Recollement Defects and MacPherson–Vilonen Boundary Residuals

1. Introduction

1.1. Motivation: Recollement Defects In MacPherson–Vilonen Gluing

Let X be a stratified space. Let $Z\subset X$ be a closed union of strata, set

$$U:=X\setminus Z,$$

and let

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

be the complementary inclusions. We write

$$D_c^b\left(U\right),D_c^b\left(X\right),D_c^b\left(Z\right)$$

for the bounded constructible derived categories with respect to the given stratifications.

The recollement formalism expresses $D_c^b\left(X\right)$ in terms of the open category $D_c^b\left(U\right)$, the closed category $D_c^b\left(Z\right)$, and the two adjoint triples

$$j_{!}⊣j^{*}⊣j_{*},i^{*}⊣i_{*}⊣i^{!}.$$

For every $M\in D_c^b\left(X\right)$, the standard reconstruction triangles are

$$j_{!}{j}^{*}M⟶M⟶i_{*}{i}^{*}M\stackrel{+1}{⟶},$$

and

$$i_{*}{i}^{!}M⟶M⟶j_{*}{j}^{*}M\stackrel{+1}{⟶}.$$

This is the categorical formalism underlying constructible sheaves, perverse sheaves, and MacPherson–Vilonen gluing [1,2,3,4].

MacPherson–Vilonen gluing gives an abelian-heart form of this reconstruction: a perverse object on X is described by open data, closed data, and boundary comparison morphisms [3,4]. The present paper isolates the stable-categorical defect calculus behind this construction. The central observation is that the closed terms

$$i_{*}{i}^{*}M\mathrm{and}{i}_{*}{i}^{!}M$$

are not merely auxiliary terms in the recollement triangles. Up to shifts, they are precisely the nontrivial twist–cotwist defects of the two open recollement adjunctions

$$j_{!}⊣j^{*},j^{*}⊣j_{*}.$$

1.2. Adjunction Defects and Sphericality

Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category, and let

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R$$

be an adjunction in $\mathcal{A}$. Let

$$\eta :{id}_{\mathcal{C}}⟶RL,ϵ:LR⟶{id}_{\mathcal{D}}$$

be the unit and counit. the twist and cotwist are

$$T_{\mathcal{C}}:=cof\left(\eta \right),T_{\mathcal{D}}:=fib\left(ϵ\right).$$

The adjunction is spherical if both $T_{\mathcal{C}}$ and $T_{\mathcal{D}}$ are equivalences. This is the convention used in spherical functor and spherical adjunction theory [5,6,7].

In genuine recollement, several of the unit and counit maps are already equivalences. For example,

$$j^{*}{j}_{!}\simeq {id}_{D_c^b\left(U\right)},j^{*}{j}_{*}\simeq {id}_{D_c^b\left(U\right)}.$$

Consequently, the corresponding twist or cotwist is zero. This is a structural feature of recollement. More generally, the abstract MacPherson–Vilonen reconstruction and orthogonality axioms themselves obstruct naive sphericality: imposing sphericality on all four raw recollement adjunctions forces the middle category to degenerate. Thus the correct invariant is not a naive “spherical recollement package,” but the collection of twist–cotwist defects carried by the recollement formalism.

1.3. Relation with the Existing Spherical and Schober Literature

The question motivating this paper is natural in light of universal sphericalization: given a MacPherson–Vilonen recollement package, what happens if one tries to apply the universal sphericalization of adjunctions to the whole linked package? Abellán–Gödicke construct the universal sphericalization of a single adjunction in a locally stable $(\infty ,2)$-category, including left and right adjoints to the inclusion of spherical adjunctions into all adjunctions [7]. Their construction is not formulated for open–closed recollement packages and does not address MacPherson–Vilonen gluing.

The schober literature contains a different spherical form of the same local perverse-sheaf phenomenon. Kapranov–Schechtman propose that, for a disk with one possible singular point, the categorical analogue of a perverse sheaf is given by a spherical functor [8]. In the usual sheaf-theoretic language, this is closer to nearby and vanishing-cycle coordinates than to the raw recollement coordinates

$$j_{!},j^{*},j_{*}.$$

The later theory of spherical adjunctions and relative S-constructions develops this spherical local model further [6]. This historical structure is consistent with the no-go result proved below: raw recollement is not the coordinate system in which sphericality is visible. the full-faithfulness identities in constructible recollement force zero twist or cotwist terms, and the abstract MV axioms already force degeneracy if all four selected adjunctions are required to be spherical. the spherical schober model is therefore better understood as arising after a change of local description, such as toward nearby/vanishing-cycle data.

Thus the contribution of the present paper is not to claim that ordinary MacPherson–Vilonen recollement is already spherical, nor that objectwise universal sphericalization automatically produces a nondegenerate spherical recollement package. The contribution is to compute the twist–cotwist defect calculus of raw recollement, identify the MacPherson–Vilonen boundary residual as an extension of the nontrivial open-adjunction defects, and prove that naive spherical MV packages degenerate unless the target or coordinates are changed.

1.4. Main Idea

The guiding chain is

$$\begin{array}{cc}\hfill \mathrm{MacPherson}--\mathrm{Vilonen}\mathrm{gluing}& ⇝\mathrm{recollement}\mathrm{adjunction}\mathrm{package},\hfill \\ \hfill \mathrm{recollement}\mathrm{adjunction}\mathrm{package}& ⇝\mathrm{twist}--\cot \mathrm{wist}\mathrm{defect}\mathrm{calculus},\hfill \\ \hfill \mathrm{twist}--\cot \mathrm{wist}\mathrm{defect}\mathrm{calculus}& ⇝\mathrm{boundary}\mathrm{residuals}.\hfill \end{array}$$

The first step records the linked adjunctions

$$j_{!}⊣j^{*}⊣j_{*},i^{*}⊣i_{*}⊣i^{!}$$

together with the orthogonality and reconstruction relations. the second step computes the twist and cotwist defects of these adjunctions. the third step identifies the MacPherson–Vilonen boundary residual as an extension of the two nontrivial open-adjunction defect terms.

