Zig-Zag Functors for Perverse Nori Motives

1. Introduction

Let X be a complex algebraic variety with a two-stratum decomposition

$$X=U\bigsqcup S,$$

where

$$j:U\hookrightarrow X,i:S\hookrightarrow X$$

are respectively the open immersion and the complementary closed immersion. The purpose of this paper is to make the zig-zag construction from [1] available inside the category of perverse Nori motives.

Recall from [1], associated with a stratification is a category

$$Z(X,S)$$

of zig-zag data and a functor

$$\mu :P\left(X\right)⟶Z(X,S)$$

from the category $P\left(X\right)$ of perverse sheaves on X. This construction is concrete and diagrammatic. It packages a perverse sheaf by its open restriction, two closed-stratum terms, and the exact sequence connecting them. For $P^{*}\in P(X-S)$, an object of $Z(X,S)$ is $P^{*}$, together with an exact sequence on the closed stratum

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)⟶K⟶C⟶H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right).$$

A morphism is a morphism of the open terms together with maps between the middle terms K and C, making the corresponding exact-sequence diagram commute. For $Q^{*}\in P\left(X\right)$, the zig-zag functor $\mu$ sends $Q^{*}$ to $j^{*}{Q}^{*}$, equipped with the exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{!}{Q}^{*}\right)⟶H^{-d}\left(i^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

The aim here is to reproduce this exact construction in the perverse Nori motivic category in the case of a two-stratum configuration. We write

$$PNM\left(X\right)$$

for the heart-level category of perverse Nori motives on X, and similarly for U and S. We construct a category

$$Z_{\mathrm{Nori}}(X,S)$$

whose objects have the same form as the zig-zag objects, but with all terms interpreted inside $PNM\left(S\right)$, and we construct a functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S)$$

defined by the same exact sequence. The main result is compatible with realization:

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

Thus the motivic zig-zag data of a perverse Nori motive realize to the classical zig-zag data of its perverse sheaf realization. This zig-zag construction is a reusable perverse-sheaf tool: it lets one read off open data, closed costalk data, closed stalk data, boundary maps, and commutative diagrams forced by functoriality and exactness. This note makes that same tool available in the perverse Nori motivic category.

1.1. The Classical Zig-Zag Construction

Let $S\subset X$ be a closed stratum of complex dimension d, and write $j:X-S\hookrightarrow X$ and $i:S\hookrightarrow X$. The construction of [1] associates to this pair a zig-zag category $Z(X,S)$. An object of $Z(X,S)$ is an object

$$P^{*}\in P(X-S)$$

together with an exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)⟶K⟶C⟶H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right).$$

A morphism

$$(P^{*},K,C)⟶({\overline{P}}^{*},\overline{K},\overline{C})$$

is a morphism $P^{*}\to {\overline{P}}^{*}$, together with maps $K\to \overline{K}$ and $C\to \overline{C}$, such that the induced diagram of exact sequences commutes.

In $\S 2$ of [1], the zig-zag functor

$$\mu :P\left(X\right)⟶Z(X,S)$$

is obtained from the standard open–closed triangle. For $Q^{*}\in P\left(X\right)$, one sets

$$\mu \left(Q^{*}\right)=\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right)\right),$$

together with the exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{!}{Q}^{*}\right)⟶H^{-d}\left(i^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

This is the classical construction lifted in this paper.

Remark 1.1 (Relation with the isolated singularity notation). 

In the isolated singularity case, the same construction is often written with an object

$$\Theta =(L,K,C,\alpha ,\beta ,\gamma )$$

and exact sequence

$$H^{n-1}\left(i^{*}{j}_{*}L\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^n\left(i^{*}{j}_{*}L\right).$$

This is the form used in applications to simple stratified spaces with one isolated singularity (see [2] as an example). The notation in this paper follows formalism developed in [1] for a general two-stratum construction. The isolated singularity notation is recovered by the corresponding shift convention for the middle degree.

1.2. Why a Nori Motivic Version Is Needed

A practical reason for using perverse sheaves is that they make local-to-global structure visible. In a two-stratum configuration, the construction of [1] does not merely assign an invariant to a perverse sheaf; it extracts a diagrammatic package from which one can read the open term, the closed costalk term, the closed stalk term, and the boundary maps connecting them. Thus the zig-zag functor is a way of seeing where a perverse sheaf lives, how it is attached across a stratum, and which commutative diagrams are forced by functoriality and exactness.

For

$$Q^{*}\in P\left(X\right),$$

this package is

$$\mu \left(Q^{*}\right)=\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

with exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

The maps in this sequence are not auxiliary decorations. They are the structural maps that record how the open and closed pieces interact.

The same kind of package is needed in motivic arguments. Perverse Nori motives provide an abelian motivic category whose realization is perverse-sheaf-theoretic, but many standard perverse-sheaf tools have not yet been explicitly written in this language. As the category becomes more useful, it is important to have direct motivic versions of constructions that, in the classical category, allow one to read off objects, maps, exact sequences, and commutative diagrams before passing to realization.

This paper supplies one such tool. We construct a motivic zig-zag category

$$Z_{\mathrm{Nori}}(X,S)$$

and a functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S)$$

by using the same objects, maps, exact sequences, and morphism diagrams as in [1], interpreted inside the perverse Nori motivic category. The resulting motivic zig-zag object records not only the terms

$$j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),$$

but also the boundary sequence and the commutative diagrams induced by morphisms. This gives a controlled way to package motivic open data, motivic closed data, and the maps between them in a form that can later be transported, compared, or used as input for residual-channel constructions.

The construction is intentionally direct. We do not introduce a different gluing formalism, and we do not replace the exact sequence above by an auxiliary presentation. The point is to erect the zig-zag category and zig-zag functor inside PNM, and then prove that realization recovers the classical construction:

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

The nontrivial point is not the formal act of naming a diagram category. The point is that the maps in the zig-zag sequence are produced motivically. Starting from $Q^{*}\in PNM\left(X\right)$, the open–closed triangle in the perverse Nori derived formalism gives the boundary sequence after applying $i^{*}$ and taking the relevant heart-level cohomology objects. Thus the maps ${\alpha }_Q,{\beta }_Q,{\gamma }_Q$ and the commutative diagrams induced by morphisms are not lifted after realization; they are formed first in $PNM\left(S\right)$, and realization identifies them with their classical counterparts.

1.3. Perverse Nori Infrastructure Used Here

We use the recent perverse Nori motivic literature as structural input. Ivorra and Morel construct categories of perverse Nori motives and develop the basic functorial formalism for perverse motives [3,4]. Their work supplies the motivic perverse categories and operations needed to speak about perverse Nori analogues of classical perverse-sheaf constructions.

Tubach proves derived-category comparison results for perverse Nori motives and mixed Hodge modules and constructs operation-compatible realization functors from Voevodsky motives to derived categories of perverse Nori motives and mixed Hodge modules [5]. This provides the realization-compatible background used here to compare the Nori zig-zag construction with the classical one.

