded as the universal central extension (by $\mathbb {Z}$ ) of the cyclic category $\Lambda $ of Connes [Reference Connes11].
Namely, in the most classical case, when $(X,N)$ is the disk $({\mathbb {D}},\{0\})$ with the origin as special point, a Milnor sheaf can be uniquely recovered from its values on Milnor disks containing $0$ . These disks form a subcategory of $M({\mathbb {D}},\{0\})$ equivalent to the paracyclic category $\Lambda _{{\infty }}$ , and our approach identifies valued perverse sheaves, with the following structures: paracyclic objects whose restriction to $\Delta ^{\operatorname {op}} \subset \Lambda _{\infty }^{\operatorname {op}}$ is a Segal [Reference Bergner5, Reference Dyckerhoff and Kapranov20] simplicial objects (see Corollary 4.3.2). Further, the equivalence of such structures with the more customary classification data (0.1) can be understood as a special instance the duplicial Dold–Kan correspondence (see §4.4).
This point of view turns out to be important for the generalization to perverse schobers. The corresponding analog of a perverse sheaf on the disk is, as mentioned above, a spherical adjunction. It turns out that any such adjunction gives, via a variant of the relative Waldhausen $S_{\bullet }$ construction [Reference Waldhausen48], rise to a paracyclic object whose restriction to $\Delta ^{\operatorname {op}}$ is 2Segal, that is, satisfies a twodimensional generalization of the Segal condition introduced in [Reference Dyckerhoff and Kapranov20]. Such data then form the local data comprising the structure of a perverse schober, as will be explained in subsequent work.
Relation to previous work
The dream of defining perverse sheaves in a way that would be at the same time topological (avoiding analysis and Dmodules) and abeliancategorical (avoiding derived categories) is of course as old as the theory of perverse sheaves itself. We should particularly mention the 1990 preprint of MacPherson [Reference MacPherson40] that introduced (in arbitrary dimension) the concept of Fary sheaves which are certain ‘cohomology theory’ data on an appropriate class of pairs $(U_+, U_)$ of opens in a stratified manifold. Our concept of a Milnor sheaf can be seen as an adaptation and a simplification of that of a Fary sheaf to the case of two real dimensions, when instead of a functor associating a graded vector space (i.e., several cohomology groups) to a pair of opens, we have a functor associating a single vector space, more in line with the idea of a ‘sheaf’.
1 Perverse sheaves on stratified surfaces
1.1 Perverse sheaves with values in abelian categories
Sheaves with values in abelian categories.
Let be an Grothendieck abelian category. In particular, has arbitrary products and projective limits.
For any topological space X, we denote by
the category of $\mathcal {A}$ valued sheaves over X. By definition, such a sheaf
is a contravariant functor from the poset of opens in X into $\mathcal {A}$ , satisfying descent. That is, for any open covering $\{U_i\}$ of an open set U, the map
is an isomorphism.
By $D(X, \mathcal {A})$ , we denote the (unbounded) derived category of . We consider it as a triangulated category.
For any continuous map $f: X\to Y$ of topological spaces, we have the standard adjoint functors
If $X,Y$ are locally compact, we also have the functors
with their standard adjunctions; cf. [Reference Kashiwara and Schapira33].
Decompositions, stratifications and exit paths
Concerning stratified spaces, we follow the terminology of [Reference Goresky and MacPherson27] part.II §1.12.
Thus, a decomposition of a topological space X is a collection ${\mathcal S}$ of locally closed subsets $S\in {\mathcal S}$ called strata such that $X=\bigsqcup _{S\in {\mathcal S}} S$ is a disjoint decomposition and the closure of a stratum is a union of strata. The set ${\mathcal S}$ acquired then a partial order $\preceq $ by inclusion of the closures, that is, $S\preceq S'$ if $S\subset \overline {S'}$ . For each $x\in X$ , we denote by $S_x\in {\mathcal S}$ the stratum containing x. A decomposed space $(X,{\mathcal S})$ is a space equipped with a decomposition.
The concept of decomposition is identical to that of an $({\mathcal S}, \preceq )$ stratification in the sense of [Reference Lurie38] Definition A.5.1. Recall that the latter defined as a continuous map $f: X\to {\mathcal S}$ , where the poset ${\mathcal S}$ is given the topology consisting of upwardly closed sets, that is, of ${\mathcal I}\subset {\mathcal S}$ such that $S\in {\mathcal I}$ implies $S'\in {\mathcal I}$ whenever $S\preceq S'$ . Explicitly, the map f is given by $f(x)=S_x$ .
Let $(X,{\mathcal S})$ be a decomposed space. We denote the inclusions of the strata by $i_S: S \to X$ . By , we denote the category of sheaves which are constructible with respect to ${\mathcal S}$ , that is, such that each is locally constant on S. By , we denote the subcategory of complexes of sheaves whose cohomology sheaves are constructible with respect to ${\mathcal S}$ .
Let us recall the concept of exit paths for $(X,{\mathcal S})$ , originally introduced by MacPherson; see [Reference Treumann45] for a more detailed treatment. For $x\in X$ , we denote by $S_x\in {\mathcal S}$ the stratum containing x. This gives a partial order $\preceq $ on X (as a set) given by $x\preceq y$ , if $S_x\preceq S_y$ , that is, $S_x\subset \overline {S_y}$ . An exit path for $(X,{\mathcal S})$ is a continuous parametrized path $\gamma : [0,1]\to X$ which is monotone with respect to $\prec $ , that is, such that for $t_1\leq t_2$ we have $\gamma (t_1)\preceq \gamma (t_2)$ . The category of exit paths $\operatorname {Exit}(X,{\mathcal S})$ has, as objects, all points $x\in X$ , with $\operatorname {Hom}_{\operatorname {Exit}(X,{\mathcal S})}(x,y)$ being the set of isotopy classes of exit paths $\gamma $ with $\gamma (0)=x$ and $\gamma (1)=y$ . Thus, $\operatorname {Exit}(X,{\mathcal S})$ can be considered as a stratified version of the fundamental groupoid of X (to which it reduces in the particular case when ${\mathcal S}$ consists of just one stratum X). By reversing the direction of the paths (or passing to the opposite category), we get the category of entrance paths ${\operatorname {Entr}}(X,{\mathcal S}) = \operatorname {Exit}(X,{\mathcal S})^{\operatorname {op}}$ .
We will use some particular types of decompositions in which one imposes various ‘conicity’ conditions describing the neighborhood of a stratum in the closure of a larger stratum:

(1) Whitney stratifications, see [Reference Goresky and MacPherson27] part II §1.2. In this case, the strata are $C^{\infty }$ manifolds.

(2) Topological stratifications, see [Reference Goresky and MacPherson26] and [Reference Treumann45] §3.1. In this case, the strata are topological manifolds.