For $M\in D_c^b\left(X\right)$, define

$${BRes}_Z\left(M\right):=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right).$$

The octahedral axiom applied to

$$j_{!}{j}^{*}M⟶M⟶j_{*}{j}^{*}M$$

gives the distinguished triangle

$$i_{*}{i}^{*}M⟶{BRes}_Z\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

The point of the paper is to refine this classical triangle by identifying its outer terms with the actual twist–cotwist defects of the open recollement adjunctions.

1.5. Main Results

The first main result is the explicit defect computation for the four recollement adjunctions.

Theorem 1 

(Twist–cotwist defects of recollement). Let

$$X=U\bigsqcup Z$$

be an open–closed decomposition of a stratified space, with inclusions

$$j:U\hookrightarrow X,i:Z\hookrightarrow X.$$

For the recollement adjunctions

$$j_{!}⊣j^{*},j^{*}⊣j_{*},i^{*}⊣i_{*},i_{*}⊣i^{!},$$

the twist–cotwist defects are as follows:

$$T_{D_c^b\left(U\right)}(j_{!}⊣j^{*})\simeq 0,T_{D_c^b\left(X\right)}(j_{!}⊣j^{*})\simeq i_{*}{i}^{*}(-)[-1],$$

$$T_{D_c^b\left(X\right)}(j^{*}⊣j_{*})\simeq i_{*}{i}^{!}(-)\left[1\right],T_{D_c^b\left(U\right)}(j^{*}⊣j_{*})\simeq 0,$$

$$T_{D_c^b\left(X\right)}(i^{*}⊣i_{*})\simeq j_{!}{j}^{*}(-)\left[1\right],T_{D_c^b\left(Z\right)}(i^{*}⊣i_{*})\simeq 0,$$

and

$$T_{D_c^b\left(Z\right)}(i_{*}⊣i^{!})\simeq 0,T_{D_c^b\left(X\right)}(i_{*}⊣i^{!})\simeq j_{*}{j}^{*}(-)[-1].$$

The second main result identifies the MacPherson–Vilonen boundary residual with an extension of the two nontrivial open-adjunction defects.

Theorem 2  

(Boundary residual from open-adjunction defects). For every

$$M\in D_c^b\left(X\right),$$

there is a functorial distinguished triangle

$$T_{D_c^b\left(X\right)}(j_{!}⊣j^{*})\left(M\right)\left[1\right]⟶{BRes}_Z\left(M\right)⟶T_{D_c^b\left(X\right)}(j^{*}⊣j_{*})\left(M\right)\stackrel{+1}{⟶}.$$

Equivalently,

$$i_{*}{i}^{*}M⟶{BRes}_Z\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

Remark 1  

(Classical and defect-theoretic content). The distinguished triangle in Theorem 2 is a classical consequence of the two recollement triangles and the octahedral axiom. the additional defect-theoretic content is the identification of its outer terms with the actual twist–cotwist defects of the two open recollement adjunctions.

The third main result records the obstruction to naive sphericality for abstract MV packages.

Theorem 3  

(Degeneracy of naive spherical MV packages). Let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right)$$

be an MV adjunction package in a locally stable $(\infty ,2)$-category. If all four selected adjunctions

$$j_{!}⊣j^{*},j^{*}⊣j_{*},i^{*}⊣i_{*},i_{*}⊣i^{!}$$

are spherical, then the zero endofunctor of the middle object

$${\mathcal{C}}_X$$

is an equivalence. In particular, in any target where the zero endofunctor is an equivalence only on the zero object, one has

$${\mathcal{C}}_X\simeq 0.$$

Remark 2  

(Consequence for sphericalization). Theorem 3 shows that the naive target $SphMV\left(\mathcal{A}\right)$ is degenerate already at the level of the abstract MV axioms. Hence a nondegenerate spherical enhancement cannot be obtained by imposing sphericality on the four raw recollement adjunctions while keeping the same MV coordinates. It must change the target, change coordinates, or add further structure. This is compatible with the schober literature, where the spherical local model is formulated in nearby/vanishing-cycle coordinates rather than in the raw $j_{!},j^{*},j_{*}$ recollement coordinates.

The fourth result is the finite-stratification version of the boundary residual construction.

Theorem 4  

(Stratified residual towers). Let

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

be a filtration by closed unions of strata. For $0\le k\le n$, set

$$U_k:=X_k\setminus X_{k-1},$$

and let

$$j_k:U_k\hookrightarrow X_k,i_k:X_{k-1}\hookrightarrow X_k$$

be the open and closed inclusions. For

$$M\in D_c^b\left(X\right),$$

put

$$M_k{:=M|}_{X_k}.$$

Define

$${BRes}_k\left(M\right):=Cone\left(j_{k!}{j}_k^{*}{M}_k\to j_{k*}{j}_k^{*}{M}_k\right).$$

Then the objects

$${BRes}_k\left(M\right),0\le k\le n,$$

form a residual tower for M. Each stage fits into a functorial distinguished triangle

$$i_{k*}{i}_k^{*}{M}_k⟶{BRes}_k\left(M\right)⟶i_{k*}{i}_k^{!}{M}_k\left[1\right]\stackrel{+1}{⟶}.$$

1.6. What Is Not Claimed

The paper does not claim that ordinary MacPherson–Vilonen recollement is already spherical. It also does not claim that applying universal sphericalization separately to the four raw recollement adjunctions automatically produces a nondegenerate spherical MV package. There are two independent obstructions.