Terenzi develops tensor and internal-Hom structures for perverse Nori motives and gives further evidence that the perverse Nori framework supports a robust functorial formalism [6]. These monoidal structures are not used directly in the construction below; the zig-zag functor only requires the four operations appearing in the open–closed sequence and their compatibility with realization.

Recent work of Pham uses perverse Nori motives in the construction of a Nori-motivic Satake equivalence [7]. This illustrates that perverse Nori motives are now being used as a serious working category in geometric representation theory. The present paper is more elementary: it isolates the zig-zag construction [1] and makes it available in this same motivic environment.

We use this literature only to the extent needed to justify the following operations and compatibilities:

$$j^{*},i^{*},i^{!},R{j}_{*},$$

the corresponding cohomology objects

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right),H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right),H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),$$

and the compatibility of these constructions with realization.

1.4. The Nori Zig-Zag Category

The motivic zig-zag category constructed in this paper is denoted

$$Z_{\mathrm{Nori}}(X,S).$$

It is defined by the same pattern as $Z(X,S)$. An object is a triple

$$(P^{*},K,C)$$

where

$$P^{*}\in PNM(X-S),K,C\in PNM\left(S\right),$$

together with an exact sequence in $PNM\left(S\right)$

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)⟶K⟶C⟶H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right).$$

A morphism is a morphism of the open terms together with maps between the two middle closed terms, making the induced exact-sequence diagram commute. Thus $Z_{\mathrm{Nori}}(X,S)$ is not an abstract substitute for $Z(X,S)$; it is the same zig-zag category formed inside the perverse Nori motivic category.

1.5. The Nori Zig-Zag Functor

For $Q^{*}\in PNM\left(X\right)$, define ${\mu }_{\mathrm{Nori}}\left(Q^{*}\right)$ to be the object of $Z_{\mathrm{Nori}}(X,S)$ whose open term is $j^{*}{Q}^{*}$ and whose closed terms are

$$H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right).$$

The maps are those appearing in the exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{!}{Q}^{*}\right)⟶H^{-d}\left(i^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

(1)

This sequence is the Nori motivic analogue of the classical zig-zag sequence defining $\mu \left(Q^{*}\right)$. On morphisms, ${\mu }_{\mathrm{Nori}}$ is defined by applying $j^{*}$, $i^{!}$, $i^{*}$, and $i^{*}R{j}_{*}{j}^{*}$ to the given morphism in $PNM\left(X\right)$. Functoriality follows from the functoriality of the open–closed triangle and of the associated long exact sequence in the perverse Nori setting.

1.6. Realization Compatibility

The realization functors

$${Real}_X:PNM\left(X\right)⟶P\left(X\right),$$

$${Real}_{X-S}:PNM(X-S)⟶P(X-S),$$

and

$${Real}_S:PNM\left(S\right)⟶P\left(S\right)$$

induce a realization functor on zig-zag categories

$${Real}_Z:Z_{\mathrm{Nori}}(X,S)⟶Z(X,S).$$

This functor sends the motivic object

$$(P^{*},K,C)$$

and its exact sequence to the realized classical object

$$\left({Real}_{X-S}\left(P^{*}\right),{Real}_S\left(K\right),{Real}_S\left(C\right)\right)$$

with the realized exact sequence. The main compatibility statement is the commutative square

More precisely, there is a natural isomorphism

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

This is the central result of the paper.

1.7. Main Theorem

The main theorem can be stated as follows.

Theorem 1.2

(Zig-zag functor for perverse Nori motives). Let $X=U\bigsqcup S$ be a two-stratum complex algebraic variety with $j:U\hookrightarrow X$ open and $i:S\hookrightarrow X$ closed, in the perverse Nori setting of Proposition 4.6. Then there is a zig-zag category

$$Z_{\mathrm{Nori}}(X,S)$$

and a functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S)$$

defined by the exact sequence modeled in $\S 2$ of [1]

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{!}{Q}^{*}\right)⟶H^{-d}\left(i^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

Moreover, realization induces a functor

$${Real}_Z:Z_{\mathrm{Nori}}(X,S)⟶Z(X,S)$$

and there is a natural isomorphism

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

The theorem says that the zig-zag construction is available in the perverse Nori motivic category and is compatible with classical realization.

1.8. Scope and Limitations

This paper relies on the availability of the operations $j^{*}$, $R{j}_{*}$, $i^{*}$, $i^{!}$, the relevant heart-level cohomology objects, and realization compatibility for these constructions. These are supplied by the perverse Nori four-operation formalism of Ivorra–Morel and the realization/comparison results of Tubach. Terenzi and Pham are used as contextual evidence for the broader development of the perverse Nori framework.

The paper does not assert essential surjectivity of realization, canonical perverse Nori lifts of classical perverse sheaves, or any Chow-theoretic, Hodge-theoretic, or rational Hodge conjecture consequence.

1.9. Organization

Section 2 fixes notation. Section 3 recalls the zig-zag category $Z(X,S)$, its objects, its morphisms, and the functor $\mu :P\left(X\right)\to Z(X,S)$. Section 4 records the perverse Nori operations and realization compatibilities used in the construction. Section 5 defines $Z_{\mathrm{Nori}}(X,S)$. Section 6 constructs ${\mu }_{\mathrm{Nori}}$. Section 7 proves the realization compatibility square. Section 8 records filtered iteration and explains the limits of the construction. Section 9 collects the main statements, and Section 10 concludes.

2. Notation and Conventions

2.1. Spaces, Strata, and Inclusions

All varieties are complex algebraic varieties unless otherwise stated. A two-stratum decomposition means

$$X=U\bigsqcup S,$$

where $U\subset X$ is Zariski open and $S=X\setminus U$ is closed. We write

$$j:U\hookrightarrow X,i:S\hookrightarrow X$$

for the open and closed immersions.

When a finite stratification is used, it is denoted

$$\mathcal{S}={\left\{S_{\lambda }\right\}}_{\lambda \in \Lambda },X=\underset{\lambda \in \Lambda }{⨆}{S}_{\lambda }.$$

The stratification is finite and algebraic. When constructibility or analytic arguments are involved, we work with a compatible Whitney stratification [8,9,10].

The main construction is two-stratum. A finite-stratification version is obtained only by choosing a filtration by closed unions of strata and iterating the two-stratum construction. No independence from the chosen filtration is asserted.

Remark 2.1 (Base field convention). 

For readability, the paper is written over $\mathbf{C}$. One may also work over a characteristic-zero field k equipped with an embedding $\sigma :k\hookrightarrow \mathbf{C}$, as in the perverse Nori literature. In that setting, $P\left(X\right)$ denotes the perverse category after passage to the associated complex analytic space. No arithmetic descent statement is used.