(3) Conical stratifications, see [Reference Lurie38] Definition A.5.5. In this case, strata are not required to be manifolds, but near a stratum S, the space X is locally identified with the product of S and the cone over another decomposed space with strata labelled by $S'\in {\mathcal S}$ with $S\prec S'$ .
It is known that these three conditions are of increasing generality, that is, (1) $\Rightarrow $ (2) $\Rightarrow $ (3).
Proposition 1.1.1. Let $(X,{\mathcal S})$ be a space with a conical stratification. The category is equivalent to (the category of covariant functors).
Proof. For topological stratifications, this is the original result of MacPherson; see [Reference Treumann45] Theorem 1.2. For conical stratifications, this follows from [Reference Lurie38] Theorem A.9.3 which gives an ${\infty }$ categorical upgrade of $\operatorname {Exit}(X,{\mathcal S})$ .
Suppose now that X is a complex manifold and ${\mathcal S}$ is a complex analytic Whitney stratification of X. By , we denote the subcategory of perverse sheaves (with respect to the middle perversity). Recall [Reference Beilinson, Bernstein and Deligne3][Reference Kashiwara and Schapira33] that iff two conditions are satisfied:

( $P^+$ ) For every $S \in {\mathcal S}$ , we have for $n> \dim _{\mathbb {C}}(S)$ ,

( $P^$ ) For every $S \in {\mathcal S}$ , we have for $n < \dim _{\mathbb {C}} (S))$ .
It is well known [Reference Beilinson, Bernstein and Deligne3] that the category is the heart of a tstructure and so is an abelian category.
The case of stratified surfaces
We specialize to the case of $\dim _{\mathbb {C}}(X)=1$ , so X is a Riemann surface, possibly noncompact and with nonempty boundary. We fix a finite subset $N \subset X$ of interior points which we refer to as special points and denote the corresponding stratification $X = N \cup (X \setminus N)$ by ${\mathcal S} = {\mathcal S}_N$ . This gives a topological stratification, and we adopt the following definition.
Definition 1.1.2. By a stratified surface, we mean a pair $(X,N)$ consisting of:

(1) A topological manifold X of real dimension $2$ , possibly noncompact and with boundary.

(2) A finite subset $N \subset X$ of interior points which we refer to as special points.
We denote by $j: X\setminus N\to X$ and $i: N\to X$ the embeddings of the strata.
Let us fix a Grothendieck abelian category . We denote by the full subcategory of complexes whose cohomology sheaves are constructible with respect to the stratification ${\mathcal S}_N$ , that is, in our case, locally constant on $X\setminus N$ .
Further, the concept of a perverse sheaf makes sense in this context and is given explicitly as follows.
Definition 1.1.3. Let $(X,N)$ be a stratified surface and a Grothendieck abelian category. An object of is called perverse if

(1) is isomorphic to $L[1]$ , where L is a local system on $X\setminus N$ with values in ,

(2) for $n> 0$ ,

(3) for $n < 0$ .
The category of perverse sheaves with respect to N will be denoted . As explained above, it is an abelian category.
1.2 Milnor disks, Milnor pairs and the purity property
We denote by ${\mathbb {D}} \subset {\mathbb {C}}$ the closed unit disk. Let $(X,N)$ be a surface X with a set of special points $N \subset X$ as in §1.1. By a closed disk, we mean a subspace $A\subset X$ homeomorphic to ${\mathbb {D}}$ .
Definition 1.2.1. A Milnor disk in $(X,N)$ is a pair $(A,A')$ , where:

(1) $A\subset X$ is a closed disk containing at most one special point.

(2) $A'\subset \partial A\simeq S^1$ is a disjoint union of finitely many closed arcs, different from $\emptyset $ and the whole $\partial A$ .
See the left of Figure 2. The concept of a Milnor disk can be compared with the following possibly more intuitive concept.
Definition 1.2.2. A Milnor pair for $(X,N)$ is a pair $(U,U')$ , $U' \subset U$ , of closed subsets of X such that

(1) U is a closed disk containing at most one special point.

(2) $U'$ is a finite, nonempty, disjoint union of closed disks $\{U_i\}_{i \in I}$ such that $K=U \setminus U'$ is contractible.
Thus, a Milnor disk can be seen as a Milnor pair $(U,U')$ with $U'$ being very thin, reducing to a union of boundary arcs; see Figure 2. Up to homotopy equivalence, there is no difference between the two concepts.
Example 1.2.3. Let X be a Riemann surface (onedimensional complex manifold), z be a holomorphic coordinate near an interior point $x\in X$ and f be a holomorphic function defined near x such that $f(x)=0$ . Then for sufficiently small $\varepsilon> \delta >0$ the pair formed by
is a Milnor pair. This explains our terminology, motivated by the concept of Milnor fibers in singularity theory. Note that the cardinality $\pi _0(U')$ is equal to ${\operatorname {ord}}_x(f)$ , the order of vanishing of f at x.
The role of Milnor disks for our purposes stems from the following:
Proposition 1.2.4 (Purity property).
Let $(X,N)$ be a stratified surface, let be a Grothendieck abelian category and let be an object of the derived constructible category . Then the following are equivalent:

(i) is a perverse sheaf.

