Perverse sheaves on Riemann surfaces as Milnor sheaves

Abstract

Constructible sheaves of abelian groups on a stratified space can be equivalently described in terms of representations of the exit-path category. In this work, we provide a similar presentation of the abelian category of perverse sheaves on a stratified surface in terms of representations of the so-called paracyclic category of the surface. The category models a hybrid exit–entrance behaviour with respect to chosen sectors of direction, placing it ‘in between’ exit and entrance path categories. In particular, this perspective yields an intrinsic definition of perverse sheaves as an abelian category without reference to derived categories and t-structures.

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 [31] 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 [2, 24] of perverse sheaves on the disk in terms of diagrams

(0.1)

with $\operatorname {id} - ab$ and $\operatorname {id} - ba$ invertible, such a categorical analog is found in the concept of a spherical adjunction; see [31].

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 t-structure 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 so-called Milnor sheaves (Theorem 3.1.13). Similarly to the description of constructible sheaves as representations of the exit path category (see [45]), 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 zero-dimensional 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 [3]. 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 self-evident in the Milnor sheaf model. Namely, it is a direct consequence of a canonical self-duality of the Milnor category obtained by swapping the boundary intervals with their complements (generalizing the well-known self-dualities 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 zero-dimensional 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 zero-dimensional 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 t-structure lies ‘in between’ the standard t-structure and its Verdier dual.

Figure 1 A morphism in $M(X,N)$ from $(A,A')$ to $(B,B')$ represented given by the isotopy H.

In particular, while the exit and entrance path categories are dual to one another, the Milnor category is self-dual: On objects, the duality is given by

$$\begin{align*}(A,A') \mapsto (A, \overline{\partial(A) \setminus A'}) \end{align*}$$

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 self-duality associates to the morphism

(0.2)

depicted in Figure 1, the morphism

(0.3)

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

(0.4)

and hence a functor

(0.5)

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. 1. The space of vanishing cycles:

   

2. 2. The space of nearby cycles:

   

3. 3. The variation map

   $$\begin{align*}a = \operatorname{var}: \Phi \to \Psi \end{align*}$$

   is the value of on the morphism (0.2).

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

2. (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 t-structure. 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 classical-looking 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 [11, 21, 36]. 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 [11].

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 [5, 20] 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 [48], rise to a paracyclic object whose restriction to $\Delta ^{\operatorname {op}}$ is 2-Segal, that is, satisfies a two-dimensional generalization of the Segal condition introduced in [20]. 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 D-modules) and abelian-categorical (avoiding derived categories) is of course as old as the theory of perverse sheaves itself. We should particularly mention the 1990 preprint of MacPherson [40] 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. [33].

Decompositions, stratifications and exit paths

Concerning stratified spaces, we follow the terminology of [27] part.II §1.1-2.

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 [38] 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 [45] 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. (1) Whitney stratifications, see [27] part II §1.2. In this case, the strata are $C^{\infty }$ -manifolds.

2. (2) Topological stratifications, see [26] and [45] §3.1. In this case, the strata are topological manifolds.

3. (3) Conical stratifications, see [38] 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 [45] Theorem 1.2. For conical stratifications, this follows from [38] 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 [3][33] 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 [3] that the category is the heart of a t-structure 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. (1) A topological manifold X of real dimension $2$ , possibly noncompact and with boundary.

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

2. (2) for $n> 0$ ,

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

2. (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.

Figure 2 A Milnor disk $(A,A')$ and a Milnor pair $(U,U')$ .

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

2. (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 (one-dimensional 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

$$\begin{align*}U=\{|z|\leq \varepsilon\}, \quad U' = \{|z| \leq \varepsilon, \,\, \Re(f(z)) \leq \delta\} \end{align*}$$

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:

1. (i) is a perverse sheaf.

2. (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:

(1.2.5)

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:

(1.2.6)

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 quasi-isomorphic 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 [33]. It is well known (loc. cit.) that is itself a perverse sheaf which, in our case, amounts to saying that is quasi-isomorphic 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

$$\begin{align*}\begin{gathered} \delta_i: [n-1] \longrightarrow [n], \,\,\, i=0, \cdots, n \quad (\text{omitting } i); \\ \sigma_j: [n+1] \longrightarrow [n] \,\,\, j=0,\cdots, n \quad (\text{repeating } j), \end{gathered} \end{align*}$$

subject to well-known relations; see, for example, [11], 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)

$$\begin{align*}\partial_i: Z_n\longrightarrow Z_{n-1}, \,\, i=0,\cdots, n; \quad s_j: Z_n\longrightarrow Z_{n+1},\,\, j=0,\cdots, n+1, \end{align*}$$

satisfying the relations dual to those among the $\delta _i$ and $\sigma _j$ . We will also use the term half-simplicial object for a contravariant functor . Thus, a half-simplicial object has only degeneracy maps but no face maps.

Definition 2.1.1 ([11] Chapter III Appendix A, [36] 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 $\delta _i: \langle n-1\rangle \to {\langle n \rangle }$ 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:

$$\begin{align*}\begin{gathered} \tau_n\delta_i = \delta_{i-1}\tau_{n-1} \text{ for } 1\leq i\leq n, \quad \tau_n\delta_0=\delta_n; \\ \tau_n\sigma_i = \sigma_{i+1}\tau_{n+1} \text{ for } 1\leq i\leq n, \quad \tau_n\sigma_0 = \sigma_n\tau_{n+1}^2; \end{gathered} \end{align*}$$

(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 [13]. 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

$$\begin{align*}\operatorname{Hom}_{\Lambda^{\infty}} ({\langle m \rangle}, {\langle n \rangle}) \longrightarrow \operatorname{Hom}_{\Lambda}({\langle m \rangle},{\langle n \rangle}) \end{align*}$$

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 half-paracyclic object we will mean a contravariant functor .

Remark 2.1.3. The categories $\Lambda $ and $\Lambda _{\infty }$ are self-dual, that is, isomorphic to their opposite categories [11] [21]. 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 self-dual 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 [4]. As pointed out in [4], this version goes back to Borsuk and Ulam [7].

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

$$\begin{align*}\operatorname{Ran}^{\leq d}(M) {\setminus}\operatorname{Ran}^{\leq d-1}(M) {{\,\, \simeq\,\, }} \operatorname{Sym}^d_{\neq}(M) \end{align*}$$

is the configuration space of unordered d-tuples 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 (infinite-dimensional) 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.

1. (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.

   

   Figure 3 An exit path in $\operatorname {Ran}(M)$ .

2. (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 [8] 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 [41] 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 [46].

The following result was proven in [9]:

Proposition 2.1.6.

1. (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 half-simplicial objects in .

2. (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 half-paracyclic 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 [11] 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 well-defined 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.

Figure 4 An exit path in $\operatorname {Ran}(M)$ with deaths.

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 [9, 10]. 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 para-disk 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 one-dimensional submanifold, that is one of the following:

1. (i) the empty set,

2. (ii) a finite nonempty union of closed intervals,

3. (iii) the full boundary circle.

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

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

2. (b) a bounded disk, if $A' = \partial A$ .

We now define morphisms between para-disks. Intuitively, such a morphism should be a certain isotopy class of paths $(A_t, A^{\prime }_t)_{t\in I}$ in the space of para-disks. We want such paths to satisfy the following dynamical requirements as t increases from $0$ to $1$ :

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

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

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

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

An example of a pant cobordism is depicted in Figure 5.

Figure 5 A pant cobordism.

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 one-dimensional submanifolds $A^{\prime }_t\subset A_t$ in the picture with paths in the space of para-disks. 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 $t\mapsto 1-t$ ) 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 para-disks $(A,A')$ for $(X,N)$ . A morphism

$$\begin{align*}f: (A_0,A^{\prime}_0) \longrightarrow (A_1, A^{\prime}_1) \end{align*}$$

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

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

  4. (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 para-disk by the condition (2) The condition (4) corresponds to the intuitive requirement (PD2) on paths in the space of para-disks 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 one-dimensional 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 quasi-isomorphism (and therefore, by purity, it reduces to an isomorphism of objects of

). Indeed, denote

$$\begin{align*}P^+ = P\cup \widetilde N^+\subset I\times {\mathbb{D}}, \,\, {\mathbb{D}}^- = \{1\}\times {\mathbb{D}} \cup \widetilde N^- \quad \subset \quad I\times{\mathbb{D}}. \end{align*}$$

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 quasi-isomorphism.

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

$$\begin{align*}{\operatorname{Entr}}(X,N) \to S(X,N),\; x \mapsto (A_x,\emptyset), \end{align*}$$

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.

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

2. (b) Let $\sigma : I\times S^1\times I\times S^1$ be the involution $(t,\theta ) \mapsto (1-t, \theta )$ . 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)

$$\begin{align*}\xi: \Lambda(X,N) \overset{\simeq}{\longrightarrow} \Lambda(X,N)^{\operatorname{op}} \end{align*}$$

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 $(H(1-t,-), P^*)$ . It is straightforward to verify that this association yields a well-defined 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 self-duality 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:

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

   1. (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 $x \in A_t$ ,

   2. (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

1. (i) f is an isomorphism, or

2. (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.