First, objectwise sphericalization changes the adjunctions and may change the underlying categories. Since an MV package is a linked diagram with shared categories

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z,$$

sphericalizing the four constituent adjunctions separately need not preserve the shared three-object diagram.

Second, the abstract MV reconstruction and orthogonality axioms already force degeneracy if all four selected adjunctions are required to be spherical. Thus the obstruction is not only a feature of constructible recollement or of full-faithfulness; it is built into the naive spherical MV target itself.

Thus this paper separates three layers:

$$\begin{array}{cc}\hfill \mathrm{recollement}\mathrm{defect}\mathrm{calculus}& \mathrm{is}\mathrm{proved}\mathrm{here},\hfill \\ \hfill \mathrm{naive}\mathrm{spherical}\mathrm{MV}\mathrm{packages}& \mathrm{are}\mathrm{shown}\mathrm{to}\mathrm{degenerate},\hfill \\ \hfill \mathrm{spherical}\mathrm{gluing}\mathrm{envelopes}& \mathrm{are}\mathrm{formulated}\mathrm{only}\mathrm{conditionally}.\hfill \end{array}$$

1.7. Relation to Nearby and Vanishing Cycles

The no-go result points toward the coordinate system used in the schober literature. Classical recollement is expressed in terms of

$$j_{!},j^{*},j_{*},$$

where the full-faithfulness identities make some adjunction defects vanish. By contrast, perverse sheaves near a singular point are often described through nearby and vanishing cycles, and the corresponding schober model uses spherical functor data [6,8]. This suggests that the passage from MacPherson–Vilonen gluing to spherical gluing should not be viewed as a direct sphericalization of the raw recollement adjunctions. Rather, it should be viewed as a change of coordinates from recollement data to nearby/vanishing-cycle data.

Beilinson’s gluing formalism, including the maximal extension construction and unipotent nearby cycles, is a natural place to seek such a coordinate change [4]. the present paper does not prove that universal sphericalization recovers Beilinson’s nearby/vanishing-cycle description. It records the residual and no-go calculations that any such comparison must respect.

1.8. Filtered, Motivic, and Realization Variants

The residual statements use only recollement triangles and the octahedral axiom. Hence they admit analogues in any six-functor setting satisfying open–closed recollement. In particular, one may formulate motivic residuals in a motivic six-functor category by

$${BRes}_{Z,\mathrm{mot}}\left(M\right):=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right).$$

Such a motivic residual construction is formal once open–closed recollement is available.

A motivic or schober-theoretic sphericalization requires more. It requires a sufficiently cocomplete locally stable $(\infty ,2)$-categorical environment, together with closure of the subcategory under the sequential colimits appearing in universal sphericalization [7]. These hypotheses are not automatic for constructible, compact, or geometric-origin motivic subcategories [9,10]. Consequently, all motivic sphericalization statements in this paper are conditional.

Remark 3  

(No unconstructed motivic or spherical lift). The paper does not assume a motivic lift, a spherical gluing envelope, or a nearby/vanishing-cycle comparison without verifying the necessary recollement, local stability, cocompleteness, closure, and compatibility hypotheses. the unconditional results are the recollement defect computations, the boundary residual triangle, the degeneracy of naive spherical MV packages, and the residual tower construction.

1.9. Outline of the Paper

Section 2 recalls adjunctions, twists, cotwists, and universal sphericalization. Section 3 defines the MacPherson–Vilonen recollement package used in the paper. Section 4 defines the naive spherical MV target and proves its abstract degeneracy. Section 5 records the conditional form of sphericalization beyond the raw MV target, including the additional shared-object and transportability hypotheses needed for a diagram-level template. Section 6 computes the twist–cotwist defects of the four recollement adjunctions and identifies the MacPherson–Vilonen boundary residual as an extension of the two open-adjunction defect terms. Section 7 proves the residual tower statement for finite stratifications. Section 8 records the filtered–graded interpretation. Section 9 proves the Verdier duality and refinement statements. Section 10 gives examples, including the basic one-point stratification. Section 11 records conditional motivic and schober-theoretic outlooks, including the possible role of nearby and vanishing-cycle coordinates.

2. Adjunctions, Twists, Cotwists, and Sphericalization

This section fixes the categorical language used in the paper. We recall adjunctions in locally stable $(\infty ,2)$-categories, define twists and cotwists as the cofiber and fiber defects of the unit and counit, and record the universal sphericalization theorem of Abellán–Gödicke [5,6,7,11]. The point needed later is negative as well as positive. Universal sphericalization is a theorem about single adjunctions, while MacPherson–Vilonen recollement is a linked package of several adjunctions sharing the same categories. Moreover, genuine recollement contains full-faithfulness identities which force some twist or cotwist defects to vanish. Thus the formalism below will be used both to compute recollement defects and to explain why naive diagram-level sphericalization degenerates.

2.1. Adjunctions in Locally Stable $(\infty ,2)$-Categories

Definition 1  

(Locally stable $(\infty ,2)$-category). An $(\infty ,2)$-category $\mathcal{A}$ is locally stable if, for every pair of objects

$$\mathcal{X},\mathcal{Y}\in \mathcal{A},$$

the mapping ∞-category

$${Map}_{\mathcal{A}}(\mathcal{X},\mathcal{Y})$$

is stable, and horizontal composition is exact in each variable.