2.2. Classical Perverse Categories

We write

$$P\left(X\right):=Perv(X,\mathbf{Q})$$

for the abelian category of rational perverse sheaves on the complex analytic space associated with X, constructible with respect to the chosen stratification. Similarly,

$$P\left(U\right):=Perv(U,\mathbf{Q}),P\left(S\right):=Perv(S,\mathbf{Q}).$$

The standard functors associated with

$$j:U\hookrightarrow X,i:S\hookrightarrow X$$

are denoted

$$j^{*},R{j}_{*},R{j}_{!},i^{*},i^{!},i_{*}.$$

When no confusion can arise, we write $j_{*}$ for $R{j}_{*}$ and $j_{!}$ for $R{j}_{!}$. Cohomology objects are denoted $H^k(-)$, following the convention used in the zig-zag construction of [1]. Thus the terms

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right),H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

are the two boundary terms appearing in the zig-zag sequence.

2.3. The Zig-Zag Convention

The phrase zig-zag construction refers throughout to the construction of Section 2 of [1]. Let $S\subset X$ be a closed stratum of complex dimension d, with

$$j:X-S\hookrightarrow X,i:S\hookrightarrow X.$$

The classical zig-zag category is denoted

$$Z(X,S).$$

An object is written

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma ),$$

where

$$P^{*}\in P(X-S),K,C\in P\left(S\right),$$

and

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

is exact in $P\left(S\right)$.

A morphism

$$\Theta ⟶\overline{\Theta }$$

is a morphism $P^{*}\to {\overline{P}}^{*}$ in $P(X-S)$, together with morphisms $K\to \overline{K}$ and $C\to \overline{C}$, such that the induced diagram of exact sequences commutes.

The classical zig-zag functor is denoted

$$\mu :P\left(X\right)⟶Z(X,S).$$

For $Q^{*}\in P\left(X\right)$,

$$\mu \left(Q^{*}\right)=\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

where

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)$$

is the exact sequence associated with the open–closed triangle.

Remark 2.2 (No auxiliary presentation). 

This paper uses the zig-zag category $Z(X,S)$, its objects, its morphisms, and the functor $\mu$ in the form above. It does not formulate the construction through an auxiliary $F,G,T$ presentation, a kernel–cokernel presentation, or a nearby–vanishing-cycle presentation.

2.4. Isolated Singularity Notation

In the isolated singularity case, as in [2], the same construction is often written

$$\Theta =(L,K,C,\alpha ,\beta ,\gamma ),$$

where L is a local system on the nonsingular part $Y^o$, with exact sequence

$$H^{n-1}\left(i^{*}{j}_{*}L\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^n\left(i^{*}{j}_{*}L\right).$$

This is the one-point version of the two-stratum notation used here. The passage between the two conventions is a shift and middle-degree normalization.

2.5. Perverse Nori notation

We write

$$PNM\left(X\right)$$

for the heart-level category of perverse Nori motives on X. This notation corresponds to the categories denoted in the literature by variants such as

$${\mathrm{M}}^{\mathrm{perv}}\left(X\right),\mathrm{M}\left(X\right),\mathrm{MPerv}\left(X\right).$$

Similarly, $D^{b}PNM\left(X\right)$ denotes the bounded derived category of $PNM\left(X\right)$.

We use the notation

$$PNM\left(U\right),PNM\left(S\right)$$

for the corresponding categories on the open and closed strata. The realization functors are denoted

$${Real}_X:PNM\left(X\right)\to P\left(X\right),$$

$${Real}_U:PNM\left(U\right)\to P\left(U\right),{Real}_S:PNM\left(S\right)\to P\left(S\right).$$

When no ambiguity is possible, the subscript is omitted.

The formal input comes primarily from the perverse Nori motive framework of Ivorra–Morel and the realization and derived-category comparison results of Tubach. Terenzi’s tensor and internal-Hom structures, and Pham’s stratified applications, provide surrounding context for the current development of the perverse Nori framework [3,4,5,6,7]. We use this literature only to form the functors and cohomology objects appearing in the zig-zag sequence and to compare them with classical realization.

2.6. Derived and Cohomological Notation

The construction is heart-level, but some terms are obtained by applying derived functors and then taking cohomology. For an object $A\in D^{b}PNM\left(S\right)$, we write

$$H^k\left(A\right)$$

for the corresponding heart-level cohomology object. Thus, for $P^{*}\in PNM(X-S)$, the expressions

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right),H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

are objects of $PNM\left(S\right)$, whenever the two-stratum pair is admissible in the sense used below. For $Q^{*}\in PNM\left(X\right)$, the expressions

$$H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right)$$

are the two closed terms appearing in the motivic zig-zag functor.

Realization is assumed to commute with the operations used in the construction:

$$j^{*},i^{*},i^{!},R{j}_{*},$$

and with the relevant cohomology objects. This compatibility is the formal input which allows the motivic zig-zag sequence to realize to the classical one.

2.7. The Nori Zig-Zag Category and Functor

The perverse Nori analogue of $Z(X,S)$ is denoted

$$Z_{\mathrm{Nori}}(X,S).$$

An object is written

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma ),$$

where

$$P^{*}\in PNM(X-S),K,C\in PNM\left(S\right),$$

and

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

is exact in $PNM\left(S\right)$. Morphisms are defined by the same commutative exact-sequence diagrams as in $Z(X,S)$.

The motivic zig-zag functor is denoted

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S).$$

For $Q^{*}\in PNM\left(X\right)$,

$${\mu }_{\mathrm{Nori}}\left(Q^{*}\right)=\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

where the maps are those of the exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

2.8. Realization of Zig-Zag Data

Realization induces a functor

$${Real}_Z:Z_{\mathrm{Nori}}(X,S)⟶Z(X,S).$$

For

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma ),$$

one sets

$${Real}_Z(\Theta )=\left({Real}_{X-S}\left(P^{*}\right),{Real}_S\left(K\right),{Real}_S\left(C\right),Real\left(\alpha \right),Real\left(\beta \right),Real\left(\gamma \right)\right),$$

using the natural identifications

$${Real}_S{H}^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\cong H^{-d-1}\left(i^{*}R{j}_{*}{Real}_{X-S}\left(P^{*}\right)\right),$$

and

$${Real}_S{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)\cong H^{-d}\left(i^{*}R{j}_{*}{Real}_{X-S}\left(P^{*}\right)\right).$$

The main compatibility proved in the paper is

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

2.9. Conventions on Claims

All statements in this paper concern the construction of the zig-zag category and zig-zag functor in the perverse Nori motivic setting, together with realization compatibility.

The paper does not assert that every classical perverse sheaf has a canonical perverse Nori lift. It does not prove that every object of $Z(X,S)$ lifts to an object of $Z_{\mathrm{Nori}}(X,S)$. It does not prove a Chow-theoretic, Hodge-theoretic, or rational Hodge conjecture statement. Such applications require additional input beyond the present note.