1. (a) The condition (Ent) is a two-step 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 $A^{\prime }_{t_0}$ .

2. (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

$$\begin{align*}\pi: {\overrightarrow{\Lambda}\!}(X,N) \longrightarrow \Lambda(X,N) \end{align*}$$

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 [23]. In fact, one can prove stronger statements, identifying $\Lambda (X,N)$ with the ${\infty }$ -categorical localization or with the Dwyer–Kan simplicial localization [16] of ${\overrightarrow {\Lambda }\!}(X,N)$ with respect to W. This can be done by adapting our proof below by using a hammock-type 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 Hom-set

(2.3.4) $$ \begin{align} \operatorname{Hom}_{{\overrightarrow{\Lambda}\!}(X,N)[W^{-1}]}(A,A'), (C,C')). \end{align} $$

By definition (cf. [23] §I.1), an element of this set is an equivalence class of zig-zags

(2.3.5) $$ \begin{align} (A, A')= (A_1, A^{\prime}_1) \buildrel {w_1}\over\leftarrow (B_1, B^{\prime}_1)\buildrel {f_1}\over\to (A_2, A^{\prime}_2) \buildrel {w_2}\over\leftarrow \cdots \buildrel {f_{n-1}} \over\to (A_n, A^{\prime}_n) = (C,C') \end{align} $$

of arbitrary length, with $w_i\in W$ . The equivalence relation on the set of such zig-zags is generated by two elementary moves:

1. (M1) For any factorization

   
   we can replace the fragment $\buildrel f_i\over \to \buildrel w_i\over \leftarrow \buildrel f_{i+1}\over \to $ with $\buildrel f_{i+1}g\over \to $ .

2. (M2) For any factorization

   
   we can replace the fragment $\buildrel f_{i-1}\over \to \buildrel w_i\over \leftarrow \buildrel f_i\over \to $ with $\buildrel hf_{i-1}\over \to $ .

These two moves imply the hammock move, which is at the basis of Dwyer–Kan localization theory [16] (except that we don’t assume that the vertical morphisms are weak equvialences):

1. (H) Any two zig-zags connected by a hammock, that is, by a commutative diagram

   
   are equivalent.

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 para-disk.

• • 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 zig-zag (2.3.5).

We claim that different choices of $(H,P)$ representing the same morphism f, give rise to equivalent zig-zags. 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:

1. (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 zig-zags connected by the move (M1).

2. (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 zig-zags connected by a hammock:

so they are equivalent by the hammock move. Therefore, the entire zig-zags 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 quasi-inverse 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:

1. (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

   
   is an equivalence in .

2. (ii) Denote by the poset of nonempty finite subsets of I, and consider the inclusion . Then the canonical map

   
   is an equivalence in .

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

$$\begin{align*}i: {\operatorname{Disk}^o}(X,N) \subset \mathfrak{O}(X),\; (U,\emptyset) \mapsto U \end{align*}$$

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 (left-bounded) derived $\infty $ -category as defined in [38, 1.3.2.8]. We equip the stable $\infty $ -category

with the t-structure

, where the t-structure on

is the one from [38, 1.3.2.19]. The heart of this t-structure is equivalent to

. Then, using the recognition principle for derived $\infty $ -categories ([38, 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. (1) is a sheaf.

2. (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 $i: {{\operatorname {Disk}^c}}(X,N) \subset {{\operatorname {Disk}^{oc}}}(X,N)$ and the subcategory of right Kan extensions along $j: {\operatorname {Disk}^o}(X,N) \subset {{\operatorname {Disk}^{oc}}}(X,N)$ . To verify the first claim, suppose that

is functor sending weak equivalences in ${{\operatorname {Disk}^c}}(X,N)$ to equivalences in

. The pointwise Kan extension formula at $U \in {\operatorname {Disk}^o}(X,N)$ exhibits

as the colimit over $i/U$ . If U contains a special point, then we may replace the category $i/U$ by the cofinal subcategory $(i/U)'$ consisting of those closed disks that contain the special point (otherwise, we set $(i/U)' = i/U$ ). The category $(i/U)'$ is filtered and hence contractible and the diagram

consists of equivalences. Hence, by Lemma A.4,

is a left Kan extension of

if and only if, for every $A \in (i/U)'$ , the map

is an equivalence. It is now an immediate consequence of the two-out-of-three property of equivalences that

is a left Kan extension of

if and only if

sends all weak equivalences to equivalences in

. The second claim regarding right Kan extensions along j follows from an essentially identical argument.

Finally, we would like to provide an explicit description of the localization ${{\operatorname {Disk}^c}}(X,N)[W^{-1}]_{\infty }$ which will provide the starting point for our discussion of Milnor disks.

Proposition 2.4.5. The functor

(2.4.6) $$ \begin{align} \pi: {{\operatorname{Disk}^c}}(X,N) \longrightarrow S(X,N),\; A \mapsto (A,\emptyset) \end{align} $$

exhibits the ordinary category $S(X,N)$ as an $\infty $ -categorical localization along the weak equivalences of ${{\operatorname {Disk}^c}}(X,N)$ , that is, identifies it with ${{\operatorname {Disk}^c}}(X,N)[W^{-1}]_{\infty }$ as an ${\infty }$ -category. In particular, for every $\infty $ -category

, the functor

is fully faithful with essential image consisting of those functors that send weak equivalences in ${{\operatorname {Disk}^c}}(X,N)$ to equivalences in

.

Proof. Let $(A,\emptyset ) \in S(X,N)$ . Suppose first that A does not contain a special point. Then we have

$$\begin{align*}(\pi/(A,\emptyset))^{\simeq} = \pi/(A,\emptyset). \end{align*}$$

Further, we claim that $\pi /(A,\emptyset )$ is contractible. To this end, consider the topological space P of continuous paths $[0,1] \to \mathring {X} \setminus N$ ending in $\mathring {A}$ . To an object $(B, \alpha : (B,\emptyset ) \to (A,\emptyset ))$ , we associate the open subset of P consisting of those paths that start in $\mathring {B}$ and lie in the same homotopy class as the class of paths that arises from the isotopy comprising $\alpha $ . This association defines a functor

which satisfies the hypothesis of Lemma A.1 thus proving the contractibility of $\pi /(A,\emptyset )$ .

Now, suppose that A does contain a special point $x \in N$ . Then we first claim that the inclusion

$$\begin{align*}j: (\pi/(A,\emptyset))^{\simeq} \subset \pi/(A,\emptyset) \end{align*}$$

is cofinal. To this end, we need to show that, given an object $b = (B, \alpha : (B,\emptyset ) \to (A,\emptyset ))$ of $\pi /(A,\emptyset )$ , the category $b/j$ is contractible. We consider the space Q of paths in $\mathring {X} \setminus {(N \setminus {x})}$ starting in $\mathring {B}$ and ending in x. To an object $b' = (B', \alpha : (B',\emptyset ) \to (A,\emptyset ))$ of $b/j$ , we associate the open subset of Q consisting of paths that lie in $\mathring {B'}$ . An application of Lemma A.1 proves the claim. Finally, an argument similar to the above shows that $(\pi /(A,\emptyset ))^{\simeq }$ is contractible so that the result follows from Proposition A.4.

As a consequence of the results of this section, we thus obtain the following:

Corollary 2.4.7. Let $(X,N)$ be a stratified surface, and let

be an $\infty $ -category. Then there is an equivalence

of $\infty $ -categories.

Remark 2.4.8. In view of Example 2.2.9, Corollary 2.4.7 recovers the presentation of the constructible derived category in terms of the exit path category. Nevertheless, the description in terms of $S(X,N)$ will be more convenient in what follows.

2.5 Verdier duality

In this section, we assume that is a stable ${\infty }$ -category.

Recall that, in Corollary 2.4.7, we have identified the $\infty $ -category of constructible sheaves on $(X,N)$ with values in any $\infty $ -category with the $\infty $ -category of presheaves on the category $S(X,N) \subset {\overrightarrow {\Lambda }\!}(X,N)$ . In this section, under the assumption that is stable, we illustrate the use of Kan extensions among subcategories of ${\overrightarrow {\Lambda }\!}(X,N)$ to provide a proof of Verdier duality. This treatment can be regarded as an introduction to the techniques to be used in §3 to establish our description of as Milnor sheaves.

Along with $S=\overrightarrow {S}=\overrightarrow {S}(X,N)$ and $\overrightarrow {B}=\overrightarrow {B}(X,N)$ defined earlier, we consider the following full subcategories of ${\overrightarrow {\Lambda }\!}(X,N)$ consisting, respectively, of the following disks:

• • ${\overrightarrow {D}}$ : disks $(A,A')$ of the form

  
  where $A' \subset \partial A$ is a single closed interval and $A \cap N \subset A'$ ,

• • $\overrightarrow {V}$ : disks $(A,A')$ of the form

  
  where $A' \subset \partial A$ is a single closed interval and $(A \setminus A') \cap N$ is a singleton.