(ii) For every Milnor disk $(A,A')$ , the relative hypercohomology vanishes for $i\neq 0$ .
We will refer to the condition (ii) as purity.
Proof of Proposition 1.2.4.
(i) $\Rightarrow $ (ii): Assume that is perverse.
Assume first that A either contains no special point or contains exactly one special point x in its interior. Note that the first possibility is really a particular case of the second, as we can always introduce a ‘dummy’ special point, where a singularity is allowed but not present. So we assume that the second possibility holds. Denote by by $i_x: \{x\} \to X$ the inclusion of the point. Note that , and so its cohomology, by Definition 1.1.3(3), is concentrated in degrees $\geq 0$ . Further, , and so its cohomology, by Definition 1.1.3(2), is concentrated in degrees $\leq 0$ . Consider now the following diagram with rows and columns being exact triangles:
Note that , a local system in degree $(1)$ and $A{\setminus }\{x\}$ is homotopy equivalent to $S^1$ . So has cohomology only in degrees $\{1,0\}$ . The long exact sequence (LES) of cohomology of the middle row of the diagram gives, using the information above, the following:
Look now at the middle column of the diagram. Since is concentrated in degree $(1)$ , in order to show that has cohomology only in degree $0$ , it suffices to show that is injective. For this, it suffices to prove that the maps induced by a and b on $H^{1}$ are injective. For a, it follows from the fact (1.2.6) that has no cohomology in degree $(1)$ . For b, we use the identification as above. Then the statement becomes that $H^0(A{\setminus }\{x\}; L)\to H^0(A';L)$ is injective which is clear.
Suppose now that the special point x lies in $\partial A$ . If $x\in A'$ , then by excision we reduce to the case when $A\cap N=\emptyset $ treated above. So let $x\in \partial A{\setminus } A'$ . In this case, the argument is similar to the above, as $A{\setminus }\{x\}$ is contractible, and so has cohomology only in degree $(1)$ .
(ii) $\Rightarrow $ (i): Vice versa, suppose that
is an object of
satisfying the purity condition. Let $A \subset X$ be a closed disk not containing any special points. Let $A' \subset \partial A$ be a disjoint union of two closed arcs so that $(A,A')$ is a Milnor disk. Since by our assumptions,
has locally constant, hence constant cohomology, it is straightforward to conclude that
By purity, this implies that
is quasiisomorphic to a single local system with values in
. This shows Condition (1) of Definition 1.1.3.
Now, let A be an closed disk that contains exactly one special point x in its interior. Let $A'\subset \partial A$ be the disjoint union of two arcs. We consider again the diagram (1.2.5), arguing now ‘in the other direction’.
That is, look at the middle column. By purity, has cohomology only in degree $0$ . But since is a single local system in degree $0$ , the complex $R\Gamma (A';F)$ has cohomology only in degree $(1)$ . Therefore, has cohomology only in degrees $\{1,0\}$ , thus establishing Condition (2) of Definition 1.1.3.
Next, look at the left column. Clearly, , as $x\notin A'$ , and so is identified with . Now, the latter can be analyzed via the top row of the diagram, which contains , with cohomology in degree $0$ and which, we claim, has cohomology only in degree $0$ . This follows from looking at the right column, where the statement reduces to the claim that $H^0(A{\setminus }\{x\};L) \to H^0(A'; L)$ is injective. Therefore, has cohomology only in degrees $\{0,1\}$ , thus establishing Condition (3) of Definition 1.1.3.
Remark 1.2.7. Assume that we are in the situation of Example 1.2.3. Then is identified with , the stalk at x of the complex of vanishing cycles for with respect to f; see [Reference Kashiwara and Schapira33]. It is well known (loc. cit.) that is itself a perverse sheaf which, in our case, amounts to saying that is quasiisomorphic to a single vector space in degree $0$ . This provides an alternative proof of purity for such Milnor pairs, at least in the classical case when is the category of vector spaces over a field.
2 The paracyclic category and constructible sheaves
In this section, we will introduce the paracyclic category $\Lambda (X,N)$ of a stratified surface and explain how the formalism of Kan extensions, applied to a directed version of $\Lambda (X,N)$ , can be used to describe the Verdier duality of the derived constructible category. The ideas and constructions introduced in this section serve as a preparation for the main part of this work, §3, where we will apply similar techniques to parametrize perverse sheaves in terms of the subcategory $M(X,N) \subset \Lambda (X,N)$ of Milnor disks.
2.1 The standard paracyclic category and the Ran space of the circle
Recall that the standard simplex category $\Delta $ has, as objects, the standard finite nonempty ordinals $[n] = \{0,1,\cdots , n\}$ , $n \geq 0$ , with morphisms being monotone maps. The morphisms of $\Delta $ are generated by the coface and codegeneracy maps
subject to wellknown relations; see, for example, [Reference Connes11], Chapter III, Appendix A, Proposition 2. We denote by $\Delta ^{\operatorname {surj}}\subset \Delta $ the subcategory with the same objects and only surjective maps as morphisms. In other words, morphisms of $\Delta ^{\operatorname {surj}}$ are generated by the $\sigma _j$ only. As usual, we call a simplicial object in a category a contravariant functor . Thus, Z consists of objects and morphisms (face and degenaracy maps)
satisfying the relations dual to those among the $\delta _i$ and $\sigma _j$ . We will also use the term halfsimplicial object for a contravariant functor . Thus, a halfsimplicial object has only degeneracy maps but no face maps.
Definition 2.1.1 ([Reference Connes11] Chapter III Appendix A, [Reference Loday36] Definition 6.1.1).
(a) The standard paracyclic category $\Lambda _{\infty }$ has the objects ${\langle n \rangle }$ , $n\geq 0$ which are in bijection with those of $\Delta $ . Its morphisms are generated by those of $\Delta $ (i.e., the $\delta _i: \langle n1\rangle \to {\langle n \rangle }$ and $ \sigma _j: \langle n+1 \rangle \to {\langle n \rangle }$ as above satisfying the same relations) together with additional automorphisms $\tau _n:{\langle n \rangle }\to {\langle n \rangle }$ which are subject to the following relations:
(b) The cyclic category $\Lambda $ is obtained from $\Lambda _{\infty }$ by imposing the additional relations $\tau _n^{n+1}=\operatorname {Id}$ .
The following proposition is well known; see [Reference Drinfeld13]. It can be expressed by saying that $\Lambda _{\infty }$ is a central extension of $\Lambda $ by $\mathbb {Z}$ .
Proposition 2.1.2. (a) The automorphisms $\tau _n^{n+1}\in \operatorname {Hom}_{\Lambda _{\infty }}({\langle n \rangle }, {\langle n \rangle })$ form a central system (i.e., define a natural transformation from the identity functor to itself).
(b) Let $p: \Lambda _{\infty }\to \Lambda $ be natural functor (identical on objects, surjective on morphisms). The fibers of each induced map
are principal homogeneous spaces with respect to the action of $\mathbb {Z}$ given by composition with powers of $\tau _m^{m+1}$ or, what by (a) is the same, by composition with powers $\tau _n^{n+1}$ .
We also denote $\Lambda _{\infty }^{\operatorname {surj}}\subset \Lambda _{\infty }$ the subcategory on the same objects with the morphisms generated by the $\sigma _j$ and $\tau _n$ only. By a paracyclic object in a category $\mathcal {A}$ , we will mean a contravariant functor . As for simplicial objects, we write $Z_n$ for the value of Z on ${\langle n \rangle }$ and $\partial _i, s_j, t_n$ for the values on $\delta _i, \sigma _j, \tau _n$ . By a halfparacyclic object we will mean a contravariant functor .
Remark 2.1.3. The categories $\Lambda $ and $\Lambda _{\infty }$ are selfdual, that is, isomorphic to their opposite categories [Reference Connes11] [Reference Elmendorf21]. In fact, by introducing the additional codegeneracies $\sigma _{n+1}= \tau _n \sigma _n\tau _{n+1}^{1}: \langle n+1\rangle \to {\langle n \rangle }$ , one can write their presentations in a manifestly selfdual way, so that cofaces and codegeneracies will be dual to each other.
A partial interpretation via the Ran space
We recall the topological version of the Ran space construction [Reference Beilinson and Drinfeld4]. As pointed out in [Reference Beilinson and Drinfeld4], this version goes back to Borsuk and Ulam [Reference Borsuk and Ulam7].
Let M be a $C^{\infty }$ manifold. The Ran space of M is the set $\operatorname {Ran}(M)$ of all finite nonempty subsets $I\subset M$ equipped with a natural (Vietoris) topology. If we choose a metric on M inducing the topology, then $\operatorname {Ran}(M)$ can be metrized using the corresponding Hausdorff distance. The space $\operatorname {Ran}(M)$ has a filtration by closed subspaces $\operatorname {Ran}^{\leq d}(M) = \{I\subset M: \, I\leq d\}$ , and the complement
is the configuration space of unordered dtuples of distinct points in M. In this way, each $\operatorname {Ran}^{\leq d}(M)$ becomes a Whitney stratified space, and $\operatorname {Ran}(M)$ can be considered as a (infinitedimensional) space with a conical stratification; see §1.1. In particular, we can speak about the category of exit paths $\operatorname {Exit}(\operatorname {Ran}(M))$ and, for a Grothendieck abelian category , about valued constructible sheaves on $\operatorname {Ran}(M)$ (with respect to the stratification by the $\operatorname {Sym}^d_{\neq }(M)$ ).
Remarks 2.1.4.