3. Classical Zig-Zag Construction

We use the notation of Section 2. This section recalls the classical zig-zag category and zig-zag functor of [1] in the form used throughout the paper. The construction is used only in its concrete form: objects, morphisms, and the functor

$$\mu :P\left(X\right)⟶Z(X,S).$$

No auxiliary $F,G,T$, kernel–cokernel, or nearby-cycle presentation is used.

3.1. The Category $Z(X,S)$

Let

$$X=U\bigsqcup S$$

be a two-stratum complex algebraic variety, with

$$j:U=X-S\hookrightarrow X,i:S\hookrightarrow X.$$

Let $d={dim}_{\mathbf{C}}S$. The category $Z(X,S)$ records open-stratum data, two closed-stratum terms, and the exact sequence connecting them.

Definition 3.1

(Classical zig-zag category). An object of $Z(X,S)$ is a sextuple

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma ),$$

where

$$P^{*}\in P(X-S),K,C\in P\left(S\right),$$

and

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

is exact in $P\left(S\right)$.

A morphism

$$\Theta ⟶\overline{\Theta }$$

from

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma )$$

to

$$\overline{\Theta }=({\overline{P}}^{*},\overline{K},\overline{C},\overline{\alpha },\overline{\beta },\overline{\gamma })$$

is a morphism

$$p:P^{*}\to {\overline{P}}^{*}$$

in $P(X-S)$, together with morphisms

$$k:K\to \overline{K},c:C\to \overline{C}$$

in $P\left(S\right)$, such that the diagram

commutes.

Lemma 3.2.

The objects and morphisms of Definition 3.1 form a category.

Proof. 

Composition is componentwise. The outer terms are functorial in $P^{*}$, and therefore the commutativity of the defining diagrams is preserved under composition. Identity morphisms are the identity maps on $P^{*}$, K, and C. Associativity follows from associativity in $P(X-S)$ and $P\left(S\right)$. □

3.2. The Zig-Zag Functor

The classical zig-zag functor is

$$\mu :P\left(X\right)⟶Z(X,S).$$

For $Q^{*}\in P\left(X\right)$, it is defined by

$$\mu \left(Q^{*}\right)=\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

where

$${\alpha }_Q,{\beta }_Q,{\gamma }_Q$$

are the maps in the exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

This sequence is obtained from the open–closed distinguished triangle

$$i_{*}{i}^{!}{Q}^{*}⟶Q^{*}⟶R{j}_{*}{j}^{*}{Q}^{*}\stackrel{+1}{\to }$$

after applying $i^{*}$ and taking the indicated cohomology objects.

On morphisms, $\mu$ is defined by applying

$$j^{*},i^{!},i^{*},i^{*}R{j}_{*}{j}^{*}$$

to a morphism in $P\left(X\right)$. Naturality of the open–closed triangle and of the associated long exact sequence gives the required commutative diagram of zig-zag sequences.

Theorem 3.3

([1]). The functor

$$\mu :P\left(X\right)⟶Z(X,S)$$

is the zig-zag functor of [1]. It has the reconstruction properties proved there: the zig-zag datum records enough information to recover the corresponding perverse-sheaf object in the sense of that theorem.

Remark 3.4 (Use of the classical theorem). 

The present paper does not reprove the classical theorem. It uses the explicit category $Z(X,S)$, its morphism diagrams, and the functor $\mu$ as the model to be internalized in perverse Nori motives.

3.3. Isolated Singularity Form

In the case of a simple stratified space

$$Y=Y^o\bigsqcup \left\{y\right\},$$

the same construction is commonly written as

$$\Theta =(L,K,C,\alpha ,\beta ,\gamma ),$$

where $L\in P\left(Y^o\right)$ and

$$H^{n-1}\left(i^{*}{j}_{*}L\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^n\left(i^{*}{j}_{*}L\right)$$

is exact. The corresponding functor sends $Q^{*}\in P\left(Y\right)$ to

$$\left(j^{*}{Q}^{*},H^n\left(i^{!}{Q}^{*}\right),H^n\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

with exact sequence

$$H^{n-1}\left(i^{*}{j}_{*}{j}^{*}{Q}^{*}\right)\to H^n\left(i^{!}{Q}^{*}\right)\to H^n\left(i^{*}{Q}^{*}\right)\to H^n\left(i^{*}{j}_{*}{j}^{*}{Q}^{*}\right).$$

This is the one-point notation used in isolated singularity applications, including [2]. The present paper uses the general two-stratum notation, and the Nori construction below specializes to this isolated singularity form.

4. Perverse Nori Input

This section records the motivic input needed to form the zig-zag category

$$Z_{\mathrm{Nori}}(X,S)$$

and the functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S).$$

We do not reconstruct Nori motives or perverse Nori motives from first principles. We use the existing perverse Nori formalism only to the extent needed to form the terms, maps, exact sequences, and realization comparisons appearing in the zig-zag construction.

4.1. Perverse Nori Motivic Categories

For a complex algebraic variety X, we write

$$PNM\left(X\right)$$

for the heart-level category of perverse Nori motives on X. The realization functor is denoted

$${Real}_X:PNM\left(X\right)⟶P\left(X\right).$$

For a two-stratum decomposition $X=U\bigsqcup S$, we similarly write

$${Real}_U:PNM\left(U\right)⟶P\left(U\right),{Real}_S:PNM\left(S\right)⟶P\left(S\right).$$

The categories $PNM(-)$ and their functorial operations are used in the sense of the perverse Nori motivic formalism. The required input comes primarily from Ivorra–Morel and Tubach; Terenzi and Pham provide surrounding context.

Remark 4.1 (No automatic lifting statement). 

The paper does not assert that every classical perverse sheaf has a canonical perverse Nori lift. The construction begins with an object of $PNM\left(X\right)$, forms its motivic zig-zag data, and compares that data with the classical zig-zag data of its realization.

4.2. Operations Used

Let

$$X=U\bigsqcup S,j:U\hookrightarrow X,i:S\hookrightarrow X.$$

The zig-zag construction uses the operations

$$j^{*},R{j}_{*},i^{*},i^{!}.$$

In the perverse Nori setting, we require the corresponding operations to exist on the relevant derived categories and the cohomology objects appearing below to lie in the perverse Nori hearts.

For

$$P^{*}\in PNM(X-S),$$

the boundary terms

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right),H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

are required to be objects of $PNM\left(S\right)$. For

$$Q^{*}\in PNM\left(X\right),$$

the closed terms

$$H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right)$$

are required to be objects of $PNM\left(S\right)$, and the sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{!}{Q}^{*}\right)⟶H^{-d}\left(i^{*}{Q}^{*}\right)⟶H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)$$

is required to be exact in $PNM\left(S\right)$.

Definition 4.2

(MV-admissible perverse Nori two-stratum pair). A two-stratum pair

$$X=U\bigsqcup S$$

is called MV-admissible in perverse Nori motives if the perverse Nori formalism supplies the operations, heart-level cohomology objects, and exact sequence listed above.