The category S is equivalent to the entrance path category of $(X,N)$ so that, for any stable $\infty $ -category , the $\infty $ -category can be identified with the category of constructible sheaves on $(X,N)$ valued in (Corollary 2.4.7). We set ${\overrightarrow {Q}} := S \cup \overrightarrow {B} \cup {\overrightarrow {D}} \cup \overrightarrow {V}$ .

Theorem 2.5.1. Let

be a stable $\infty $ -category. And let

be a functor. Then the following are equivalent:

1. (i) satisfies the following conditions:

   1. (1) is a right Kan extension of , and

   2. (2) is a left Kan extension of .

2. (ii)

   1. (1) maps weak equivalences in $\overrightarrow {B}$ to equivalences in ,

   2. (2) is a left Kan extension of , and

   3. (a) is a right Kan extension of .

Before we provide a proof of the theorem, we explain its implications. The following lemma generalizes one of the statements of Corollary 2.3.8.

Lemma 2.5.2. Let $B \subset \Lambda (X,N)$ denote the full subcategory spanned by the bounded disks. Then the restriction $\overrightarrow {B} \to B$ of the canonical functor ${\overrightarrow {\Lambda }\!}(X,N) \to \Lambda (X,N)$ exhibits B as an $\infty $ -categorical localization of $\overrightarrow {B}$ along the weak equivalences.

Proof. This follows immediately from Proposition A.4.

Corollary 2.5.3. There is a canonical equivalence of stable $\infty $ -categories

identifying constructible sheaves and constructible cosheaves valued in

.

Proof. Let

be the full ( ${\infty }$ -)subcategories consisting of functors satisfying the conditions (i) and (ii) of Theorem 2.5.1, respectively. By Proposition A.3, the restriction functor

is an equivalence. For the same reason, the restriction functor

is an equivalence. Since (i) and (ii) are equivalent, and $\overrightarrow {B}[W^{-1}]_{\infty } \simeq B$ by Lemma 2.5.2, we obtain an equivalence

by composing an inverse of p with q. The equivalence $B \simeq S^{\operatorname {op}}$ induced by the duality $\xi $ then yields the desired result.

Proof of Theorem 2.5.1.

Let be a functor.

We will provide concrete interpretations of the Kan extension conditions in (i) and (ii) so as the claimed equivalence will become an apparent consequence of the stability of the $\infty $ -category .

We begin with (i): For every $(A,A') \in {\overrightarrow {D}}$ , the category $(A,A')/S$ is empty so that is a right Kan extension of if and only if, for every $(A,A') \in {\overrightarrow {D}}$ , we have .

Suppose now that

is a left Kan extension of

. We first determine the value of

at

as determined by the pointwise formula (A.2). The overcategory $S \cup {\overrightarrow {D}}/(A,A')$ admits an $\infty $ -cofinal subcategory depicted by

(2.5.4)

where the morphisms around the boundary of the $2$ -simplex are given by rotation by the smallest possible angle so that a full turn is obtained by traversing the boundary once. Thus, the value of

at $(A,A')$ is determined by the colimit cone (i.e., biCartesian cube)

(2.5.5)

In particular, since

, this biCartesian cube induces an equivalence

Similarly, for a disk of the form

the overcategory $S \cup {\overrightarrow {D}}/(A,A')$ admits an $\infty $ -cofinal subcategory depicted by

(2.5.6)

again exhibiting an equivalence

More precisely, we observe that any weak equivalence

in $\overrightarrow {B}$ induces an equivalence

in

since the induced map relating the above $\infty $ -cofinal subcategories (2.5.6) and (2.5.4) becomes a pointwise equivalence upon applying

.

We next describe the value of

at

To this end, we argue that the overcategory $(S \cup {\overrightarrow {D}})/(A,A')$ contains an $\infty $ -cofinal subcategory of the form

In particular, the value of

at $(A,A')$ is determined by the colimit cone (i.e., biCartesian square)

(2.5.7)

Since the top-right object is a zero object, this diagram exhibits

as a cofiber (cone) of the morphism

Finally, it remains to characterize the value of

at

Similarly, as in Step 1, the overcategory $S \cup {\overrightarrow {D}} \cup \overrightarrow {V}/(A,A')$ admits an $\infty $ -cofinal subcategory depicted by

where the morphisms around the boundary of the $2$ -simplex are given by rotation by the smallest possible angle. Thus, the value of

at $(A,A')$ is determined by the colimit cone (i.e., biCartesian cube)

(2.5.8)

In conclusion, we may characterize the functors

satisfying the Kan extension conditions of (i) as those functors for which

and further the square (2.5.7) as well as the cubes (2.5.5) and (2.5.8) are biCartesian.

We now discuss the Kan extension conditions of (ii). A similar argumentation as the one for (i) show that a functor satisfies the conditions of (ii)(2) and (ii)(3) if and only if

1. (1) ,

2. (2) the cubes (2.5.5) and (2.5.8) are limit cones, and hence biCartesian,

3. (3) the square

   (2.5.9)

   

   is biCartesian.

Thus, to finish the proof, we have to argue why, assuming further (ii)(1), equation (2.5.7) being biCartesian is equivalent to equation (2.5.9) being biCartesian (in the presence of the remaining conditions). To show this, consider the commutative diagram in ${\overrightarrow {Q}}$ depicted by

(2.5.10)

consisting of three stacked cubes, and further, the diagram in obtained by applying . The middle cube is identical to equation (2.5.8) which is biCartesian. Furthermore, the cube given by the composite of the three cubes coincides may be decomposed as

(2.5.11)

where the top cube coincides with equation (2.5.5) and the bottom cube is biCartesian since all vertical maps are weak equivalences which, by (ii)(1), are mapped to equivalences by . Thus, the composite cube is biCartesian as well. The front face of the top cube in equation (2.5.10) is biCartesian since it contains two parallel arrows that are equivalences. Proposition A.4 implies that the top cube is biCartesian if and only if its back face, which coincides with equation (2.5.7), is biCartesian. By the same argument, the bottom cube will be biCartesian if and only if its front face, which coincides with equation (2.5.9), is biCartesian. As a consequence, the two-out-of-three property for the pasting of biCartesian cubes (Proposition A.6) implies that equation (2.5.7) is biCartesian if and only if equation (2.5.9) is biCartesian, concluding our argument.

3 Milnor sheaves

By Corollary 2.4.7, the $\infty $ -category of constructible sheaves

with values in a stable $\infty $ -category

may be parametrized in terms of standard disks: There is an equivalence

(3.0.1)

If

is the derived category of an abelian category

, then this equivalence restricts to an equivalence

In other words, the equivalence (3.0.1) is compatible with the standard t-structure on

. In this section, we provide yet another parametrization of

, in terms of Milnor disks, which is in the same sense compatible with the perverse t-structure. In particular, it provides an intrinsically abelian description of the category of perverse sheaves.

3.1 Constructible sheaves as Milnor sheaves

Let $(X,N)$ be a stratified surface, and let ${\overrightarrow {\Lambda }\!}(X,N)$ denote its directed paracyclic category. A collared cut of an object $(A,A') \in {\overrightarrow {\Lambda }\!}(X,N)$ consists of

• • a cut $\alpha $ , by which we mean an embedding $\alpha : I \to A$ with $\alpha ^{-1}(\partial A) = \{0,1\}$ . We denote the two connected components of the complement if $\alpha (I)$ in A by $U_1$ and $U_2$ .

• • a collar for $\alpha $ , by which we mean a continuous map $G: [-1,1] \times I \to A$ such that

  1. – $G(0,t) = \alpha (t)$ ,

  2. – for every $s \in [-1,1]$ , the map $G(s,-)$ is a cut,

  3. – $G([-1,1] \times I) \cap \partial A' = \emptyset $ ,

  4. – $G(\{-1,1\} \times I) \cap N = \emptyset $ .

  We denote $C = G([-1,1] \times I)$ and $A_1 = U_1 \cup C$ and $A_2 = U_2 \cup C$ .

Associated to a collared cut, there is a commutative square

(3.1.1)

in ${\overrightarrow {\Lambda }\!}(X,N)$ .

Definition 3.1.2. Let be a pointed $\infty $ -category. A functor is called a Milnor cosheaf if

1. (1) maps weak equivalences in $\overrightarrow {M}(X,N)$ to equivalences in ,

2. (2) maps objects of the form

   

   to a zero object,

3. (3) for every object $(A,A')$ of $\overrightarrow {M}(X,N)$ and for every collared cut of $(A,A')$ , such that the associated diagram (3.1.1) takes values in $\overrightarrow {M}(X,N)$ , maps equation (3.1.1) to a coCartesian square in .

Dually, is called a Milnor sheaf if is a Milnor cosheaf. We denote by the $\infty $ -category of Milnor cosheaves and by the $\infty $ -category of Milnor sheaves defined as full subcategories of the respective functor categories.