(a) An exit path in $\operatorname {Ran}(M)$ can be seen as a history of a colony of bacteria living in M which can move and multiply (by splitting) but not merge together, and cannot die; see Figure 3.

(b) A constructible sheaf on $\operatorname {Ran}(M)$ assigns to any finite nonempty $I\subset M$ an object (the stalk). When I ‘evolves’ into J by moving and splitting, we have a morphism (the generalization map).
Let us focus, in particular, on the Ran spaces of the real line ${\mathbb {R}}$ and the circle $S^1$ .
Example 2.1.5. It goes back to Bott [Reference Bott8] that $\operatorname {Ran}^{\leq 3}(S^1)$ is homeomorphic to the $3$ sphere $S^3$ . Further, inside this sphere $\operatorname {Ran}^{\leq 1}(S^1)=S^1$ is embedded as a trefoil knot, and $\operatorname {Ran}^{\leq 2}(S^1)$ is a Moebius band bounding this knot. See [Reference Mostovoy41] for a beautiful treatment using elliptic functions. The topology and homotopy type of $\operatorname {Ran}^{\leq d}(S^1)$ for higher d was studied in [Reference Tuffley46].
The following result was proven in [Reference Cepek9]:
Proposition 2.1.6.

(a) The category $\operatorname {Exit}(\operatorname {Ran}({\mathbb {R}}))$ is equivalent to $(\Delta ^{\operatorname {surj}})^{\operatorname {op}}$ . In particular, valued constructible sheaves on $\operatorname {Ran}({\mathbb {R}})$ can be identified with halfsimplicial objects in .

(b) The category $\operatorname {Exit}(\operatorname {Ran}(S^1))$ is equivalent to $(\Lambda _{\infty }^{\operatorname {surj}})^{\operatorname {op}}$ . In particular, valued constructible sheaves on $\operatorname {Ran}(S^1)$ can be identified with halfparacyclic objects in .
Proof. (a) An exit path $\gamma $ in any $\operatorname {Ran}(M)$ going from I to J gives, for any $x\in I$ , a tree of descendents of x which terminates in a subset of J. This gives a surjection $a_{\gamma }: J\to I$ (the ‘ancestry map’). Isotopic exit paths lead to the same surjection. If $M={\mathbb {R}}$ , then the order of ${\mathbb {R}}$ makes both I and J into nonempty finite ordinals and the surjection $a_{\gamma }$ is monotone.
(b) Recall from [Reference Connes11] Chapter III, Appendix A the geometric definition of the cyclic category $\Lambda $ . For this, we identify ${\langle n \rangle }$ with the set of $(n+1)$ st roots of $1$ in the standard circle $S^1$ . Then $\operatorname {Hom}_{\Lambda }({\langle m \rangle }, {\langle n \rangle })$ is the set of connected components of the space of degree $1$ monotone maps $f: (S^1,{\langle m \rangle })\to (S^1, {\langle n \rangle })$ . Each such connected component has the homotopy type of $S^1$ , and $\operatorname {Hom}_{\Lambda _{\infty }}({\langle m \rangle }, {\langle n \rangle })$ is obtained by passing to the universal coverings of these components. That is, $\operatorname {Hom}_{\Lambda _{\infty }}({\langle m \rangle }, {\langle n \rangle })$ is the set of isotopy classes of data $(f,s)$ consisting of f as above together with a homotopy s between f and the identity (as maps $S^1\to S^1$ ). Note now that for $M=S^1$ , an exit path $\gamma $ as in (a) gives not only a surjection $a_{\gamma }$ but a welldefined isotopy class of pairs $(f,s)$ , where $f:(S^1,J)\to (S^1, I)$ is a monotone degree $1$ map and s is homotopy of f to the identity.
Remark 2.1.7. One would like to extend the approach with the Ran spaces so as to realize the full categories $\Delta , \Lambda _{\infty }$ or functors out of them in terms of some categories of exit paths or constructible sheaves. For this, in the language of Remark 2.1.4(a), we would need to modify the concept of an exit path as a history of a colony of bacteria so as to allow the bacteria to die; see Figure 4. Then for such a ‘history with deaths’ evolving from I to J we will still have the ancestry map $J\to I$ but it need not be surjective, as some lines may die out.
To account for such ‘exit paths with deaths’, one needs to consider constructible sheaves on $\operatorname {Ran}(M)$ equipped with an additional monotone structure which is a system of maps given for any nested pair $I\subset J\subset S^1$ of nonempty finite sets and transitive in nested triples.
We do not pursue this approach further but note that our point of view based on Milnor disks $(A,A')$ has $A'$ , a finite union of intervals in the circle $\partial A{{\,\, \simeq \,\, }} S^1$ , playing the role of a finite subset $I\in \operatorname {Ran}(\partial A)$ .
A systematic approach to the matter discussed in Remark 2.1.7 via ‘unital’ Ran spaces was developed in [Reference Cepek9, Reference Cepek10]. The author recovers the paracyclic category and Joyal’s categories $\Theta _n$ as unital exit path categories associated to the Ran spaces of $S^1$ and ${\mathbb {R}}^n$ , respectively.
2.2 The paracyclic category of a stratified surface
Let $(X,N)$ be a stratified surface as defined above. Throughout this text, we will assume that, if $X \cong S^2$ , then $N \ge 2$ . In this section, we introduce the paracyclic category $\Lambda (X,N)$ of $(X,N)$ which can be seen as a certain amalgamation of the copies of $\Lambda ^{\infty }$ associated with the circles of directions at all the points $x\in X$
Pant cobordisms and the paracyclic category
We will use the notation $I=[0,1]$ for the closed unit interval and, as before, ${\mathbb {D}}$ for the closed unit disk.
Definition 2.2.1. By a paradisk in $(X,N)$ , we mean a pair $(A,A')$ , where $ A \subset X$ is a closed disk such that $A \cap N \le 1$ and $A' \subset \partial A \cong S^1$ is a compact onedimensional submanifold, that is one of the following:

(i) the empty set,

(ii) a finite nonempty union of closed intervals,

(iii) the full boundary circle.
Thus, a Milnor disk is a particular case of a paradisk corresponding to the possibility (ii) of Definition 2.2.1. In the other two cases, a paradisk $(A,A')$ will be called:

(a) a standard disk, if $A' = \emptyset $ ,

(b) a bounded disk, if $A' = \partial A$ .
We now define morphisms between paradisks. Intuitively, such a morphism should be a certain isotopy class of paths $(A_t, A^{\prime }_t)_{t\in I}$ in the space of paradisks. We want such paths to satisfy the following dynamical requirements as t increases from $0$ to $1$ :

(PD1) The components $A^{\prime }_t$ can merge together and can appear ex nihilo (growing out of single points) but cannot split.