Definition 2  

(Adjunction). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. An adjunction in $\mathcal{A}$ is a tuple

$$(\mathcal{C},\mathcal{D},L,R,\eta ,ϵ)$$

consisting of objects

$$\mathcal{C},\mathcal{D}\in \mathcal{A},$$

morphisms

$$L:\mathcal{C}\to \mathcal{D},R:\mathcal{D}\to \mathcal{C},$$

and 2-morphisms

$$\eta :{id}_{\mathcal{C}}\to RL,ϵ:LR\to {id}_{\mathcal{D}},$$

such that the composites

$$L\stackrel{L\eta }{\to }LRL\stackrel{ϵL}{\to }L$$

and

$$R\stackrel{\eta R}{\to }RLR\stackrel{Rϵ}{\to }R$$

are equivalent to

$${id}_L\mathrm{and}{id}_R,$$

respectively. We write

$$L⊣R.$$

Remark 4  

(Round-trip maps). For an adjunction $L⊣R$, the unit and counit compare the two round trips

$$\mathcal{C}\stackrel{L}{\to }\mathcal{D}\stackrel{R}{\to }\mathcal{C},\mathcal{D}\stackrel{R}{\to }\mathcal{C}\stackrel{L}{\to }\mathcal{D}$$

with the corresponding identity functors. Thus

$$\eta :{id}_{\mathcal{C}}\to RL$$

measures the defect of the first round trip from the identity on $\mathcal{C}$, while

$$ϵ:LR\to {id}_{\mathcal{D}}$$

measures the defect of the second round trip from the identity on $\mathcal{D}$.

Definition 3  

(The category of adjunctions). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. We denote by

$$Adj\left(\mathcal{A}\right)$$

the $(\infty ,2)$-category of adjunctions in $\mathcal{A}$. Its objects are adjunctions

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R,$$

and its morphisms are adjointable squares of adjunctions.

2.2. Twist and Cotwist as Adjunction Defects

Definition 4  

(Twist and cotwist). Let

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R$$

be an adjunction in a locally stable $(\infty ,2)$-category $\mathcal{A}$, with unit

$$\eta :{id}_{\mathcal{C}}\to RL$$

and counit

$$ϵ:LR\to {id}_{\mathcal{D}}.$$

The twist of $L⊣R$ on $\mathcal{C}$ is

$$T_{\mathcal{C}}(L⊣R):=cof\left(\eta \right)=cof\left({id}_{\mathcal{C}}\to RL\right).$$

The cotwist of $L⊣R$ on $\mathcal{D}$ is

$$T_{\mathcal{D}}(L⊣R):=fib\left(ϵ\right)=fib\left(LR\to {id}_{\mathcal{D}}\right).$$

Remark 5  

(Adjunction defects). The functors

$$T_{\mathcal{C}}(L⊣R)\mathrm{and}{T}_{\mathcal{D}}(L⊣R)$$

are the two adjunction defects associated to $L⊣R$. They satisfy

$$T_{\mathcal{C}}(L⊣R)=0⟺\eta :{id}_{\mathcal{C}}\to RL\mathrm{is}\mathrm{an}\mathrm{equivalence},$$

and

$$T_{\mathcal{D}}(L⊣R)=0⟺ϵ:LR\to {id}_{\mathcal{D}}\mathrm{is}\mathrm{an}\mathrm{equivalence}.$$

Sphericality is stronger than nonvanishing. It requires both defect endofunctors to be equivalences.

Definition 5  

(Bicartesian form). The twist and cotwist of Definition 4 are determined by bicartesian squares

$$\begin{array}{cc}\hfill {id}_{\mathcal{C}}[r,"\eta "]\left[d\right]& RL\left[d\right]\\ \hfill 0\left[r\right]& T_{\mathcal{C}}(L⊣R)\end{array}$$

and

$$\begin{array}{cc}\hfill T_{\mathcal{D}}(L⊣R)\left[r\right]\left[d\right]& 0\left[d\right]\\ \hfill LR[r,"ϵ{"}^{\prime }]& {id}_{\mathcal{D}}.\end{array}$$

2.3. Spherical Adjunctions

Definition 6  

(Spherical adjunction). Let

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R$$

be an adjunction in a locally stable $(\infty ,2)$-category $\mathcal{A}$. the adjunction $L⊣R$ is spherical if the endofunctors

$$T_{\mathcal{C}}(L⊣R):\mathcal{C}\to \mathcal{C},T_{\mathcal{D}}(L⊣R):\mathcal{D}\to \mathcal{D}$$

are equivalences.

Definition 7  

(Category of spherical adjunctions). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. We write

$$Sph\left(\mathcal{A}\right)\subset Adj\left(\mathcal{A}\right)$$

for the full subcategory spanned by spherical adjunctions.

Remark 6  

(Relation with spherical functors). Spherical adjunctions generalize the categorical structures appearing in spherical functor theory and in local models for perverse schobers [5,6,8].

Lemma 1  

(Zero defect obstruction). Let

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R$$

be an adjunction in a locally stable $(\infty ,2)$-category $\mathcal{A}$. Suppose

$$T_{\mathcal{C}}(L⊣R)\simeq 0.$$

If $L⊣R$ is spherical, then the zero endofunctor

$$0:\mathcal{C}\to \mathcal{C}$$

is an equivalence. In particular, in any target where the zero endofunctor is an equivalence only on the zero object, one has

$$\mathcal{C}\simeq 0.$$

Similarly, if

$$T_{\mathcal{D}}(L⊣R)\simeq 0$$

and $L⊣R$ is spherical, then

$$\mathcal{D}\simeq 0$$

under the same nondegeneracy condition on the target.