Definition 3.1.3. Let $(A,\emptyset )$ be an object of $S(X,N)$ . We denote by $\Lambda _A$ the subcategory of $\overrightarrow {M}(X,N)$ with objects $(A,A')$ and morphisms, represented by an isotopy $H: I \times {\mathbb {D}} \to X$ such that, for every $t \in I$ , $H_t({\mathbb {D}}) = A$ . We further denote by $\Lambda _A^+ = \Lambda _A \cup (A,\emptyset )$ obtained by adjoining the initial object $(A,\emptyset )$ .

Remark 3.1.4. For every $(A, \emptyset )$ , the category $\Lambda _{A}$ is equivalent to the paracyclic category $\Lambda _{\infty }$ .

We say that a functor is locally Segal if, for every $(A,\emptyset ) \in S(X,N)$ , with $\partial A \cap N = \emptyset $ , the object is a Segal object, that is, the restriction along an embedding $\Delta \to \Lambda _{A}$ is Segal.

Proposition 3.1.5. Let be a stable $\infty $ -category, and let be a functor. Then is a Milnor cosheaf if and only if the following hold:

1. (1) maps weak equivalences to equivalences in .

2. (2) is locally Segal.

3. (3) For every $x \in N$ , maps any square of the form

   

   to a coCartesian square in .

Proof. Condition (1) appears directly in the definition of a Milnor cosheaf.

Every Milnor cosheaf satisfies the conditions (2) and (3) since, using the version (3.3.2) of the Segal conditions, the respective coCartesian squares all arise from collared cuts.

Suppose now that satisfies (2) and (3). Let $(A,\emptyset ) \in \overrightarrow {M}(X,N)$ with $A \cap N = \emptyset $ . Then the local Segal conditions imply that equation (3.1.1) is coCartesian for every cut which only intersects one of the boundary intervals. If the cut $\alpha $ intersects two boundary intervals, then it is straightforward to deduce that equation (3.1.1) is coCartesian by considering cuts $\alpha _1$ of $(A_1, A_1 \cap A')$ and $\alpha _2$ of $(A_2,A_2 \cap A')$ which are obtained by sliding the endpoint of $\alpha $ out of the boundary interval towards the two possible directions (In the language of [20], this amounts to the statement that every $1$ -Segal object is $2$ -Segal).

By the exact same argumentation, we deduce that equation (3.1.1) is coCartesian for $(A,\emptyset ) \in \overrightarrow {M}(X,N)$ with $A \cap N = \{x\}$ as long as the cut $\alpha $ runs through the special point x. It remains to verify the coCartesianess of equation (3.1.1) for a cut $\alpha $ which does not run through x. But this case can be reduced to equation (3) by induction on the number of boundary intervals: The induction step is obtained by introducing one additional cut which runs either through the special point x or lies completely in the component of $A \setminus \alpha (I)$ which does not contain x.

We denote by

$$\begin{align*}{\overrightarrow{M}^+\!}(X,N) \subset {\overrightarrow{\Lambda}\!}(X,N) \end{align*}$$

the full subcategory spanned by the standard and Milnor disks and by

$$\begin{align*}{\overrightarrow{\Lambda}\!}(X,N)_{\le n} \subset {\overrightarrow{M}^+\!}(X,N) \end{align*}$$

the full subcategory consisting of objects $(A,A')$ such that $A'$ has at most n connected components.

Further, we denote by

$$\begin{align*}{\overrightarrow{D}}(X,N) \subset {\overrightarrow{\Lambda}\!}(X,N) \end{align*}$$

the full subcategory of objects $(A,A') \in {\overrightarrow {\Lambda }\!}(X,N)$ of the form

Theorem 3.1.6. Let be a functor. Then the following are equivalent:

1. (1) and is a left Kan extension of .

2. (2) is a Milnor cosheaf and is a right Kan extension of .

Proof. Suppose that is a left Kan extension of . To show that is a Milnor cosheaf, we verify conditions (2) and (3) of Proposition 3.1.5. Let $(A,A') \in \overrightarrow {M}(X,N)$ with $A \cap N$ empty. Then the inclusion

$$\begin{align*}{\Lambda\!^{+}\!}_{A, \le 1}/(A,A') \subset (S(X,N) \cup {\overrightarrow{D}}(X,N))/(A,A') \end{align*}$$

is an equivalence of categories and hence $\infty $ -cofinal. In particular, by Proposition 3.3.3, is a Segal object.

Now, let

such that $A \cap N = \{x\} \subset A \setminus A'$ is a singleton. Then the inclusion

$$\begin{align*}(S(A,\{x\}) \cup {\overrightarrow{D}}(A,\{x\})/(A,A') \subset (S(X,N) \cup {\overrightarrow{D}}(X,N))/(A,A'), \end{align*}$$

where the first undercategory is taken in $\overrightarrow {M}(U,\{x\})$ , is an equivalence, in particular $\infty $ -cofinal. Now, the category $(S(A,\{x\}) \cup D(A,\{x\})/(A,\varphi )$ is equivalent to the category depicted by

where the automorphisms $\mathbb {Z}$ correspond to the disk moving around the special point x so that it is equivalent to the category depicted by

This latter category contains the $\infty $ -cofinal subcategory

so that the pointwise left Kan extension condition for

is thus equivalent to the square

(3.1.7)

being coCartesian. For a more general $(A,A') \in M(X,N)$ with $A \cap N = \{x\} \subset A \setminus A'$ , by a similar argument, the category $(S(X,N) \cup D(X,N))/(A,A')$ contains a cofinal subcategory C of the form

(3.1.8)

where

is given by the constant isotopy and the morphisms

enter the special point and map the unique interval to the ith interval of A (with respect to some chosen order). We have a functor from the category

to $\operatorname {Cat}/C$ by associating to $0$ the subcategory

of C and to $i> 0$ the subcategory

An application of [37, 4.2.3.10], using that equation (3.1.7) is a pushout, implies that the pointwise left Kan condition for $(A,A')$ is equivalent to the diagram

being a colimit cone. Here, the maps

are morphisms in $\Lambda _{A}$ which move the interval into the various intervals comprising $A'$ . In particular, this implies that the diagram

is a left Kan extension of its restriction to

so that

satisfies the Segal conditions by Proposition 3.3.3. We have thus shown that

is locally Segal.

A similar argument shows that the value of at a disk $(A,A')$ with $A' \cap N = \{x\}$ is determined by the colimit cone

(3.1.9)

Further, the Segal conditions for at a disk $(A_0,A_0')$ , obtained by moving $(A,A')$ away from the special point x so that $A_0 \cap N = \emptyset $ , imply that the diagram

(3.1.10)

is a colimit cone. Since the map

is, as a map between zero objects, an equivalence, we deduce from the induced map on colimit cones that the map

is an equivalence as well. In particular,

maps weak equivalences to equivalences in

.

Condition (3) follows by applying [37, 4.2.3.10] to equation (3.1.8) for $n=2$ with respect to the functor from the category

into $\operatorname {Cat}/C$ which associates to $0$ the subcategory

to $1$ the subcategory

and to $2$ the subcategory

The above statements imply that

is a Milnor sheaf. It remains to show that

is a right Kan extension of

. To this end, let

with $A \cap N = \{x\}$ a singleton. Then it is easily seen that the inclusion

$$\begin{align*}(A,\emptyset)/\Lambda_{A} \subset (A,\emptyset)/\overrightarrow{M}(X,N), \end{align*}$$

where the left-hand overcategory is taken in the category ${\Lambda \!^{+}\!}_{A}$ , is $\infty $ -coinitial. Thus, by Proposition 3.3.3 below, the value of

at $(A,\emptyset )$ is given by right Kan extension of

.

Finally, consider

with $A \cap N$ empty. Again, we consider the inclusion

$$\begin{align*}j: (A,\emptyset)/\Lambda_{A} \subset (A,\emptyset)/\overrightarrow{M}(X,N). \end{align*}$$

We claim that j is $\infty $ -coinitial. To this end, we have to verify, for every $f: (A,\emptyset ) \to (A_1,A_1') \in (A,\emptyset )/M(X,N)$ , that $j/f$ is contractible. This statement is clear if $A_1 \cap N = \emptyset $ . Suppose now that $A_1 \cap N = \{x\} \subset A_1 \setminus A^{\prime }_1$ . In this case, we proceed by exhibiting a contractible $\infty $ -cofinal subcategory of $j/f$ : Fix an object $a_0$ of $j/f$ whose underlying disk $(B,B')$ has $|\pi _0(B')| = |\pi _0(A^{\prime }_1)|+1$ boundary components and such that the map $(B, B') \to (A_1,A^{\prime }_1)$ includes $|\pi _0(A^{\prime }_1)|-1$ intervals of $B'$ into respective intervals of $A^{\prime }_1$ and includes the two intervals adjacent to the entry location of x into the remaining interval of $A^{\prime }_1$ . There are objects $\{a_i | i \in \mathbb {Z}\}$ of $j/f$ which differ from $a_0$ in that the entry point of x lies i segments in $S^1 \setminus B'$ away from the entry point of x for $a_0$ . For $i \in \mathbb {Z}$ , we denote by $a_i^+$ and $a_i^-$ the two objects of $j/f$ obtained by omitting one of the intervals of $B'$ adjacent to the entry point of x. The full subcategory of $j/f$ spanned by these objects has the form:

It is now straightforward to verify that this subcategory is cofinal in $j/f$ and, since it is further contractible, the claim follows. Finally, the contractibility of $j/f$ in the remaining case where $x \in A_1' \cap N$ is immediate.