(PD2) A special point $x\in N$ can enter the interior of $A_t$ (i.e., $A_t$ can ‘run it over’) only through the complement $A_t{\setminus } A^{\prime }_t$ and exit $A_t$ only through $A^{\prime }_t$ .
To implement this formally, we represent paths in the space of paradisks via maps $I\times {\mathbb {D}}\to X$ . We start with formalizing the merging behavior of the components $A_t$ as in (PD1).
Definition 2.2.2.

(1) Let $P\subset I\times S^1$ be a subset. For any $t\in I$ , we denote by $P_t = P \cap (\{t\} \times S^1)$ the slice of P over t. We can view $P_t$ as a subset in $S^1$ .

(2) By a pant cobordism, we will mean a closed twodimensional (topological) submanifold $P \subset I \times S^1$ with boundary such that:

(2a) The slices $P_0, P_1\subset S^1$ are compact onedimensional submanifolds with boundary, as in Definition 2.2.1.

(2b) The inclusion $P_1 \subset P$ is a homotopy equivalence.

An example of a pant cobordism is depicted in Figure 5.
Remarks 2.2.3. (a) Strictly speaking, a pant cobordism P is a manifold with corners, not just boundary, the corners being the boundary points of $P_0$ and $P_1$ , as one can see in Figure 5. Since we consider P as a topological manifold, we ignore this subtlety.
(b) Intuitively, the slices $P_t\subset S^1$ correspond to the onedimensional submanifolds $A^{\prime }_t\subset A_t$ in the picture with paths in the space of paradisks. Of course, for some values of t such slices may not be of the form allowed in Definition 2.2.1, in particular, they may have, as components, single points (which can then disappear or grow to become intervals) Nevertheless, the condition (2b) of Definition 2.2.2 corresponds to the requirement (PD1) on the paths. In this way, a pant cobordism can (after time reversal $t\mapsto 1t$ ) be seen as a thickened version of an ‘exit path with deaths’ from Remark 2.1.7.
Definition 2.2.4. The paracyclic category $\Lambda (X,N)$ of $(X,N)$ is the category with objects being paradisks $(A,A')$ for $(X,N)$ . A morphism
in $\Lambda (X,N)$ consists of

• a pant cobordism $P\subset I\times S^1$ .

• a continuous map $H: I \times {\mathbb {D}} \to X$ , which we also consider as a family of maps $H_t: {\mathbb {D}} \to X$ , $t\in I$ such that

(1) H is an isotopy, that is, each $H_t$ is an embedding,

(2) for $i \in \{0,1\}$ , the embedding $H_i$ induces homeomorphisms ${\mathbb {D}} \cong A_i$ and $P_i \cong A^{\prime }_i$ ,

(3) for every $t \in I$ , we have $H_t({\mathbb {D}}) \cap N \le 1$ ,

(4) for every $t_0 \in I$ and $x \in H_{t_0}(P_{t_0}) \cap N$ , there exists $\varepsilon>0$ such, for every $t_0 \le t \le t + \varepsilon $ , $x \notin H_{t}({\mathbb {D}} \setminus P_{t_0})$ .


• two such data $(H,P)$ , $(H',P')$ define the same morphism if there exists a homeomorphism $\varphi : I \times {\mathbb {D}} \to I \times {\mathbb {D}}$ such that $\varphi P$ induces a homeomorphism with $P'$ , together with a homotopy $\alpha : I^2 \times {\mathbb {D}} \to X$ with $\alpha _0 = H$ and $\alpha _1 = H'$ such that, for every $s \in I$ , $\alpha _s$ satisfies the above conditions.
We denote by $S(X,N) \subset \Lambda (X,N)$ the full subcategory of standard disks, by $B(X,N) \subset \Lambda (X,N)$ the full subcategory of bounded disks and by $M(X,N) \subset \Lambda (X,N)$ the full subcategory of Milnor disks. We refer to $M(X,N)$ as the Milnor category of $(X,N)$ .
Remarks 2.2.5. (a) Given a morphism f with a representative $(P,H)$ , we have, for any $t\in I$ , a closed disk $A_t=H_t({\mathbb {D}})\subset X$ and a closed subset $A^{\prime }_t = H_t(P_t) \subset \partial A_t$ . The pair $(A_t, A^{\prime }_t)$ depends only on f. For generic values of t, the slice $P_t$ belongs to one of the three types described in Definition 2.2.1 and so $(A_t, A^{\prime }_t)$ is a paradisk by the condition (2) The condition (4) corresponds to the intuitive requirement (PD2) on paths in the space of paradisks while (PD1) corresponds, as mentioned above, to the condition (2b) of Definition 2.2.2 of a pant cobordism.
(b) Our assumption that if $X \cong S^2$ , then $N \ge 2$ implies that the mapping spaces which appear implicitly in our definition of $\Lambda (X,N)$ have contractible components so that it is justified to consider it as an ordinary category (rather than an ${\infty }$ category).
Example 2.2.6. The category $M({\mathbb {C}},\emptyset )$ of Milnor disks in $({\mathbb {C}},0)$ is equivalent to the paracyclic category $\Lambda _{\infty }$ . This is shown similarly to the proof of Proposition 2.1.6. Further, the category $\Lambda ({\mathbb {C}},\emptyset )$ is equivalent to the category obtained from $\Lambda _{\infty }$ by adjoining an initial and a final objects which correspond to the objects
respectively.
The Milnor category and perverse sheaves
The role of the category $M(X,N)$ for our purposes is explained by the following.
Proposition 2.2.7. Let be a perverse sheaf on $(X,N)$ with values in a Grothendieck abelian category . Then the correspondence extends to a functor .
Proof. Let $f: (A_0, A^{\prime }_0)\to (A_1, A^{\prime }_1)$ be a morphism between two Milnor disks represented by a pair $(P,H)$ as in Definition 2.2.4. Let $\widetilde N= H^{1}(N)\subset I\times {\mathbb {D}}$ . Because of condition (1) of that definition, $\widetilde N$ is a onedimensional topological submanifold with boundary, that is, a disjoint union of closed curvilinear intervals in the cylinder $I\times D$ , each of them projecting to I in an injective way. We orient these curves following the increase of $t\in I$ .
Let $\widetilde N^+\subset \widetilde N$ be the union of components that terminate (in the sense of the above orientation) on P. Let $\widetilde N^\subset \widetilde N$ be the union of components that terminate on $\{1\}\times {\mathbb {D}}$ . Thus, $\widetilde N^+ \cup \widetilde N^ =\widetilde N$ and $\widetilde N^+\cap \widetilde N^$ is the union of components that terminate on the slice $P_1$ .
Further, let
. It is a complex of sheaves on $I\times D$ constructible with respect to the stratification given by $\widetilde N$ . By Proposition 1.2.4,
Consider the diagram of restrictions
We claim that $\rho _1$ is a quasiisomorphism (and therefore, by purity, it reduces to an isomorphism of objects of
). Indeed, denote
Because of the condition (2b) of Definition 2.2.2 and the entry–exit condition (4) of Definition 2.2.4, the inclusion of the slice $P_1\subset P^+$ is a homotopy equivalence, and the inclusion $\{1\}\times {\mathbb {D}}{\hookrightarrow } {\mathbb {D}}^$ is a homotopy equivalence as well. This means that each of the two restriction morphisms
whose composition is $\rho _1$ , is a quasiisomorphism.
We now define the value of the functor on f, that is, the morphism to be given by $\rho _2\rho _1^{1}$ . The necessary verifications are left to the reader.
Remark 2.2.8. In a similar way, utilizing the $\infty $ category of spans, one can show that the association extends to an ${\infty }$ functor from $\Lambda (X,N)$ to , the ${\infty }$ categorical enhancement of the derived category of ; see §A.3.
Example 2.2.9. The categories $S(X,N)$ of standard disks and $B(X,N)$ of bounded disks are equivalent to ${\operatorname {Entr}}(X,N)$ and $\operatorname {Exit}(X,N)$ , the categories of entrance and exit paths of the stratified space $(X,N)$ respectively. The first equivalence has the form
where $A_x \subset X$ is a disk containing x such that $A_x \cap N = \emptyset $ if $x \notin N$ . The second equivalence is defined in the dual way.
The paracyclic duality
Next, we describe an identification of $\Lambda (X,N)$ with its opposite category $\Lambda (X,N)^{\operatorname {op}}$ which will play an important role in interpreting the Verdier duality for perverse sheaves. We start with the following remarks. For a closed subset Z of a topological space Y, we denote by $\mathring {Z}$ the interior of Z. The next two propositions are then clear.
Proposition 2.2.10.