Proof. 

If $L⊣R$ is spherical, then

$$T_{\mathcal{C}}(L⊣R):\mathcal{C}\to \mathcal{C}$$

is an equivalence. If this endofunctor is also equivalent to the zero endofunctor, then the zero endofunctor is an equivalence of $\mathcal{C}$. the stated conclusion follows in any target where this can occur only when $\mathcal{C}\simeq 0$. the proof for $\mathcal{D}$ is identical. □

Remark 7  

(Use for recollement). Lemma 1 is the elementary obstruction behind the degeneracy of naive spherical recollement packages. In a genuine recollement, identities such as

$$j^{*}{j}_{!}\simeq id\mathrm{and}{j}^{*}{j}_{*}\simeq id$$

force some twist or cotwist defects to vanish. Requiring the corresponding recollement adjunctions to be spherical therefore imposes a zero-functor equivalence on the open or closed categories.

2.4. Universal Sphericalization

Definition 8  

(Sphericalization hypotheses). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. We say that $\mathcal{A}$ satisfies the sphericalization hypotheses if the sequential colimits and inverse limits used in the construction of universal sphericalization exist in the relevant mapping ∞-categories and are preserved by horizontal composition in each variable.

Theorem 5  

(Universal sphericalization). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category satisfying the sphericalization hypotheses of Definition 8. Then the inclusion

$$\iota :Sph\left(\mathcal{A}\right)\hookrightarrow Adj\left(\mathcal{A}\right)$$

admits a left adjoint

$$S:Adj\left(\mathcal{A}\right)\to Sph\left(\mathcal{A}\right)$$

and a right adjoint

$$L^{\vee }:Adj\left(\mathcal{A}\right)\to Sph\left(\mathcal{A}\right).$$

For an adjunction

$$F=(L:\mathcal{C}\rightleftarrows \mathcal{D}:R),$$

the left sphericalization $S\left(F\right)$ is computed by the directed colimit of the iterated twist–cotwist system

$$F⟶T\left(F\right)⟶T^2\left(F\right)⟶\dots$$

in $Adj\left(\mathcal{A}\right)$. the right sphericalization $L^{\vee }\left(F\right)$ is computed dually by the corresponding inverse limit system. the left adjoint S satisfies

$${Map}_{Sph\left(\mathcal{A}\right)}\left(S\left(F\right),G\right)\simeq {Map}_{Adj\left(\mathcal{A}\right)}\left(F,\iota G\right)$$

for every

$$F\in Adj\left(\mathcal{A}\right)\mathrm{and}G\in Sph\left(\mathcal{A}\right).$$

Proof. 

This is the universal sphericalization theorem of [7]. the proof is not repeated here. □

Remark 8  

(Sphericalization changes the adjunction). The sphericalization $S\left(F\right)$ is not, in general, obtained by leaving the objects

$$\mathcal{C},\mathcal{D}$$

fixed and modifying only the unit and counit. It is computed as a colimit in

$$Adj\left(\mathcal{A}\right).$$

Consequently, when several adjunctions share the same source or target category, sphericalizing them separately need not preserve the shared-category diagram. This observation is essential for recollement, where the categories

$$D_c^b\left(U\right),D_c^b\left(X\right),D_c^b\left(Z\right)$$

are shared by several adjunctions simultaneously.

Remark 9  

(Use in this paper). Theorem 5 is used only under the hypotheses of Definition 8. the unconditional results of this paper are the recollement defect computations, the boundary residual triangle, the degeneracy of naive spherical recollement packages, and the residual tower construction. Any spherical enhancement of MacPherson–Vilonen gluing is treated as a conditional construction requiring additional shared-object, cocompleteness, and transportability hypotheses.

2.5. Coherent Twist–Cotwist Morphisms

Proposition 1  

(Twist–cotwist morphism of adjunctions). Let

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R$$

be an adjunction in a locally stable $(\infty ,2)$-category $\mathcal{A}$. Let

$$T_{\mathcal{C}}:=cof({id}_{\mathcal{C}}\to RL),T_{\mathcal{D}}:=fib(LR\to {id}_{\mathcal{D}})$$

be its twist and cotwist. Then there is a commutative square

$$\begin{array}{cc}\hfill \mathcal{C}[r,"T_{\mathcal{C}}"][d,"L{"}^{\prime }]& \mathcal{C}[d,"L"]\\ \hfill \mathcal{D}[r,"T_{\mathcal{D}}{"}^{\prime }]& \mathcal{D}\end{array}$$

which is vertically adjointable. Hence the pair

$$(T_{\mathcal{C}},T_{\mathcal{D}})$$

defines an endomorphism of the adjunction $L⊣R$ in

$$Adj\left(\mathcal{A}\right).$$

Proof. 

The assertion is the adjointability statement for the twist–cotwist square proved in [7]. the defining bicartesian squares for $T_{\mathcal{C}}$ and $T_{\mathcal{D}}$ give canonical equivalences

$$T_{\mathcal{D}}L\simeq L{T}_{\mathcal{C}},R{T}_{\mathcal{D}}\simeq T_{\mathcal{C}}R.$$

These equivalences identify the relevant mate with an equivalence. Therefore the square is vertically adjointable and determines a morphism of adjunctions. □

Corollary 1 

(Iterated coherent endomorphism). Let

$$F\in Adj\left(\mathcal{A}\right).$$

The twist–cotwist construction gives a functorial sequence

$$F⟶T\left(F\right)⟶T^2\left(F\right)⟶\dots$$

in $Adj\left(\mathcal{A}\right)$. If the corresponding colimit exists, it is the left sphericalization $S\left(F\right)$.