Therefore, by Proposition 3.3.3, the value of at $(A,\emptyset )$ is also given by right Kan extension of so that, in conclusion, is a right Kan extension of .

The converse implication (2) $\Rightarrow $ (1) is a consequence of the above argumentation and the converse implication $(2) \Rightarrow (1)$ of Proposition 3.3.3.

Remark 3.1.11. In the context of Theorem 3.1.6, let

denote the full subcategory consisting of those functors that satisfy the equivalent conditions (1) and (2). By arguments analogous to the ones in the proof of Proposition 3.1.5, it can be shown that the objects of

are precisely the cyclic cosheaves, namely functors

such that

1. (1) maps objects of the form

   

   to zero objects in ,

2. (2) for every object $(A,A')$ and for every collared cut of $(A,A')$ , maps the associated square (3.1.1) to a coCartesian square in .

Corollary 3.1.12. Let $(X,N)$ be a stratified surface and

a stable $\infty $ -category. Then there are equivalences of stable $\infty $ -categories

given by restriction along $\overrightarrow {M}(X,N) \subset {\overrightarrow {M}^+\!}(X,N)$ and $S(X,N) \subset {\overrightarrow {M}^+\!}(X,N)$ , respectively. In particular, via the equivalence

from Corollary 2.4.7, there is a canonical equivalence

Proof. We apply Theorem 4.3.2.15 of [37]. The fact that $\rho _1$ is an equivalence is then an immediate consequence of the equivalence Theorem 3.1.6. The functor $\rho _2$ is an equivalence by Theorem 3.1.6 combined with the observation that a functor is a right Kan extension of its restriction to $S(X,N)$ if and only if , that is, two successive applications of loc. cit.

Theorem 3.1.13. Let

be the derived $\infty $ -category of an abelian category

. Then the equivalence

from Corollary 3.1.12 restricts to an equivalence

identifying perverse sheaves on $(X,N)$ with Milnor sheaves valued in

.

Proof. Under the equivalence

the value of a Milnor sheaf

on a Milnor disk $(A,A') \in \overrightarrow {M}(X,N)$ is equivalent to the value of the corresponding constructible sheaf

on a Milnor pair $(U,U')$ , where U is a sufficiently small open disk containing the closed disk A and $U'$ is a union of open disks where each disk contains one of the intervals comprising $A'$ . Thus, by Proposition 1.2.4, a constructible sheaf

is perverse if and only if

takes values in

. Further, since all horizontal morphisms that arise in the Milnor sheaf conditions admit sections, they are Cartesian in

if and only if they are Cartesian in

. This proves the claim.

Corollary 3.1.14. Let

be an abelian category. Then we have an natural equivalence

where $M(X,N) \subset \Lambda (X,N)$ is the full subcategory of the (undirected) paracyclic category of $(X,N)$ spanned by the Milnor disks.

Proof. This follows from the observation that $\overrightarrow {M}(X,N) \to M(X,N)$ is a localization along the weak equivalences (Corollary 2.3.8).

3.2 Verdier duality for perverse sheaves

Proposition 3.2.1. Let

be an abelian category. Then the self-duality

$$\begin{align*}\xi: \Lambda(X,N) \longrightarrow \Lambda(X,N)^{\operatorname{op}} \end{align*}$$

induces an equivalence

between Milnor sheaves and cosheaves.

Proof. The Milnor sheaf conditions (in terms of face maps) get swapped with the dual conditions (in terms of degeneracy maps); cf. the proof of Proposition 3.3.4.

Remark 3.2.2. Suppose is an abelian category with exact duality $\delta $ . Then the resulting antiequivalence $\delta \circ \xi ^*$ of can be identified with the Verdier self-duality of . Note that, even more classically, we may understand the perfect pairing between and as an elementary instance of Lefschetz duality for manifolds with boundary.

3.3 Paracyclic Segal objects

Let

be an $\infty $ -category with finite colimits. A cosimplicial object

is called a Segal object, if it satisfies the Segal conditions: for every $n \ge 1$ , the map

(3.3.1) $$ \begin{align} X_1 \amalg_{X_0} \dots \amalg_{X_0} X_1 \longrightarrow X_n \end{align} $$

induced by the inclusions $[1] \cong \{i,i+1\} \subset [n]$ is an equivalence. Equivalently, X is a Segal object if, for every $1 \le m < n$ , the square

(3.3.2)

induced by the diagram

is a pushout square in

.

Proposition 3.3.3. Let

be a stable $\infty $ -category, and let ${\Lambda \!^{+}\!}$ be the augmented paracyclic category obtained from $\Lambda $ by adjoining an initial object $\emptyset $ . Let

denote the full subcategory spanned by $\emptyset $ and $\langle 0 \rangle $ . Then for a functor

the following conditions are equivalent:

1. (1) is a left Kan extension of .

2. (2) satisfies the Segal conditions and is a right Kan extension of .

Proof. We make two observations:

• • The inclusion

  $$\begin{align*}\Delta \subset \Lambda \end{align*}$$
  is coinitial: The category $\Delta /{\langle n \rangle }$ is the category of simplices of the simplicial object $\operatorname {Hom}_{\Lambda }(-,{\langle n \rangle })|\Delta ^{\operatorname {op}}$ whose geometric realization is homeomorphic to $|\Delta ^n| \times {\mathbb {R}}$ .

• • The inclusion

  $$\begin{align*}(\Delta^+)_{\le 1}/[n] \subset ({\Lambda\!^{+}\!})_{\le 1}/{\langle n \rangle} \end{align*}$$
  is an equivalence and hence cofinal.

Therefore, we have reduced the proof of Proposition 3.3.3 to the statement of Proposition 3.3.4 below.

Proposition 3.3.4. Let

be a stable $\infty $ -category, and let $\Delta ^+$ be the augmented simplex category obtained from $\Delta $ by adjoining an initial object $\emptyset $ . Let

denote the full subcategory spanned by $\emptyset $ and $[0]$ . Let

be an augmented cosimplicial object in

. Then the following conditions are equivalent:

1. (1) X is a left Kan extension of .

2. (2) $X|\Delta $ satisfies the Segal conditions and X is a right Kan extension of $X|\Delta $ .

Proof. (1) $\Rightarrow $ (2): Suppose that X is a left Kan extension of

. The pointwise formula for Kan extensions implies that, for every $n \ge 1$ , $X_n$ is a colimit of the restriction of X to

. We define a functor f from the poset

$$\begin{align*}{\mathcal I} = \{0,1\} \leftarrow \{1\} \rightarrow \{1,2\} \leftarrow \{2\} \rightarrow \dots \leftarrow \{n-1\} \rightarrow \{n-1,n\} \end{align*}$$

to

sending a set I to the nerve of the subposet of

consisting of those maps with image contained in I. By [37, 4.2.3.10], we may compute the colimit of

as the colimit of the diagram

yielding the nth Segal condition.

To show that X is a right Kan extension of $X|\Delta $ , first note that, since $X|\Delta $ is Segal, by Lemma 3.3.5 below, it is a right Kan extension of $X|(\Delta _{\le 1})$ . Therefore, it suffices to show that X is a right Kan extension of $X|(\Delta _{\le 1})$ . By the pointwise criterion, this is equivalent to the statement that X maps the diagram

in $\Delta ^+$ to a pullback square in

. But, since X is a left Kan extension of

, it maps the square to a pushout square in

so that the statement follows since

is stable.

(2) $\Rightarrow $ (1): Suppose that $X|\Delta $ satisfies the Segal conditions. Then, by the above arguments, X is left Kan extension of

if and only if it maps the square

to a pushout square. But, by the last part of the argument of (1) $\Rightarrow $ (2), this is equivalent to X being a right Kan extension of $X|\Delta $ , concluding the argument.

Lemma 3.3.5. Let be a stable $\infty $ -category, and let be a cosimplicial object in . Let $\Delta _{\le 1} \subset \Delta $ denote the full subcategory spanned by the objects $[0]$ and $[1]$ . Then Y is a Segal object if and only if Y is a right Kan extension of its restriction $Y|(\Delta _{\le 1})$ .