(a) For a paradisk $(A,A')\subset X$ the pair $(A,A')^*:= (A, \partial A{\setminus }( \mathring {A'}))$ is again a paradisk.

(b) Let $\sigma : I\times S^1\times I\times S^1$ be the involution $(t,\theta ) \mapsto (1t, \theta )$ . For a pant cobordism $P\subset I\times S^1$ , the subset $P^* = \sigma (I\times S^1) {\setminus }\mathring { P}$ is again a pant cobordism.
Proposition 2.2.11. Let $i: \Lambda (X,N)' \subset \Lambda (X,N)$ denote the full subcategory consisting of those Milnor disks $(A,A')$ such that $\partial A \cap N = \emptyset $ . Then the inclusion i is an equivalence of categories.
Proposition 2.2.12. We have a perfect duality (which we call the paracyclic duality)
defined on objects by the association $(A,A')\mapsto (A,A')^*$ .
Proof. Using Proposition 2.2.11, it suffices to define a duality on the equivalent subcategories $\xi ': \Lambda (X,N)' \overset {\simeq }{\longrightarrow } \Lambda (X,N)^{\prime \operatorname {op}}$ , which is given on objects by the desired formula $(A,A')\mapsto (A,A')^*$ .
To do this, suppose we have a morphism f represented by $(H,P)$ ; note that we may assume, replacing $(H,P)$ by an equivalent representative if needed, that special points enter in $I \times S^1 \setminus P$ and exit in $\mathring {P}$ . Then we define $\xi (f)$ to be represented by $(H(1t,), P^*)$ . It is straightforward to verify that this association yields a welldefined functor squaring to the identity, that is, giving a perfect duality.
Note, that the paracyclic duality $\xi $ interchanges the subcategories $S(X,N)$ and $B(X,N)$ , identifying them as opposite to one another, and restricts to a selfduality of $M(X,N)$ .
2.3 The directed paracyclic category and its localization
Let $(X,N)$ be as before. In this section, we exhibit $\Lambda (X,N)$ as a localization of another category ${\overrightarrow {\Lambda }\!}(X,N)$ which we call the directed paracyclic category. This latter category turns out to be more suitable for the use of Kan extensions.
Definition 2.3.1. We define the directed paracyclic category ${\overrightarrow {\Lambda }\!}(X,N)$ exactly as in Definition 2.3.1 but replacing condition (4) by the following:

(Ent) For every $x \in N$ , we have

(Ent1) if $x \in A_{t_0} = H_{t_0}({\mathbb {D}})$ for $t_0 \in I$ , then, for all $t \ge t_0$ , we have $x \in A_t$ ,

(Ent2) if $x \in A^{\prime }_{t_0}= H_{t_0}(P_{t_0})$ for $t_0 \in I$ , then, for all $t \ge t_0$ , we have $x \in A^{\prime }_t$ .

A morphism $f: (A,A') \to (B,B')$ in ${\overrightarrow {\Lambda }\!}(X,N)$ is called a weak equivalence if either

(i) f is an isomorphism, or

(ii) f can be represented by a pair $(P,H)$ such that $H_0^{1}(A') \subset P$ is a homotopy equivalence and $H^{1}(N) \subset P$ .
We denote $W \subset \operatorname {Mor}({\overrightarrow {\Lambda }\!}(X,N))$ the set of weak equivalences.
Remarks 2.3.2.

(a) The condition (Ent) is a twostep version of the entrance path condition: If a special point x enters $A_{t_0}$ , then it stays in all the $A_t$ for all $t\geq t_0$ , and similarly for $A^{\prime }_{t_0}$ .