Proof. 

Proposition 1 gives the first morphism

$$F\to T\left(F\right)$$

in $Adj\left(\mathcal{A}\right)$. Applying the same construction inductively gives

$$T^m\left(F\right)\to T^{m+1}\left(F\right)$$

for every $m\ge 0$. the universal sphericalization theorem identifies the colimit of this sequence with $S\left(F\right)$. □

Remark 10  

(No automatic diagram-level sphericalization). Proposition 1 is a statement about one adjunction and its endomorphism in

$$Adj\left(\mathcal{A}\right).$$

It does not imply that a finite diagram of adjunctions with shared objects remains a diagram of the same shape after each constituent adjunction is sphericalized separately. In particular, it does not by itself produce a sphericalized recollement diagram.

3. Macpherson–Vilonen Adjunction Packages

This section fixes the open–closed recollement package used throughout the paper. For an open–closed decomposition $X=U\bigsqcup Z$, the constructible categories over U, X, and Z are linked by the adjoint triples

$$j_{!}⊣j^{*}⊣j_{*},i^{*}⊣i_{*}⊣i^{!},$$

together with orthogonality and reconstruction triangles. This is the stable-categorical source of MacPherson–Vilonen gluing: in the perverse heart, objects are described by open data, closed data, and boundary comparison morphisms [1,2,3,4]. We formulate the package intrinsically as recollement data in a locally stable $(\infty ,2)$-category. the definition is given as a fiber-product style datum of adjunctions with specified orthogonality and reconstruction cofiber sequences; zero relations are interpreted inside stable mapping categories.

3.1. Open–Closed Decompositions and Recollement

Definition 9  

(Open–closed decomposition). Let X be a stratified space. An open–closed decomposition of X is a pair

$$U\subset X,Z:=X\setminus U,$$

where U is an open union of strata and Z is a closed union of strata. We write

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

for the inclusions.

Definition 10  

(Constructible recollement). Let

$$X=U\bigsqcup Z$$

be an open–closed decomposition as in Definition 9. The constructible derived categories

$$D_c^b\left(U\right),D_c^b\left(X\right),D_c^b\left(Z\right)$$

satisfy constructible recollement if the six functors

$$j_{!},j^{*},j_{*},i^{*},i_{*},i^{!}$$

are defined and form adjoint triples

$$j_{!}⊣j^{*}⊣j_{*},i^{*}⊣i_{*}⊣i^{!},$$

with orthogonality relations

$$j^{*}{i}_{*}=0,i^{*}{j}_{!}=0,i^{!}{j}_{*}=0,$$

and with reconstruction triangles

$$j_{!}{j}^{*}M⟶M⟶i_{*}{i}^{*}M\stackrel{+1}{⟶},$$

and

$$i_{*}{i}^{!}M⟶M⟶j_{*}{j}^{*}M\stackrel{+1}{⟶}$$

for every

$$M\in D_c^b\left(X\right).$$

Remark 11  

(Classical source). The recollement relations in Definition 10 are standard in constructible and perverse sheaf theory [1,2]. the MacPherson–Vilonen construction gives an abelian-heart gluing description of perverse sheaves from the same open–closed structure [3,4].

Definition 11  

(Lower, upper, and closed reconstruction functors). Let $X=U\bigsqcup Z$ be an open–closed decomposition, and let

$$M\in D_c^b\left(X\right).$$

The lower open reconstruction of M is

$$j_{!}{j}^{*}M.$$

The upper open reconstruction of M is

$$j_{*}{j}^{*}M.$$

The lower closed residual term is

$$i_{*}{i}^{*}M.$$

The upper closed residual term is

$$i_{*}{i}^{!}M.$$

Remark 12  

(Source of residuals). The triangles of Definition 10 place M between its lower and upper open reconstructions. the closed terms

$$i_{*}{i}^{*}M\mathrm{and}{i}_{*}{i}^{!}M$$

measure the two complementary failures of open reconstruction from $j^{*}M$. These terms become the outer terms of the boundary residual triangle in Section 6.

3.2. The Abstract MV Recollement Datum

Definition 12  

(Zero composite). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. For objects

$$\mathcal{C},\mathcal{D}\in \mathcal{A},$$

the mapping ∞-category

$${Map}_{\mathcal{A}}(\mathcal{C},\mathcal{D})$$

is stable. Hence it has a zero object. A morphism

$$F:\mathcal{C}\to \mathcal{D}$$

is called zero if it is equivalent to this zero object in

$${Map}_{\mathcal{A}}(\mathcal{C},\mathcal{D}).$$

Definition 13  

(Endofunctor cofiber sequence). Let $\mathcal{A}$ be locally stable and let

$$\mathcal{C}\in \mathcal{A}.$$

A cofiber sequence of endofunctors of $\mathcal{C}$ is a cofiber sequence in the stable mapping category

$${Map}_{\mathcal{A}}(\mathcal{C},\mathcal{C}).$$

Thus a diagram

$$F⟶G⟶H$$

of endofunctors of $\mathcal{C}$ is a cofiber sequence if it is a distinguished triangle in

$${Map}_{\mathcal{A}}(\mathcal{C},\mathcal{C}).$$

Definition 14  

(The MV datum). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. An abstract MV datum in $\mathcal{A}$ consists of three objects