Remark 4.3 (Meaning of admissibility). 

The admissibility condition is not a gluing or reconstruction theorem. It is the minimal condition needed to write down the zig-zag object in the perverse Nori heart. In the applications considered here, this condition is supplied by the available perverse Nori six-functor and realization formalism.

4.3. Realization Compatibility

The realization functors are required to commute with the operations used in the construction. Thus we use natural isomorphisms

$${Real}_U\circ j^{*}\cong j^{*}\circ {Real}_X,$$

$${Real}_S\circ i^{*}\cong i^{*}\circ {Real}_X,{Real}_S\circ i^{!}\cong i^{!}\circ {Real}_X,$$

and

$${Real}_X\circ R{j}_{*}\cong R{j}_{*}\circ {Real}_U.$$

After applying $i^{*}$ and taking the relevant cohomology objects, these give identifications

$${Real}_S{H}^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\cong H^{-d-1}\left(i^{*}R{j}_{*}{Real}_U\left(P^{*}\right)\right),$$

and

$${Real}_S{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)\cong H^{-d}\left(i^{*}R{j}_{*}{Real}_U\left(P^{*}\right)\right).$$

Similarly, for $Q^{*}\in PNM\left(X\right)$, realization identifies the motivic exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\to H^{-d}\left(i^{!}{Q}^{*}\right)\to H^{-d}\left(i^{*}{Q}^{*}\right)\to H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)$$

with the classical zig-zag sequence associated to ${Real}_X\left(Q^{*}\right)$.

Definition 4.4

(Realization-compatible MV-admissibility). An MV-admissible perverse Nori two-stratum pair is realization-compatible if realization commutes with $j^{*}$, $i^{*}$, $i^{!}$, $R{j}_{*}$, the relevant cohomology objects, and the exact sequence defining the zig-zag functor.

Remark 4.5 (Use of the modern Nori literature). 

The role of the modern perverse Nori literature is to provide the functorial and realization-compatible environment in which Definitions 4.2 and 4.4 can be verified. The paper uses only the specific operations that occur in the zig-zag sequence.

Proposition 4.6

(Availability of the Nori zig-zag input). For the two-stratum algebraic pairs considered in this paper, the perverse Nori formalism supplies the operations

$$j^{*},R{j}_{*},i^{*},i^{!},$$

the heart-level cohomology objects appearing in the zig-zag sequence, and the realization compatibilities required in Definitions 4.2 and 4.4. Consequently, the pair $X=U\bigsqcup S$ is MV-admissible and realization-compatible in the sense used below.

Proof. 

The four operations on perverse Nori motives and the relevant perverse heart formalism are supplied by Ivorra–Morel [4]. The operation-compatible realization and comparison with the classical constructible and Hodge-theoretic formalisms are supplied by Tubach [5]. Applying these results to the open immersion j and closed immersion i gives the operations $j^{*},R{j}_{*},i^{*},i^{!}$ in the perverse Nori setting and identifies their realizations with the corresponding classical functors. Taking the indicated heart-level cohomology objects gives the terms appearing in the zig-zag sequence, and exactness of the resulting sequence is the heart-level long exact sequence associated with the open–closed triangle. The realization compatibilities identify this sequence with the classical sequence used to define $\mu$. □

4.4. Exactness

The category $Z_{\mathrm{Nori}}(X,S)$ is an abelian heart-level construction. Exactness is therefore always exactness in

$$PNM\left(S\right).$$

In particular, an object of $Z_{\mathrm{Nori}}(X,S)$ contains an exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right),$$

and the sequence defining ${\mu }_{\mathrm{Nori}}\left(Q^{*}\right)$ is also required to be exact in $PNM\left(S\right)$.

The realization functor is exact on the relevant hearts. Hence exact sequences in $PNM\left(S\right)$ realize to exact sequences in $P\left(S\right)$.

Remark 4.7 (Role of exactness). 

Exactness is the abelian input that makes the zig-zag category and its morphism diagrams meaningful. No Chow groups, cycle-class maps, Hodge classes, or rational Hodge conjecture statements enter this construction.

5. The Nori Zig-Zag Category

We now define the perverse Nori motivic analogue of the zig-zag category $Z(X,S)$. The definition is obtained by replacing the classical perverse categories

$$P(X-S),P\left(S\right)$$

with

$$PNM(X-S),PNM\left(S\right),$$

and by using the same exact-sequence data as in [1].

5.1. Objects and Morphisms

Definition 5.1

(Nori zig-zag category). Let $X=U\bigsqcup S$ be a realization-compatible MV-admissible perverse Nori two-stratum pair. The category

$$Z_{\mathrm{Nori}}(X,S)$$

is defined as follows.

An object is a sextuple

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma ),$$

where

$$P^{*}\in PNM(X-S),K,C\in PNM\left(S\right),$$

and

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

is exact in $PNM\left(S\right)$.

A morphism

$$\Theta ⟶\overline{\Theta }$$

from

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma )$$

to

$$\overline{\Theta }=({\overline{P}}^{*},\overline{K},\overline{C},\overline{\alpha },\overline{\beta },\overline{\gamma })$$

is a triple

$$(p,k,c),$$

where

$$p:P^{*}\to {\overline{P}}^{*}\mathrm{in}PNM(X-S),$$

and

$$k:K\to \overline{K},c:C\to \overline{C}\mathrm{in}PNM\left(S\right),$$

such that the diagram 

commutes.

Proposition 5.2

(Category structure). The objects and morphisms of Definition 5.1 form a category.

Proof. 

Composition is defined componentwise. Given morphisms

$$(p,k,c):\Theta \to \overline{\Theta }$$

and

$$(\overline{p},\overline{k},\overline{c}):\overline{\Theta }\to \widehat{\Theta },$$

their composite is

$$(\overline{p}p,\overline{k}k,\overline{c}c).$$

The outer terms in the exact-sequence diagram are functorial in the open term $P^{*}$, so the commutativity of the two defining diagrams implies the commutativity of the composite diagram. Identity morphisms are the identities on $P^{*}$, K, and C, and associativity follows from associativity in $PNM(X-S)$ and $PNM\left(S\right)$. □

5.2. Realization of Nori Zig-Zag Objects

The realization functors induce a functor

$${Real}_Z:Z_{\mathrm{Nori}}(X,S)⟶Z(X,S).$$

For an object

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma )\in Z_{\mathrm{Nori}}(X,S),$$

define

$${Real}_Z(\Theta )=\left({Real}_{X-S}\left(P^{*}\right),{Real}_S\left(K\right),{Real}_S\left(C\right),Real\left(\alpha \right),Real\left(\beta \right),Real\left(\gamma \right)\right),$$

using the natural identifications

$${Real}_S{H}^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\cong H^{-d-1}\left(i^{*}R{j}_{*}{Real}_{X-S}\left(P^{*}\right)\right)$$

and

$${Real}_S{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)\cong H^{-d}\left(i^{*}R{j}_{*}{Real}_{X-S}\left(P^{*}\right)\right).$$

Proposition 5.3

(Realization of zig-zag data). The assignment above defines a functor

$${Real}_Z:Z_{\mathrm{Nori}}(X,S)⟶Z(X,S).$$

Proof. 