(b) The condition (ii) in the definition of a weak equivalence means that a special point x is allowed to enter $A^{\prime }_{t_0}\subset A_{t_0}$ from the outside of $A_{t_0}$ and stay there for all $t\geq t_0$ .
We also denote by $\overrightarrow {S}(X,N), \overrightarrow {B}(X,N), \overrightarrow {M}(X,N) \subset {\overrightarrow {\Lambda }\!}(X,N)$ the full subcategories of standard disks, bounded disks and Milnor disks, respectively.
Proposition 2.3.3. The natural morphism
exhibits $\Lambda (X,N)$ as a localization of ${\overrightarrow {\Lambda }\!}(X,N)$ along W.
Here, by ‘localization’ we mean ${\overrightarrow {\Lambda }\!}(X,N)[W^{1}]$ , the Gabriel–Zisman localization in the sense of ordinary categories [Reference Gabriel and Zisman23]. In fact, one can prove stronger statements, identifying $\Lambda (X,N)$ with the ${\infty }$ categorical localization or with the Dwyer–Kan simplicial localization [Reference Dwyer and Kan16] of ${\overrightarrow {\Lambda }\!}(X,N)$ with respect to W. This can be done by adapting our proof below by using a hammocktype model for the Dwyer–Kan localization. We will not need this generalization for our purposes except for a very particular case in Lemma 2.5.2 below, which is easily proved directly.
Proof. Recall that in $\Lambda (X,N)$ a special point x is allowed to exit $A_{t_0}$ through $A^{\prime }_{t_0}$ . This process is inverse to entering $A_{t_0}$ through $A^{\prime }_{t_0}$ from the outside which is, according to Remark 2.3.2(b), a general form of a weak equivalence (apart from an isomorphism). Indeed, the composite process (entering $A_{t_0}$ through $A^{\prime }_{t_0}$ from the outside and then bouncing back to the original position) is connected to the identity by a homotopy $\alpha $ as in Definition 2.2.4.
Therefore, the functor $\pi $ inverts weak equivalences and we obtain an induced functor $\overline {\pi }: {\overrightarrow {\Lambda }\!}(X,N)[W^{1}] \to \Lambda (X,N)$ . We claim that $\overline \pi $ is an equivalence. To this end, we study a typical Homset
By definition (cf. [Reference Gabriel and Zisman23] §I.1), an element of this set is an equivalence class of zigzags
of arbitrary length, with $w_i\in W$ . The equivalence relation on the set of such zigzags is generated by two elementary moves:

(M1) For any factorization

(M2) For any factorization
These two moves imply the hammock move, which is at the basis of Dwyer–Kan localization theory [Reference Dwyer and Kan16] (except that we don’t assume that the vertical morphisms are weak equvialences):

(H) Any two zigzags connected by a hammock, that is, by a commutative diagram
We now compare this with $\operatorname {Hom}_{\Lambda (X,N)}((A,A'), (C,C'))$ . An element f of this latter set is an equivalence class of pairs $(P,H)$ as in Definition 2.2.4. As usual, we write $A_t= H_t({\mathbb {D}})$ , $A^{\prime }_t=H_t(P_t)$ . Without loss of generality, we can assume that:

• P is smooth as a manifold with corners, that is, the part of $\partial P$ lying over the open interval $(0,1)\subset I$ is smooth.

• The projection of this part of $\partial P$ to $(0,1)$ is a Morse function. This implies that for all but finitely many values of t (which we call critical values) the slice $P_t$ has one of the three forms listed in Definition 2.2.1 and therefore $(A_t, A^{\prime }_t)$ is a paradisk.

• The moments $t_1 <\cdots < t_n$ , $t_i\in I$ , of exit of special points $x\in N$ out of $A_t$ (happening through $A^{\prime }_t$ ) are noncritical.
Let $t^{\prime }_i>t_i$ , $i=1,\cdots , n$ , be sufficently close. As explained in the beginning of the proof, the restriction of $(P,H)$ to the preimage of each interval $[t_i, t^{\prime }_i]$ can be seen as an inverse of a weak equivalence in ${\overrightarrow {\Lambda }\!}(X,N)$ . while the restriction to each interval in the complement of the union of the $[t_i, t^{\prime }_i]$ , is a morphism in ${\overrightarrow {\Lambda }\!}(X,N)$ . Therefore, we can associate to $(H,P)$ a zigzag (2.3.5).
We claim that different choices of $(H,P)$ representing the same morphism f, give rise to equivalent zigzags. Any two such different choices are, by Definition 2.2.4, related by a reparemetrization $\varphi : I\times {\mathbb {D}} \to I \times {\mathbb {D}}$ and a homotopy $\alpha : I^2 \times {\mathbb {D}} \to X$ . By choosing $\alpha $ generic enough, we see that any two choices are connected by a sequence of the following moves and their inverses:

(M’1) replacing a representative $(P,H)$ with a representative $(P, \widetilde H)$ which, locally around $t \in I$ , avoids the special point contained in $A_t'$ :
(2.3.6)Denote by $\buildrel w_i\over \longleftarrow $ the slice of $(P,H)$ from the moment of exit of x until shortly afterwards and by $\buildrel f_i\over \longrightarrow $ the slice from shortly before exit to the moment of exit; see (2.3.6). We see that we have three morphisms $g, w_i, f_i \ {\overrightarrow {\Lambda }\!}(X,N)$ and a factorization $w_i g=f_i$ in ${\overrightarrow {\Lambda }\!}(X,N)$ represented by an appropriate homotopy $\alpha $ . Therefore, the move (M $'$ 1) yields two zigzags connected by the move (M1). 
(M’2) replacing a representative $(P,H)$ with a representative $(\widetilde P, H)$ , where $\widetilde P$ is obtained by deforming P in a suitable way locally around one of the exit moments $t_i$ so that two intervals in $A^{\prime }_{t_i}$ are replaced by one:
(2.3.7).
Making four slices of each the two cobordisms as in (2.3.7), we get two zigzags connected by a hammock:
so they are equivalent by the hammock move. Therefore, the entire zigzags corresponding to $(P,H)$ and $(\widetilde P, H)$ are equivalent as well.
In this way, we define a functor ${\overrightarrow {\Lambda }\!}(X,N)[W^{1}]\to \Lambda (X,N)$ which is easily seen to be quasiinverse to $\overline \pi $ .
Corollary 2.3.8. The functor $\pi $ from Proposition 2.3.3 induces an equivalence $\overrightarrow {S} (X,N) \simeq S(X,N)$ and localizations $\overrightarrow {M}(X,N) \to M(X,N)$ , $\overrightarrow {B}(X,N) \to B(X,N)$ .
2.4 Constructible sheaves with values in ${\infty }$ categories
Let $(X,N)$ be a stratified surface, let $\mathfrak {O}(X)$ denote the poset of open subsets of X and let be an $\infty $ category. The following is an ${\infty }$ categorical analog of the discussion for abelian categories in §1.1.
Lemma 2.4.1. Given a functor , an open subset $U \subset X$ , and an open cover of U, the following conditions are equivalent:

(i) Denote by the poset of open subsets $V \subset X$ such that $V \subset U_i$ for some $i \in I$ . Then the canonical map