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z,$$

six morphisms

$$j_{!}:{\mathcal{C}}_U\to {\mathcal{C}}_X,j^{*}:{\mathcal{C}}_X\to {\mathcal{C}}_U,j_{*}:{\mathcal{C}}_U\to {\mathcal{C}}_X,$$

and

$$i^{*}:{\mathcal{C}}_X\to {\mathcal{C}}_Z,i_{*}:{\mathcal{C}}_Z\to {\mathcal{C}}_X,i^{!}:{\mathcal{C}}_X\to {\mathcal{C}}_Z,$$

together with adjunction data

$$j_{!}⊣j^{*},j^{*}⊣j_{*},i^{*}⊣i_{*},i_{*}⊣i^{!}.$$

Definition 15  

(Reconstruction data in the MV datum). Let the notation be as in Definition 14. Reconstruction data consist of the following two cofiber sequences in

$${Map}_{\mathcal{A}}({\mathcal{C}}_X,{\mathcal{C}}_X):$$

$$j_{!}{j}^{*}⟶{id}_{{\mathcal{C}}_X}⟶i_{*}{i}^{*},$$

and

$$i_{*}{i}^{!}⟶{id}_{{\mathcal{C}}_X}⟶j_{*}{j}^{*}.$$

Remark 13  

(Why an intrinsic datum is used). A MacPherson–Vilonen package is not a single adjunction. It is a linked system of adjunctions sharing the middle category

$${\mathcal{C}}_X.$$

The formulation above avoids treating zero relations as formal symbols in an abstract diagram. Instead, zero morphisms and reconstruction triangles are interpreted inside the stable mapping categories of $\mathcal{A}$.

Remark 14  

(Generators and relations). The present definition may be regarded as a generators-and-relations description only after the ambient locally stable target has been fixed. the zero relations are interpreted in stable mapping categories, and the reconstruction relations are imposed as cofiber sequences of endofunctors of ${\mathcal{C}}_X$. This gives a well-formed definition in any locally stable $(\infty ,2)$-category.

3.3. Definition of $MVAdj\left(\mathcal{A}\right)$

Definition 16  

(MacPherson–Vilonen adjunction package). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. A MacPherson–Vilonen adjunction package in $\mathcal{A}$ is an abstract MV datum

$$\mathfrak{A}=\left({\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z;j_{!},j^{*},j_{*};i^{*},i_{*},i^{!}\right)$$

as in Definition 14, satisfying the following additional conditions.

First, the orthogonality relations hold:

$$j^{*}{i}_{*}=0,i^{*}{j}_{!}=0,i^{!}{j}_{*}=0.$$

Second, the reconstruction data of Definition 15 are imposed:

$$j_{!}{j}^{*}⟶{id}_{{\mathcal{C}}_X}⟶i_{*}{i}^{*},$$

and

$$i_{*}{i}^{!}⟶{id}_{{\mathcal{C}}_X}⟶j_{*}{j}^{*}.$$

Definition 17  

(The category of MV adjunction packages). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. We denote by

$$MVAdj\left(\mathcal{A}\right)$$

the $(\infty ,2)$-category whose objects are MacPherson–Vilonen adjunction packages in $\mathcal{A}$, and whose morphisms are morphisms of the underlying six-functor data preserving the adjunctions, orthogonality relations, and reconstruction cofiber sequences.

Definition 18  

(Underlying categories of an MV package). For

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right),$$

write

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z$$

for the three objects appearing in the datum

$$\mathfrak{A}=\left({\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z;j_{!},j^{*},j_{*};i^{*},i_{*},i^{!}\right).$$

We write

$$j_{!},j^{*},j_{*},i^{*},i_{*},i^{!}$$

also for the corresponding morphisms in $\mathcal{A}$, when no confusion is possible.

Definition 19  

(Selected adjunctions). Let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right).$$

The selected, or constituent, recollement adjunctions of $\mathfrak{A}$ are

$$j_{!}:{\mathcal{C}}_U\rightleftarrows {\mathcal{C}}_X:j^{*},j^{*}:{\mathcal{C}}_X\rightleftarrows {\mathcal{C}}_U:j_{*},$$

and

$$i^{*}:{\mathcal{C}}_X\rightleftarrows {\mathcal{C}}_Z:i_{*},i_{*}:{\mathcal{C}}_Z\rightleftarrows {\mathcal{C}}_X:i^{!}.$$

These are the four adjunctions whose twist–cotwist defects are computed in Section 6.

Remark 15  

(Fiber-product description). Equivalently, $MVAdj\left(\mathcal{A}\right)$ may be constructed as a full subcategory of the fiber product of four copies of $Adj\left(\mathcal{A}\right)$,

$$Adj\left(\mathcal{A}\right){\times }_{{\mathcal{A}}^3}Adj\left(\mathcal{A}\right){\times }_{{\mathcal{A}}^3}Adj\left(\mathcal{A}\right){\times }_{{\mathcal{A}}^3}Adj\left(\mathcal{A}\right),$$

where the fiber product imposes the shared objects

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z.$$

The full subcategory condition then imposes the three orthogonality relations and the two reconstruction cofiber sequences of Definition 16.

Remark 16  

(Soundness and coverage). Definition 16 is a structural definition. It specifies the adjunctions, orthogonality relations, and reconstruction cofiber sequences present in MacPherson–Vilonen recollement. It does not assert that every diagram of adjunctions in $\mathcal{A}$ is an MV package, nor that every schober-theoretic gluing datum arises from such a package. These are separate coverage questions.