Proof. Suppose Y satisfies the Segal conditions. We need to verify that, for every $n \ge 2$ , $Y_n$ is a limit of $Y|([n]/\Delta _{\le 1})$ . We prove the statement by induction on n starting with $n=2$ . Consider the commutative diagram in $\Delta $ depicted by

Since all horizontal and vertical composites yields the identity on the respective object, Y maps all $2x1$ and $1x2$ rectangles to biCartesian squares in

. The Segal condition for $n=2$ is equivalent to Y mapping the top-left square to a pushout, and hence biCartesian, square. The pointwise condition on Y being a right Kan extension is equivalent to Y mapping the bottom-right square to a pullback, hence biCartesian, square. But, by the two-out-of-three property for biCartesian squares ([37, 4.4.2.1]), the top-left square is biCartesian if and only if the bottom-right square is biCartesian. Therefore, for $n=2$ , the Segal condition is equivalent to the corresponding pointwise Kan extension criterion for $Y_2$ .

Assume that the nth Segal condition is equivalent to the pointwise Kan extension formula for $Y_n$ . Consider the diagram

in $\Delta $ . A similar argument to the case $n=2$ implies the equivalence of the $(n+1)$ st Segal condition and the pointwise Kan extension formula for $Y_{n+1}$ , concluding the argument.

4 Perverse sheaves on $({\mathbb {C}}, \{0\})$

In this chapter, we consider the classical case when $X = {\mathbb {C}}$ is the complex plane and $N=\{0\}$ . The corresponding category of perverse sheaves is well known, but our approach provides a new point of view on it which will be crucial in the further work on categorical generalization to perverse schobers. In what follows, we compare the two approaches and discuss the concepts they lead to.

4.1 The classical $(\Phi , \Psi )$ -description

Let be a Grothendieck abelian category. The following result goes back to the early days of the theory of perverse sheaves [2, 24]. It was originally formulated for perverse sheaves of vector spaces, but the proof given in [24] generalizes easily to the -valued case.

Proposition 4.1.1. The category is equivalent to the category of data $(\Phi , \Psi , a,b)$ , where $\Phi $ and $\Psi $ are objects of and

(4.1.2)

are morphisms such that the monodromy transformations

(4.1.3) $$ \begin{align} T_{\Psi}: = \operatorname{Id}_{\Psi} - ab \quad \text{and} \quad T_{\Phi}: \operatorname{Id}_{\Phi}-ba \quad \end{align} $$

are isomorphisms. In fact, $T_{\Psi }$ being an isomorphism is equivalent to $T_{\Phi }$ being an isomorphism.

For a given perverse sheaf , the corresponding objects $\Phi =\Phi (F)$ and $\Psi =\Psi (F)$ are called the objects of vanishing and nearby cycles of F. We will now describe the relationship between the classification data in Proposition 4.1.1 and our description of perverse sheaves as Milnor sheaves from Corollary 3.1.14.

4.2 From a Milnor sheaf to vanishing and nearby cycles

Let

be a Milnor sheaf. We will explain how to most directly extract from

the classification data (4.1.2) and verify conditions (4.1.3). First, we define

is any disk that does not contain the origin $0$ . Further, we set

is any disk containing $0$ in its interior. The descent conditions force rotation of $(A,A')$ by $\pi $ to be multiplication by $-1$ : In the local model explained in §3.3, this automorphism corresponds to the paracyclic shift on the Čech nerve of $0 \to \Psi [1]$ . The monodromy transformation $T_{\Psi }$ is obtained by moving $(A,A')$ as a rigid body (parallel to itself) in a circle around the origin $0 \in {\mathbb {C}}$ . The monodromy $T_{\Phi }$ is induced by rotating $(B,B')$ by an angle of $2\pi $ around the center of the disk B. The map

$$\begin{align*}a: \Phi \longrightarrow \Psi \end{align*}$$

is obtained from the morphism in $M({\mathbb {C}},\{0\})$ that is represented by a bordism of the form

(4.2.1)

while the morphism b corresponds to the dual of equation (4.2.1):

(4.2.2)

To obtain the relations (4.1.3), we investigate the descent condition for

Namely, introducing the depicted cut, the corresponding descent condition (3.1.1) provides a direct sum decomposition

. We then directly observe that, with respect to that decomposition, the transformation induced on

by rotating $(C,C')$ around its center by $2 \pi $ , is given by the matrix

(4.2.3) $$ \begin{align} Q = \left( \begin{array}{rr} T_{\Phi} & 0 \\ 0 & T_{\Psi} \end{array} \right). \end{align} $$

On the other hand, this transformation comes equipped with a square root, induced by rotating $(C,C')$ around its center by $\pi $ . A somewhat more careful analysis shows that, in terms of the above direct sum decomposition, this transformation can be described by the matrix

(4.2.4) $$ \begin{align} P = \left( \begin{array}{rr} -\operatorname{id} & b \\ -a & \operatorname{id} \end{array} \right). \end{align} $$

Now, the relation $P^2 = Q$ implies the desired relations (4.1.3). Note that, in order to extract the above data, various choices have to be made – the advantage of the description of

lies in the intrinsic nature of the parametrizing category $M({\mathbb {C}},\{0\})$ of Milnor disks.

4.3 The equivalence of classical and Milnor sheaf descriptions

In this section, we elaborate on the discussion in §4.2 to provide a direct argument for why these descriptions are equivalent. This can, of course, also be indirectly deduced by combining our Corollary 3.1.14 and [24], but it is nevertheless interesting to provide an explicit dictionary.

The Milnor sheaf description

Proposition 4.3.1. Let

be an abelian category. Let ${\mathbb {D}} \subset {\mathbb {C}}$ be the unit disk. Then the restriction along $\Lambda _{{\mathbb {D}}} \subset M^+({\mathbb {C}},\{0\})$ induces a fully faithful functor

with essential image given by those paracyclic objects whose underlying simplicial object satisfies the Segal conditions.

Proof. For notational convenience, we replace by and prove the cosheaf version of the statement. By Corollary 2.3.8, the category $M^+$ may be described as the localization of its directed variant $\overrightarrow {M}^+$ . In the statement of the proposition, we may therefore replace the category by the equivalent category , where here, the superscript $\sharp $ also contains the requirement that weak equivalences be sent to isomorphisms in . We now focus on the following collections of objects of $\overrightarrow {M}^+$ (and the subcategories they span):

• • $\overrightarrow {M}_0$ : all objects $(A,A')$ , where $0 \in A \setminus A'$ ,

• • $\overrightarrow {M}_1$ : $\overrightarrow {M}_0$ together with all objects of the form

  

• • $\overrightarrow {M}_2$ : $\overrightarrow {M}_1$ together with all objects $(A,A')$ such that $0 \in A'$ ,

• • $\overrightarrow {M}_3$ : $\overrightarrow {M}_2$ together with all objects of the form

  

The fact that the restriction functor of the proposition is an equivalence now follows from the statement that the functors can be characterized by the following conditions:

1. (1) The paracyclic object satisfies the Segal conditions.

2. (2) is obtained from its restriction to $\overrightarrow {M}_0$ via a sequence of left (resp. right) Kan extensions as indicated in

   

   The details are left to the reader.

Corollary 4.3.2. The category of Milnor sheaves on $({\mathbb {C}},\{0\})$ with values in , and therefore the category of perverse sheaves on $({\mathbb {C}},\{0\})$ , is equivalent to the category of paracyclic objects in whose underlying simplicial object satisfies the Segal conditions.

In what follows, we provide the relation to the more traditional classification of Proposition 4.1.1 by means of a paracyclic nerve construction which can also be regarded as a special instance of a duplicial variant of the Dold–Kan correspondence established in [17].

Paracyclic structures on the nerve of a Picard groupoid

To compare Propositions 4.1.1 and 4.3.1 in a direct way, we assume for simplicity that is the category of abelian groups. It is classical that a simplicial set is Segal if and only if it is isomorphic to the nerve of a small category. The categories relevant for us are are Picard groupoids of a particular type.

We recall (cf. [12]) that a Picard groupoid is a symmetric monoidal category in which each object is invertible with respect to $\otimes $ and each morphism is invertible with respect to the composition.

Example 4.3.3. Let $E^{\bullet }$ be a two-term complex of abelian groups situated in degrees $[-1,0]$ . It will be suggestive for us to write $E^{\bullet }$ as $\{\Psi \buildrel b\over \to \Phi \}$ with $\Phi $ in degree $0$ and $\Psi $ in degree $(-1)$ . To such a datum, one associates a Picard groupoid $[E^{\bullet }] = [\Psi \buildrel b\over \to \Phi ]$ with

$$\begin{align*}\begin{gathered} \operatorname{Ob}\, [\Psi\buildrel b\over\to \Phi] \,\,=\,\,\Phi; \\ \operatorname{Hom}(\varphi', \varphi) \,\,=\,\,\bigl\{ \psi\in\Psi\, \bigl| \, b(\psi) = \varphi-\varphi' \bigr\}. \end{gathered} \end{align*}$$

Composition of morphisms is given by addition of the $\psi $ . The tensor product of objects is given by addition of the $\varphi $ . We note that the set of all morphisms in $[\Psi \buildrel b\over \to \Phi ]$ (i.e., the disjoint union of all the $ \operatorname {Hom}(\varphi , \varphi ') $ ) can be described as

$$\begin{align*}\operatorname{Mor} \, [\Psi\buildrel b\over\to \Phi] \,\,=\,\, \Psi \oplus \Phi, \end{align*}$$

with the source and target maps $s,t: \operatorname {Mor}\to \operatorname {Ob}$ given by

(4.3.4) $$ \begin{align} s(\psi,\varphi)= \varphi - b(\psi) , \quad t(\psi,\varphi) = \varphi. \end{align} $$

See [12] for more details.