(ii) Denote by the poset of nonempty finite subsets of I, and consider the inclusion . Then the canonical map
Proof. The inclusion is $\infty $ cofinal.
A
valued sheaf on X is a functor
such that, for every open $U \subset X$ and every open cover
of U, the equivalent conditions of Lemma 2.4.1 hold. We denote by
the full subcategory spanned by the
valued sheaves on X.
Let $({\operatorname {Disk}^o}(X,N, \leq )$ be the poset of standard pairs $(U,\emptyset )$ ordered by inclusion. We will consider it as a category. A morphism in $({\operatorname {Disk}^o}(X,N, \leq )$ (i.e., an inclusion $U_1\subset U_2$ of standard disks) will be called a weak equivalence, if $N\cap U_1=N\cap U_2$ . We denote by W the set of weak equivalences. The map
identifies ${\operatorname {Disk}^o}(X,N)$ with a full subposet of $\mathfrak {O}(X)$ . A sheaf in is called constructible if its restriction maps weak equivalences to equivalences in . We denote the full subcategory of spanned by the constructible sheaves by .
Remark 2.4.2. Let
be an abelian category with enough injectives, and let
denote the corresponding (leftbounded) derived $\infty $ category as defined in [Reference Lurie38, 1.3.2.8]. We equip the stable $\infty $ category
with the tstructure
, where the tstructure on
is the one from [Reference Lurie38, 1.3.2.19]. The heart of this tstructure is equivalent to
. Then, using the recognition principle for derived $\infty $ categories ([Reference Lurie38, 1.3.3.7]), we obtain an equivalence of $\infty $ categories
In particular, the $\infty $ category
is really an enhancement of the ordinary derived category of complexes of
valued sheaves. Further, this equivalence identifies our constructible category
with the more traditional derived constructible category, defined as the full subcategory of
spanned by objects with constructible cohomology sheaves.
We denote by ${\operatorname {Disk}^o}(X,N)[W^{1}]_{\infty }$ the $\infty $ categorical localization of ${\operatorname {Disk}^o}(X,N)$ along the weak equivalences W. In particular, we may identify
with the full subcategory spanned by those functors that map weak equivalences in ${\operatorname {Disk}^o}(X,N)$ to equivalences in
.
Proposition 2.4.3. The functor
is an equivalence of $\infty $ categories.
Proof. Let be a presheaf on X such that sends weak equivalences to equivalences in . We claim that the following conditions are equivalent:

(1) is a sheaf.

(2) is a right Kan extension of .
The claim immediately implies the statement of the proposition. The reason why this statement is not completely formal is that in condition (2), we do not assume that the restriction of to ${\operatorname {Disk}^o}(X,N)^{\operatorname {op}}$ satisfies a descent condition. We rather need to convince ourselves that this is automatic due to the assumption that is constructible.
(1) $\Rightarrow $ (2): Suppose that is a sheaf. We need to show that, for every open $U \subset X$ , is the limit of the diagram . We interpret the set as an open cover of U so that this statement follows immediately from the hypothesis that is a sheaf.
(2) $\Rightarrow $ (1): Suppose that
is a right Kan extension of
. Let $U \subset X$ be an open subset, and let
be an open cover of U. Let
(resp.
) denote the subposet of
consisting of those opens V (resp. $V \in {\operatorname {Disk}^o}(X,N)$ ) such that $V \subset U_i$ for some
. We need to show that the map
is an equivalence. Since
is a right Kan extension of
, it suffices to show that the composite
is an equivalence. Via the pointwise formula for
, we deduce that it suffices to show that
is a right Kan extension along $i^{\operatorname {op}}$ , where
To this end, let $D \in {\operatorname {Disk}^o}(X,N)$ with $D \subset U$ . We need to show that
is a limit of
. Denote
, and introduce the category
with

• the set of objects of is the set of objects of ,

• a morphism between objects V and $V'$ of is a homotopy class of paths $\gamma $ in $\operatorname {Emb}(V,D)$ such that $\gamma (0)$ is the embedding $V \subset D$ , $\gamma (1)$ is a homeomorphism $V \cong V'$ and, if $\gamma (t)(V)$ contains the special point for some t, then $\gamma (t')(V)$ contains the special point for all $t' \ge t$ .
Denote by
the natural functor. We will show that $\pi $ is an $\infty $ cofinal localization at the set of weak equivalences in
.
Step 1. $\pi $ is $\infty $ cofinal. To show this claim, we need to show that, for every
, the category $V/\pi $ is weakly contractible. To this end, we consider the space $E = P'\operatorname {Emb}(V,D)$ of paths $\gamma $ in $\operatorname {Emb}(V,D)$ that satisfy: If $\alpha (t)(V)$ contains the special point, then $\alpha (t')(V)$ contains the special point for all $t' \ge t$ . We then deduce that $V/\pi $ is weakly contractible, by applying Lemma A.1 to the functor
where $U([\gamma ])$ is the open subset of E consisting of paths that end in an embedding $V {\hookrightarrow } V'$ and whose associated homotopy class, obtained by composing with any path of embeddings from $V {\hookrightarrow } V'$ to $V \cong V'$ , agrees with $\gamma $ .
Step 2. For , denote by $j: (V/\pi )^{\cong } \subset V/\pi $ the inclusion of the full subcategory spanned by the isomorphisms in . By a similar argument as in Step 1, using Lemma A.1, it follows that j is $\infty $ coinitial. It is then that, for every $\infty $ category with limits, the unit $\operatorname {id} \to \pi _*\pi ^*$ is an equivalence, and the counit $\pi ^*\pi _* \to \operatorname {id}$ is an equivalence on those functors that map weak equivalences to equivalences. This implies that $\pi ^*$ is fully faithful with essential image consisting precisely of these latter functors .
Now, equipped with this statement, we show that is a limit of . Namely, by assumption, maps weak equivalences to equivalences so that it is equivalent to for some functor . Since $\pi $ is $\infty $ cofinal, we may compute the limit of as the limit of . But now the category has an initial object given by a disk $D' \subset D$ so that, if D contains a special point, then $D'$ also contains the special point. In any case, we have that $D' \subset D$ is a weak equivalence. Therefore, we obtain the desired equivalence .
In our treatment of Milnor sheaves, it will be important to have a good control on the boundary of disks which is why we now switch from open disks to closed disks. Let ${{\operatorname {Disk}^{oc}}}(X,N)$ denote the poset of all open and closed disks in X containing at most one special point. We denote by ${\operatorname {Disk}^o}(X,N) \subset {{\operatorname {Disk}^{oc}}}(X,N)$ and ${{\operatorname {Disk}^c}}(X,N) \subset {{\operatorname {Disk}^{oc}}}(X,N)$ the subsets of open and closed disks, respectively. The poset ${{\operatorname {Disk}^{oc}}}(X,N)$ comes equipped with a set of weak equivalences W given by those inclusions of disks that preserve the number of special points.
Proposition 2.4.4. Let $(X,N)$ be a stratified surface, and let
be an $\infty $ category. There are equivalences of ${\infty }$ categories
Proof. We claim that the subcategory
can be identified with the subcategory of left Kan extensions along