Let

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma )$$

be an object of $Z_{\mathrm{Nori}}(X,S)$. Its defining sequence is exact in $PNM\left(S\right)$. Since realization is exact on the relevant heart and commutes with the operations used to form the boundary terms, the realized sequence is exact in $P\left(S\right)$. Hence ${Real}_Z(\Theta )$ is an object of $Z(X,S)$.

Now let

$$(p,k,c):\Theta \to \overline{\Theta }$$

be a morphism. Applying realization to $p,k,c$ gives morphisms

$${Real}_{X-S}\left(p\right),{Real}_S\left(k\right),{Real}_S\left(c\right).$$

By realization compatibility, the outer vertical maps in the defining diagram realize to the corresponding classical maps induced by ${Real}_{X-S}\left(p\right)$. Since the motivic diagram commutes, the realized diagram commutes. Thus realization sends morphisms of Nori zig-zag objects to morphisms of classical zig-zag objects. Preservation of identities and composition follows from functoriality of realization. □

Remark 5.4 (Faithfulness to the classical construction). 

The category $Z_{\mathrm{Nori}}(X,S)$ is not a new auxiliary diagram category. It is the zig-zag category of [1] formed inside the perverse Nori motivic category: the objects, morphisms, and exact-sequence diagrams are the same, while the ambient abelian categories are replaced by $PNM(X-S)$ and $PNM\left(S\right)$.

6. The Nori Zig-Zag Functor

We use the notation and conventions of Section 2. Let

$$X=U\bigsqcup S$$

be a realization-compatible MV-admissible perverse Nori two-stratum pair. In Section 5 we defined the category

$$Z_{\mathrm{Nori}}(X,S).$$

We now construct the motivic zig-zag functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S).$$

6.1. Definition on Objects

Let

$$Q^{*}\in PNM\left(X\right).$$

The open–closed triangle

$$i_{*}{i}^{!}{Q}^{*}⟶Q^{*}⟶R{j}_{*}{j}^{*}{Q}^{*}\stackrel{+1}{\to }$$

induces, after applying $i^{*}$ and taking the relevant cohomology objects, an exact sequence in $PNM\left(S\right)$:

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

Define

$${\mu }_{\mathrm{Nori}}\left(Q^{*}\right):=\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right).$$

Proposition 6.1

(Well-definedness on objects). For every

$$Q^{*}\in PNM\left(X\right),$$

the datum

$${\mu }_{\mathrm{Nori}}\left(Q^{*}\right)$$

is an object of $Z_{\mathrm{Nori}}(X,S)$.

Proof. 

By MV-admissibility in the perverse Nori setting, the objects

$$j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right)$$

belong to the required hearts, the boundary terms

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right),H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)$$

belong to $PNM\left(S\right)$, and the displayed sequence is exact in $PNM\left(S\right)$. Hence the datum satisfies the defining condition for an object of $Z_{\mathrm{Nori}}(X,S)$. □

6.2. Definition on Morphisms

Let

$$f:Q^{*}⟶{\overline{Q}}^{*}$$

be a morphism in $PNM\left(X\right)$. Applying

$$j^{*},i^{!},i^{*},i^{*}R{j}_{*}{j}^{*}$$

to f, and then taking the relevant cohomology objects, gives morphisms between the corresponding terms of the zig-zag sequence. Naturality of the open–closed triangle and of the associated long exact sequence gives a commutative diagram

Together with the open-term morphism

$$j^{*}f:j^{*}{Q}^{*}\to j^{*}{\overline{Q}}^{*},$$

this defines a morphism

$${\mu }_{\mathrm{Nori}}\left(f\right):{\mu }_{\mathrm{Nori}}\left(Q^{*}\right)⟶{\mu }_{\mathrm{Nori}}\left({\overline{Q}}^{*}\right)$$

in $Z_{\mathrm{Nori}}(X,S)$.

Theorem 6.2

(The Nori zig-zag functor). The assignments above define a functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S).$$

Proof. 

The assignment on objects is defined above. The assignment on morphisms is induced by functoriality of the operations

$$j^{*},i^{!},i^{*},i^{*}R{j}_{*}{j}^{*}$$

and by naturality of the long exact cohomology sequence. Identity morphisms and compositions are preserved because all operations used in the construction are functorial. Hence ${\mu }_{\mathrm{Nori}}$ is a functor. □

6.3. Isolated Singularity Form

In the isolated singularity normalization, the same functor is written

$${\mu }_{\mathrm{Nori}}\left(Q^{*}\right)=\left(j^{*}{Q}^{*},H^n\left(i^{!}{Q}^{*}\right),H^n\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

with exact sequence

$$H^{n-1}\left(i^{*}{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^n\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^n\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^n\left(i^{*}{j}_{*}{j}^{*}{Q}^{*}\right).$$

This is the one-point simple-stratification form of the construction; the paper otherwise uses the general two-stratum notation.

7. Realization Compatibility

We now prove that the Nori zig-zag functor realizes to the classical zig-zag functor. Throughout this section, $X=U\bigsqcup S$ is a realization-compatible MV-admissible perverse Nori two-stratum pair.

7.1. Realization of the Defining Sequence

Let

$$Q^{*}\in PNM\left(X\right).$$

The defining sequence of ${\mu }_{\mathrm{Nori}}\left(Q^{*}\right)$ is

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

By realization compatibility, applying ${Real}_S$ identifies this sequence with

$$\begin{array}{ccc}\hfill H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Real}_X\left(Q^{*}\right)\right)& ⟶\hfill & \hfill H^{-d}\left(i^{!}{Real}_X\left(Q^{*}\right)\right)\\ & & \hfill ⟶& H^{-d}\left(i^{*}{Real}_X\left(Q^{*}\right)\right)\hfill \\ & & \hfill ⟶& H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Real}_X\left(Q^{*}\right)\right).\hfill \end{array}$$

This is precisely the exact sequence defining the classical zig-zag object

$$\mu \left({Real}_X\left(Q^{*}\right)\right).$$

Lemma 7.1

(Realization of Nori zig-zag objects). For every

$$Q^{*}\in PNM\left(X\right),$$

there is a natural isomorphism in $Z(X,S)$

$${Real}_Z\left({\mu }_{\mathrm{Nori}}\left(Q^{*}\right)\right)\cong \mu \left({Real}_X\left(Q^{*}\right)\right).$$

Proof. 