The nerve $N [\Psi \buildrel b\over \to \Phi ]$ is a simplicial abelian group with n-simplices

(4.3.5) $$ \begin{align} N_n [\Psi\buildrel b\over\to \Phi] \,\,=\,\, \Psi^{\oplus n}\oplus \Phi. \end{align} $$

Passing from a two-term complex $\{ \Psi \buildrel b\over \to \Phi \}$ to the simplicial object $N [\Psi \buildrel b\over \to \Phi ]$ is a particular case of the Dold–Kan correspondence between nonpositively graded cochain complexes of abelian groups and simplicial abelian groups; see §4.4 below.

Proposition 4.3.6. Let $b: \Psi \to \Phi $ be a morphism of abelian groups. Then the following are in bijection:

1. (i) Morphisms $a: \Phi \to \Psi $ such that the data $(\Phi , \Psi , a,b)$ satisfy the conditions of Proposition 4.1.1, that is, define a perverse sheaf .

2. (ii) Extensions of the structure of a simplicial abelian on $ N [\Psi \buildrel b\over \to \Phi ]$ to that of a paracyclic abelian group, that is, systems of automorphisms $t_n \in \operatorname {Aut}\bigl (N_n [\Psi \buildrel b\over \to \Phi ]\bigr )$ (actions of the $\tau _n\in \operatorname {Aut}_{\Lambda _{\infty }}{\langle n \rangle } $ ) satisfying the relations dual to those imposed in Definition 2.1.1(a).

Under this bijection, the automorphism $t_n^{n+1}$ corresponds, via the identification (4.3.5), to the direct sum $T_{\Psi }^{\oplus n}\oplus T_{\Phi }$ of the monodromies.

Proof. Explicitly, the convention (4.3.4) on labelling the source and target of a morphism implies that the simplicial face and degeneracy operators on $N_n [\Psi \buildrel b\over \to \Phi ]$ are given by

$$\begin{align*}\partial_i: \Psi^{\oplus n}\oplus \Phi \to \Psi^{\oplus(n-1)} \oplus \Phi, \quad (\psi_1,\cdots, \psi_n;\varphi ) \mapsto \begin{cases} ( \psi_2,\cdots, \psi_n; \varphi), & i=0, \\ ( \psi_1, \cdots, \psi_i+\psi_{i+1}, \cdots\psi_n;\varphi), & 1\leq i < n; \\ (\psi_1, \cdots, \psi_{n-1}; \varphi-b(\psi_n)), & i=n; \end{cases} \end{align*}$$

$$\begin{align*}s_j: \Psi^{\oplus n} \oplus \Phi \to \Psi^{\oplus (n+1)}\oplus \Phi, \quad (\psi_1,\cdots, \psi_n; \varphi) \mapsto (\psi_1, \cdots, \psi_{j-1},0, \psi_{j+1}, \cdots \psi_n; \varphi), \,\, j=0,\cdots, n. \end{align*}$$

Now, let $a: \Phi \to \Psi $ be as in (a). For each $n\geq 0$ , define an endomorphism $t_n$ of $\Psi ^{\oplus n}\oplus \Phi $ by

(4.3.7) $$ \begin{align} t_n(\psi_1,\cdots, \psi_n, \varphi) \,=\, \bigl(-\psi_1-\cdots - \psi_n + a(\varphi), \psi_2, \cdots, \psi_{n-1}; \varphi-b(\psi_n)\bigr). \end{align} $$

We then check directly that the relations dual to those of Definition 2.1.1(a) are satisfied. We also check that $t_n^{n+1} = T_{\Psi }^{\oplus n}\oplus T_{\Phi }$ which implies that $t_n$ is invertible.

Conversely, suppose we have automorphisms $t_n$ as in (b). The relation $\partial _0t_n=\partial _n$ implies that $t_n$ has the form

$$\begin{align*}t_n(\psi_n, \cdots, \psi_n; \varphi) \,\,=\,\,\biggl( -\sum_{i=1}^n x^{(n)}_i(\psi_i) + a_n(\varphi); \, \psi_2, \cdots, \psi_{n-1}, \varphi-b(\psi_n)\biggr) \end{align*}$$

for some linear maps $x^{(n)}_i: \Psi \to \Psi $ and $a_n: \Phi \to \Psi $ . We denote $a_1=a$ and will prove that

(4.3.8) $$ \begin{align} x^{(n)}_i=\operatorname{Id}, \quad a_n=a, \quad \forall \, n, \,\, i=1,\cdots, n, \end{align} $$

that is, that all the $t_n$ are given by the formula (4.3.7). This will imply the invertibility of $T_{\Psi }=\operatorname {Id}-ab$ and $T_{\Phi }=\operatorname {Id}-ba$ by identifying $t_n^{n+1}$ as above.

The equalities (4.3.8) are proved recursively, using the relations of $\Lambda ^{\infty }$ . To start, the relation $\partial _1t_2=t_1\partial _0$ implies that

$$\begin{align*}\partial_1t_2(\psi_1, \psi_2; \varphi) \,=\, (-x^{(2)}_1\psi_1 - x^{(2)}_2\psi_2 + a_2\varphi + \psi_1; \, \varphi-b\psi_2) \end{align*}$$

is equal to

$$\begin{align*}t_1\partial_0(\psi_1, \psi_2; \varphi) \,=\, (-x^{(1)}_1\psi_2+a\varphi;\, \varphi-b\psi_2), \end{align*}$$

which entails

$$\begin{align*}x^{(2)}_2 = x^{(1)}_1, \,\,\, x^{(2)}_1=\operatorname{Id}. \end{align*}$$

The relation $\partial _2t_2=t_1\partial _1$ then implies that

$$\begin{align*}\partial_1 t_2(\psi_1, \psi_2;\psi) \,=\, (-x^{(2)}_1\psi_1-x^{(2)}_2\psi_2 + a_2\varphi; \, \varphi-b\psi_2-b\psi_1) \end{align*}$$

is equal to

$$\begin{align*}t_1 \partial_1(\psi_1, \psi_2;\psi) \,=\, (-x^{(1)}_1\psi_1 -x^{(1)}_1 \psi_2 + a(\varphi);\, \varphi-b\psi_2-b\psi_1), \end{align*}$$

which entails

$$\begin{align*}x^{(2)}_1 = x^{(2)}_2 = x^{(1)}_1, \,\,\, a_2=a. \end{align*}$$

Since we already know that $x^{(2)}_1=\operatorname {Id}$ , we see that $x^{(2)}_2=x^{(1)}_1=\operatorname {Id}$ . Continuing like this, we prove equation (4.3.8).

Remark 4.3.9. One can consider paracyclic structures on the nerves of more general Picard groupoids, not necessarily those corresponding to two-term complexes. It would be interesting to understand the relation of such structures to perverse sheaflike objects. We recall [30] that Picard groupoids correspond to spectra (stable homotopy types in the sense of homotopy topology) which have only two nontrivial homotopy groups in adjacent degrees, say only $\pi _0$ and $\pi _1$ or only $\pi _1$ and $\pi _2$ .

More generally, unstable homotopy types with only $\pi _1$ and $\pi _2$ nontrivial, are described by crossed modules (see, e.g., [42]), which are two-term complexes of possibly nonabelian groups

$$\begin{align*}G^{\bullet} \,=\,\bigl\{ G^{-1} \buildrel\partial\over\longrightarrow G^0\bigr\} \end{align*}$$

with a compatible action of $G^0$ on $G^{-1}$ . A crossed module $G^{\bullet }$ gives rise to a non-abelian Picard groupoid (also known as a $2$ -group) $[G^{\bullet }]$ , defined similarly to Example 4.3.3. One can ask about the meaning of paracyclic structures on the nerve of $[G^{\bullet }]$ and the possibility of defining perverse sheaves of nonabelian groups in one complex dimension.

4.4 Relation to the duplicial Dold–Kan correspondence

The classical Dold–Kan

Let

be an abelian category and

be the (abelian) category of cochain complexes over

situated in degrees $\leq 0$ . As usual, by

we denote the category of simplicial objects of

. The Dold–Kan correspondence (see, e.g., [25]) is the pair of mutually quasi-inverse (in particular, adjoint) equivalences of categories

defined as follows. The functor ${C_{\operatorname {DK}}}$ , called the normalized chain complex functor, takes

to the complex ${C_{\operatorname {DK}}}(A_{\bullet })$ with

$$\begin{align*}{C_{\operatorname{DK}}}^{-n}(A_{\bullet}) \,=\,\bigcap_{i=1}^{n}\, \operatorname{Ker} \{\partial_i: A_n\longrightarrow A_{n-1}\}, \quad n\geq 0, \end{align*}$$

with the differential given by the remaining face map $\partial _0$ .