The open term realizes as

$${Real}_U\left(j^{*}{Q}^{*}\right)\cong j^{*}{Real}_X\left(Q^{*}\right).$$

The two closed middle terms realize as

$${Real}_S{H}^{-d}\left(i^{!}{Q}^{*}\right)\cong H^{-d}\left(i^{!}{Real}_X\left(Q^{*}\right)\right),$$

and

$${Real}_S{H}^{-d}\left(i^{*}{Q}^{*}\right)\cong H^{-d}\left(i^{*}{Real}_X\left(Q^{*}\right)\right).$$

The two boundary terms realize as

$${Real}_S{H}^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\cong H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Real}_X\left(Q^{*}\right)\right),$$

and

$${Real}_S{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\cong H^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Real}_X\left(Q^{*}\right)\right).$$

Under these identifications, the realized maps

$$Real\left({\alpha }_Q\right),Real\left({\beta }_Q\right),Real\left({\gamma }_Q\right)$$

are the maps in the classical zig-zag sequence for ${Real}_X\left(Q^{*}\right)$. Hence the realized Nori zig-zag object is naturally isomorphic to the classical one. □

7.2. The Realization Square

The compatibility is expressed by the square

Theorem 7.2

(Realization compatibility of the Nori zig-zag functor). There is a natural isomorphism of functors

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

Proof. 

On objects, the required isomorphism is Lemma 7.1. It remains to check naturality.

Let

$$f:Q^{*}\to {\overline{Q}}^{*}$$

be a morphism in $PNM\left(X\right)$. The morphism

$${\mu }_{\mathrm{Nori}}\left(f\right):{\mu }_{\mathrm{Nori}}\left(Q^{*}\right)\to {\mu }_{\mathrm{Nori}}\left({\overline{Q}}^{*}\right)$$

is obtained by applying

$$j^{*},i^{!},i^{*},i^{*}R{j}_{*}{j}^{*}$$

to f, followed by the relevant cohomology functors. Since realization commutes with these operations, the realized morphism

$${Real}_Z\left({\mu }_{\mathrm{Nori}}\left(f\right)\right)$$

is identified with the classical morphism of zig-zag data induced by

$${Real}_X\left(f\right):{Real}_X\left(Q^{*}\right)\to {Real}_X\left({\overline{Q}}^{*}\right).$$

This is precisely

$$\mu \left({Real}_X\left(f\right)\right).$$

Thus the objectwise isomorphisms are natural, and the two composite functors are naturally isomorphic. □

Corollary 7.3

(Realization preserves the zig-zag diagram). For every morphism

$$f:Q^{*}\to {\overline{Q}}^{*}$$

in $PNM\left(X\right)$, the commutative diagram of exact sequences defining

$${\mu }_{\mathrm{Nori}}\left(f\right)$$

realizes to the commutative diagram of exact sequences defining

$$\mu \left({Real}_X\left(f\right)\right).$$

Proof. 

This is the morphism-level content of Theorem 7.2. □

8. Filtered Iteration and Failure Modes

We use the notation and conventions of Section 2. The preceding sections construct the zig-zag category and zig-zag functor for a two-stratum decomposition

$$X=U\bigsqcup S.$$

This section records how the construction may be reused along a chosen finite filtration and clarifies the limits of the construction.

8.1. Filtered Iteration

Let

$$X=\underset{\lambda \in \Lambda }{⨆}{S}_{\lambda }$$

be a finite algebraic stratification. A closed filtration compatible with the stratification is a chain of closed subvarieties

$$⌀=X_{-1}\subset X_0\subset X_1\subset \dots \subset X_m=X$$

such that each $X_r$ is a union of strata. Setting

$$U_r:=X_r\setminus X_{r-1},$$

we obtain a sequence of two-stratum decompositions

$$X_r=U_r\bigsqcup X_{r-1},j_r:U_r\hookrightarrow X_r,i_r:X_{r-1}\hookrightarrow X_r.$$

Definition 8.1

(Filtered MV-admissible Nori datum). A filtration

$$⌀=X_{-1}\subset X_0\subset \dots \subset X_m=X$$

by closed unions of strata is called filtered MV-admissible in perverse Nori motives if each two-stratum pair

$$X_r=U_r\bigsqcup X_{r-1}$$

is MV-admissible and realization-compatible in the sense of Definitions 4.2 and 4.4.

For each such step, the two-stratum construction gives a category

$$Z_{\mathrm{Nori}}(X_r,X_{r-1})$$

and a functor

$${\mu }_{\mathrm{Nori},r}:PNM\left(X_r\right)⟶Z_{\mathrm{Nori}}(X_r,X_{r-1}).$$

For $Q_r^{*}\in PNM\left(X_r\right)$, the functor records the exact sequence

$$H^{-d_r-1}\left(i_r^{*}R{j}_{r,*}{j}_r^{*}{Q}_r^{*}\right)⟶H^{-d_r}\left(i_r^{!}{Q}_r^{*}\right)⟶H^{-d_r}\left(i_r^{*}{Q}_r^{*}\right)⟶H^{-d_r}\left(i_r^{*}R{j}_{r,*}{j}_r^{*}{Q}_r^{*}\right),$$

where $d_r$ denotes the degree normalization for the r-th two-stratum step.

Definition 8.2

(Filtered Nori zig-zag extraction). For a filtered MV-admissible Nori datum $X_{•}$, the filtered Nori zig-zag extraction of an object

$$Q^{*}\in PNM\left(X\right)$$

is the filtration-dependent diagrammatic package obtained by applying the two-stratum functors

$${\mu }_{\mathrm{Nori},r}:PNM\left(X_r\right)⟶Z_{\mathrm{Nori}}(X_r,X_{r-1})$$

successively along the chosen filtration.

This construction is not a new stratified theory. It is the repeated use of the two-stratum zig-zag tool along a chosen filtration. At each stage it extracts the open term, the two closed terms, and the boundary exact sequence for that step.

8.2. Realization of Filtered Extraction

For every r, the two-stratum realization theorem gives a natural isomorphism

$${Real}_{Z,r}\circ {\mu }_{\mathrm{Nori},r}\cong {\mu }_r\circ {Real}_{X_r},$$

where

$${\mu }_r:P\left(X_r\right)⟶Z(X_r,X_{r-1})$$

is the classical zig-zag functor for the pair $X_r=U_r\bigsqcup X_{r-1}$.

Theorem 8.3

(Realization compatibility for filtered extraction). Let $X_{•}$ be a filtered MV-admissible Nori datum. Then the filtered Nori zig-zag extraction realizes to the corresponding filtered classical zig-zag extraction along the same filtration.

Proof. 

Apply Theorem 7.2 at each two-stratum step. The filtered extraction is obtained by composing these stepwise constructions along the chosen filtration, and the realization isomorphisms are natural at each stage. Hence their composite identifies the filtered Nori extraction with the filtered classical extraction. □

Remark 8.4 (Filtered rather than canonical). 

The filtered statement is relative to a chosen filtration $X_{•}$. It does not assert that different filtrations produce canonically equivalent iterated zig-zag packages. Such a statement would require additional comparison data and is not part of this note.

8.3. Failure Modes

The construction in this paper is small but specific. It requires exactly the data needed to write the zig-zag exact sequence in the perverse Nori heart and to compare that sequence with classical realization. Several nearby-looking constructions do not suffice.

Classical zig-zag theory alone does not construct Nori data. It constructs

$$Z(X,S)\mathrm{and}\mu :P\left(X\right)\to Z(X,S)$$

inside the category of perverse sheaves. To construct

$$Z_{\mathrm{Nori}}(X,S)\mathrm{and}{\mu }_{\mathrm{Nori}}:PNM\left(X\right)\to Z_{\mathrm{Nori}}(X,S),$$

one must know that the terms

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right),H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right),H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right)$$

exist in the relevant perverse Nori hearts and that the defining exact sequence exists in $PNM\left(S\right)$.

Derived functoriality alone is also not enough. The zig-zag category is an abelian heart-level category of exact sequences. A derived functorial formalism supplies objects in derived categories, but the construction also requires the relevant cohomology objects to lie in $PNM\left(S\right)$ and to form an exact sequence there.

Likewise, the categories

$$PNM\left(X\right),PNM\left(U\right),PNM\left(S\right)$$

do not by themselves determine $Z_{\mathrm{Nori}}(X,S)$. An object of $Z_{\mathrm{Nori}}(X,S)$ is not merely a triple

$$(P^{*},K,C);$$

it is such a triple together with the exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)⟶K⟶C⟶H^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right).$$

Without the boundary terms and exactness condition, one has only an arbitrary diagram category.

Finally, realization is not an essential-surjectivity statement. The functor

$${Real}_Z:Z_{\mathrm{Nori}}(X,S)\to Z(X,S)$$

sends motivic zig-zag objects to classical zig-zag objects, but this paper does not prove that every object of $Z(X,S)$ lifts to $Z_{\mathrm{Nori}}(X,S)$.

Remark 8.5 (Structural principle). 

The construction requires the zig-zag exact sequence in the perverse Nori heart and realization compatibility for that sequence. Omitting either ingredient gives only a partial or unrelated construction.

9. Main Theorem Package

We collect the main statements proved above. Throughout, $X=U\bigsqcup S$ is a realization-compatible MV-admissible perverse Nori two-stratum pair.

Theorem 9.1

(Nori zig-zag category). There is a category

$$Z_{\mathrm{Nori}}(X,S)$$

whose objects are sextuples

$$\Theta =(P^{*},K,C,\alpha ,\beta ,\gamma ),$$

where

$$P^{*}\in PNM(X-S),K,C\in PNM\left(S\right),$$

and

$$H^{-d-1}\left(i^{*}R{j}_{*}{P}^{*}\right)\stackrel{\alpha }{\to }K\stackrel{\beta }{\to }C\stackrel{\gamma }{\to }{H}^{-d}\left(i^{*}R{j}_{*}{P}^{*}\right)$$

is exact in $PNM\left(S\right)$. Morphisms are morphisms of open terms together with maps between the two closed middle terms making the corresponding exact-sequence diagram commute.

Proof. 

This is Definition 5.1 and Proposition 5.2. □

Theorem 9.2

(Nori zig-zag functor). There is a functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S).$$

For $Q^{*}\in PNM\left(X\right)$, it is given by

$${\mu }_{\mathrm{Nori}}\left(Q^{*}\right)=\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

where

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)$$

is exact in $PNM\left(S\right)$.

Proof. 

This is Theorem 6.2. □

Theorem 9.3

(Realization compatibility). Realization induces a functor

$${Real}_Z:Z_{\mathrm{Nori}}(X,S)⟶Z(X,S),$$

and there is a natural isomorphism

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

Equivalently, the square 

commutes up to natural isomorphism.

Proof. 

The functor ${Real}_Z$ is constructed in Proposition 5.3. The natural isomorphism is Theorem 7.2. □

Corollary 9.4

(Filtered iteration). Let

$$⌀=X_{-1}\subset X_0\subset \dots \subset X_m=X$$

be a filtered MV-admissible Nori datum. Then the two-stratum construction can be applied successively to the pairs

$$X_r=(X_r-X_{r-1})\bigsqcup X_{r-1}.$$

The resulting filtered Nori zig-zag extraction realizes to the corresponding filtered classical zig-zag extraction along the same filtration.

Proof. 

This is Theorem 8.3. □

Remark 9.5 (Summary and nonclaims). 

The paper proves the construction of

$$Z_{\mathrm{Nori}}(X,S),{\mu }_{\mathrm{Nori}}:PNM\left(X\right)\to Z_{\mathrm{Nori}}(X,S),$$

and the realization comparison

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

It does not prove that every classical perverse sheaf has a canonical perverse Nori lift, that every object of $Z(X,S)$ lifts to $Z_{\mathrm{Nori}}(X,S)$, or that there is a new reconstruction equivalence in perverse Nori motives. It also does not prove any statement about Chow groups, algebraic cycles, Hodge classes, or the rational Hodge conjecture.

10. Conclusion

The zig-zag construction of [1] provides a compact way to package open perverse-sheaf data, closed costalk data, closed stalk data, and the boundary maps connecting them. This paper constructs the same package in the perverse Nori motivic category.

For a two-stratum decomposition

$$X=U\bigsqcup S,$$

we define the category

$$Z_{\mathrm{Nori}}(X,S)$$

of perverse Nori zig-zag objects and the functor

$${\mu }_{\mathrm{Nori}}:PNM\left(X\right)⟶Z_{\mathrm{Nori}}(X,S).$$

For $Q^{*}\in PNM\left(X\right)$, this functor records the package

$$\left(j^{*}{Q}^{*},H^{-d}\left(i^{!}{Q}^{*}\right),H^{-d}\left(i^{*}{Q}^{*}\right),{\alpha }_Q,{\beta }_Q,{\gamma }_Q\right),$$

with exact sequence

$$H^{-d-1}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right)\stackrel{{\alpha }_Q}{\to }{H}^{-d}\left(i^{!}{Q}^{*}\right)\stackrel{{\beta }_Q}{\to }{H}^{-d}\left(i^{*}{Q}^{*}\right)\stackrel{{\gamma }_Q}{\to }{H}^{-d}\left(i^{*}R{j}_{*}{j}^{*}{Q}^{*}\right).$$

The construction is functorial in $Q^{*}$: a morphism of perverse Nori motives induces a morphism of zig-zag objects, equivalently a commutative diagram of exact sequences on the closed stratum. The main theorem proves that this construction is compatible with realization:

$${Real}_Z\circ {\mu }_{\mathrm{Nori}}\cong \mu \circ {Real}_X.$$

Thus the motivic zig-zag data of a perverse Nori motive realize to the classical zig-zag data of the corresponding perverse sheaf.