The functor $\operatorname {N_{DK}}$ , called the Dold–Kan nerve, takes a complex into the simplicial object $\operatorname {N_{DK}}(E^{\bullet })$ with

$$\begin{align*}\operatorname{N_{DK}}(E^{\bullet})_n \,=\, Z^0(\Delta^n, E^{\bullet}), \end{align*}$$

the object of degree $0$ simplicial (hyper)cocycles on $\Delta ^n$ with values in $E^{\bullet }$ . That is, denoting $\Delta ^n_m$ the set of m-simplices of $\Delta ^n$ ,

$$\begin{align*}Z^0(\Delta^n, E^{\bullet}) \,\subset \, \prod_{m\geq 0} (E^{-m})^{\Delta^n_m} \end{align*}$$

is given by the following ‘end’ condition: The action of the morphism induced by each $d_E: E^{-m} \to E^{-m+1}$ is equal to the action of the morphism induced by $\sum (-1)^i\partial _i: \Delta ^n_{m+1} \to \Delta ^n_m$ .

Examples 4.4.1.

1. (a) Let . An element of $ Z^0(\Delta ^n, E^{\bullet })$ is in this case a rule $\gamma $ associating:

   1. (0) To each vertex $e_i$ , $0\leq i\leq n$ , of $\Delta ^n$ , an element $\gamma _i\in E^0$ .

   2. (1) To each edge (possibly degenerate) $e_{ij}$ , $0\leq i\leq j\leq n$ , of $\Delta ^n$ , an element $\gamma _{ij}\in E^{-1}$ so that $d_E(\gamma _{ij}) = e_j-e_i$ .

   3. (2) To each two-face (possibly degenerate) $e_{ijk}$ , $0\leq i\leq j \leq k\leq n$ , of $\Delta ^n$ , an element $\gamma _{ijk}\in E^{-2}$ so that $d_E(\gamma _{ijk}) = \gamma _{jk} - \gamma _{ik} + \gamma _{ij}$ .

   ( $\cdots $ ) And so on.

2. (b) In particular, if $E^{\bullet } = \{E^{-1}\to E^0\}$ is a two-term complex of abelian groups, then $\operatorname {N_{DK}}(E^{\bullet }) = \operatorname {N}[E^{\bullet }]$ is the usual nerve of the Picard groupoid $[E^{\bullet }]$ .

Proposition 4.3.6 extends verbatim to the following.

Proposition 4.4.2. Let be any abelian category and $b: \Psi \to \Phi $ be a morphism in . Then the following are in bijection:

1. (i) Morphisms $a: \Phi \to \Psi $ such that the data $(\Phi , \Psi , a,b)$ satisfy the conditions of Proposition 4.1.1, that is, define a perverse sheaf .

2. (ii) Extensions of the simplicial object structure on $\operatorname {N_{DK}}\{\Psi \buildrel b\over \to \Phi \}$ to a structure of a paracyclic object.

The duplicial Dwyer–Kan correspodence

Let be an abelian category and . Proposition 4.4.2 leads to the following question: What is the meaning of a paracyclic structure on $\operatorname {N_{DK}}(E^{\bullet })$ extending the given simplicial structure? An answer to that can be given by the results of Dwyer–Kan [17] which we recall.

We define the duplex category $\Xi $ to have objects ${\langle n \rangle }$ , $n \in \mathbb {N}:= \mathbb {Z}_{\geq 0} $ . A morphism from ${\langle m \rangle }$ to ${\langle n \rangle }$ consists of a weakly monotone map $f: \mathbb {N} \to \mathbb {N}$ satisfying the following periodicity condition: For all $i \in \mathbb {N}$ , we have $f(i + m +1) = f(i) + n + 1$ . The simplex category $\Delta $ is naturally a subcategory of $\Xi $ obtained by restricting to those morphisms between ${\langle m \rangle }$ and ${\langle n \rangle }$ that map the interval $[0,m]$ to $[0,n]$ . A duplicial object in a category is a functor .

We recall [21] that the paracyclic category $\Lambda _{\infty }$ can be defined in a very similar way, except we consider weakly mononote maps $f: \mathbb {Z}\to \mathbb {Z}$ (instead of $\mathbb {N}\to \mathbb {N}$ ) satisfying the same periodicity condition. In particular, the shift map

$$\begin{align*}\tau_n: {\langle n \rangle} \longrightarrow {\langle n \rangle},\quad i \mapsto i+1 \end{align*}$$

is invertible as en element of $\operatorname {Hom}_{\Lambda _{\infty }}({\langle n \rangle }, {\langle n \rangle })$ (with i running in $\mathbb {Z}$ ) but is not invertible as an element of $\operatorname {Hom}_{\Xi }({\langle n \rangle }, {\langle n \rangle })$ (with i running in $\mathbb {N}$ ). In fact, comparing [17] and [21] leads to the following.

Proposition 4.4.3. $\Lambda _{\infty }\simeq \Xi [\tau _n^{-1}|\, n\geq 0] $ is identified with the localization of $\Xi $ with respect to the morphisms $\tau _n$ , $n\geq 0$ .

In fact, the powers $\tau _n^{n+1}$ forming a central system (a natural transformation from $\operatorname {Id}_{\Xi }$ to itself), it is easy to see that the ${\infty }$ -categorical localization $ \Xi [\tau _n^{-1}|\, n\geq 0]_{\infty }$ is also identified with $\Lambda _{\infty }$ . In particular, $\Xi $ , like $\Lambda _{\infty }$ , is generated by the coface and codegeneracy morphisms

$$\begin{align*}\begin{gathered} \delta^{(n)}_i: \langle n-1\rangle \to {\langle n \rangle}, \quad (n\geq 1, \,\, 0\leq i\leq n), \\ \sigma^{(n)}_i: {\langle n \rangle} \to \langle n-1\rangle, \quad (n\geq 1, \,\, 0\leq i\leq n), \end{gathered} \end{align*}$$

satisfying the same quadratic relations as in $\Lambda _{\infty }$ . The morphisms

$$\begin{align*}\delta^{(n)}_i, \,\, 0\leq i\leq n, \quad \sigma^{(n)}_i,\,\, 0\leq i\leq n-1 \end{align*}$$

generate the simplex category $\Delta \subset \Xi $ . The shift map is expressed as $\tau _n=\delta ^{(n-1}_0 \sigma ^{(n)}_{n-1}$ . Accordingly, a duplicial object $Y_{\bullet }$ in a category can be identified with a sequence of objects $X_0, X_1, \dots $ equipped with face and degeneracy maps

$$\begin{align*}\begin{gathered} \partial_i: X_n \longrightarrow X_{n-1}\quad (n \ge 1, 0 \le i \le n), \\ s_i: X_{n-1} \longrightarrow X_{n}\quad (n \ge 1, 0 \le i \le n), \end{gathered} \end{align*}$$

subject to relations dual to those among the $\delta ^{(n)}_i, \sigma ^{(n)}_i$ . The action of $\tau _n$ is then $t_n = s_{n+1}\partial _0: Y_n\to Y_n$ . A paracyclic object is a duplicial object such that all the $t_n$ are isomorphisms.

Following Dwyer–Kan, we call a connective ducomplex in

a diagram

satisfying $d^2=0$ , $\delta ^2=0$ and no further relations. We denote

the category of connective ducoplexes in

.

Theorem 4.4.4 (Dwyer–Kan).

(a) There is an equivalence of categories

given by associating to a duplicial abelian group $A_{\bullet }$ the ducomplex $B^{\bullet }$ with

$$ \begin{align*} B^{-n} & = \bigcap_{ i=1}^n \, \operatorname{Ker}\{\partial_i: A_n\to A_{n-1}\}, \quad n\geq 0, \\ d & = \partial_0: B^{-n} \to B^{-n+1}, \\ \delta & = \sum_{i=0}^{n} (-1)^i s_i : B^{-n+1} \to B^{-n}. \end{align*} $$

(b) Under this equivalence, paracyclic objects correspond to ducomplexes satisfying

$$\begin{align*}\operatorname{Id}_{B^{-n}} + (-1)^n (d\delta-\delta d): B^{-n} \longrightarrow B^{-n} \quad \text {is invertible for any } n\geq 0. \end{align*}$$

Proof. Part (a) is Theorem 3.5 of [17]. Part (b) follows from the interpretation of $t_n^{n+1}$ in terms of ducomplexes given in Proposition 6.5 of [17].

The equivalence between the descriptions of the category of perverse sheaves on $({\mathbb {C}},\{0\})$ from Proposition 4.3.1 and Proposition 4.1.1, respectively, is then a consequence of restricting the equivalence from Theorem 4.4.4 to paracyclic Segal objects.