3.4. Classical MV Gluing as an Example

Proposition 2  

(Constructible sheaves define MV packages). Let

$$X=U\bigsqcup Z$$

be an open–closed decomposition of a stratified space. Then the assignment

$${\mathcal{C}}_U:=D_c^b\left(U\right),{\mathcal{C}}_X:=D_c^b\left(X\right),{\mathcal{C}}_Z:=D_c^b\left(Z\right),$$

together with the six functors

$$j_{!},j^{*},j_{*},i^{*},i_{*},i^{!},$$

defines an object

$${\mathfrak{A}}_X\in MVAdj\left(\mathrm{St}\right).$$

Proof. 

The constructible derived categories

$$D_c^b\left(U\right),D_c^b\left(X\right),D_c^b\left(Z\right)$$

are stable. the functors

$$j_{!},j^{*},j_{*},i^{*},i_{*},i^{!}$$

are exact functors between stable categories. the adjunctions

$$j_{!}⊣j^{*}⊣j_{*},i^{*}⊣i_{*}⊣i^{!}$$

hold by the standard six-functor formalism for constructible sheaves [1,2]. the orthogonality relations

$$j^{*}{i}_{*}=0,i^{*}{j}_{!}=0,i^{!}{j}_{*}=0$$

follow from the disjointness of U and Z. the reconstruction triangles are the standard recollement triangles. Hence the displayed data satisfy Definition 16. □

Remark 17  

(Perverse heart). If $D_c^b\left(X\right)$ is equipped with the perverse t-structure, then the abelian heart

$$Perv\left(X\right)$$

inherits the MacPherson–Vilonen gluing description from the same recollement data [3,4]. Thus Proposition 2 is the stable-categorical ambient form of classical MV gluing.

Remark 18  

(Use in later sections). The object

$${\mathfrak{A}}_X\in MVAdj\left(\mathrm{St}\right)$$

is the raw recollement adjunction package. Its selected constituent adjunctions are not expected to be spherical. Instead, Section 6 computes their twist–cotwist defects, and Section 4 explains why imposing sphericality on all four constituent adjunctions gives a degenerate notion for genuine recollement.

4. Spherical MV Packages

This section isolates the obstruction to a naive sphericalization target for MacPherson–Vilonen recollement packages. the package of Section 3 contains four linked adjunctions sharing the three objects

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z.$$

Universal sphericalization applies to a single adjunction under the hypotheses of Theorem 5 [7]. It does not automatically preserve a diagram of several adjunctions with shared objects. Moreover, genuine recollement contains full-faithfulness identities, such as

$$j^{*}{j}_{!}\simeq id,j^{*}{j}_{*}\simeq id,$$

which force several twist–cotwist defects to vanish [1,2]. Thus the full subcategory of MV packages whose four constituent recollement adjunctions are all spherical is, in general, a degenerate target. We keep this naive target as a precise comparison object, and prove that it does not contain ordinary constructible recollement except in degenerate cases.

4.1. Selected Adjunctions

Definition 20  

(Selected adjunction set). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category, and let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right)$$

be an MV adjunction package. Write

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z$$

for the three underlying objects of $\mathfrak{A}$, as in Definition 18. the selected adjunction set of $\mathfrak{A}$ is

$$\mathcal{J}\left(\mathfrak{A}\right):=\left\{j_{!}⊣j^{*},j^{*}⊣j_{*},i^{*}⊣i_{*},i_{*}⊣i^{!}\right\}.$$

Explicitly,

$$j_{!}:{\mathcal{C}}_U\rightleftarrows {\mathcal{C}}_X:j^{*},j^{*}:{\mathcal{C}}_X\rightleftarrows {\mathcal{C}}_U:j_{*},$$

and

$$i^{*}:{\mathcal{C}}_X\rightleftarrows {\mathcal{C}}_Z:i_{*},i_{*}:{\mathcal{C}}_Z\rightleftarrows {\mathcal{C}}_X:i^{!}.$$

Remark 19  

(Choice of selected adjunctions). Definition 20 selects all four constituent recollement adjunctions. This is useful for bookkeeping the full recollement defect calculus. It should not be read as asserting that all four adjunctions can be made spherical inside the original recollement category without degeneracy. A nondegenerate spherical gluing theory may instead select a smaller family

$${\mathcal{J}}_0\left(\mathfrak{A}\right)\subseteq \mathcal{J}\left(\mathfrak{A}\right)$$

or may change coordinates, for example toward nearby and vanishing-cycle data.

Definition 21  

(Twists and cotwists of selected adjunctions). Let

$$a=\left(L_a:{\mathcal{C}}_a\rightleftarrows {\mathcal{D}}_a:R_a\right)$$

be an element of $\mathcal{J}\left(\mathfrak{A}\right)$. Let

$${\eta }_a:{id}_{{\mathcal{C}}_a}\to R_a{L}_a,{ϵ}_a:L_a{R}_a\to {id}_{{\mathcal{D}}_a}$$

be its unit and counit. Define

$$T_{{\mathcal{C}}_a}\left(a\right):=cof\left({\eta }_a\right),T_{{\mathcal{D}}_a}\left(a\right):=fib\left({ϵ}_a\right).$$

The family

$${\left\{T_{{\mathcal{C}}_a}\left(a\right),T_{{\mathcal{D}}_a}\left(a\right)\right\}}_{a\in \mathcal{J}\left(\mathfrak{A}\right)}$$

is called the twist–cotwist family of $\mathfrak{A}$.

Remark 20  

(Residual interpretation). For each $a\in \mathcal{J}\left(\mathfrak{A}\right)$, the functors

$$T_{{\mathcal{C}}_a}\left(a\right),T_{{\mathcal{D}}_a}\left(a\right)$$

are the two adjunction defects of a. In genuine recollement, some of these defects vanish because some unit or counit morphisms are equivalences. the nonzero defects are computed in Section 6; in particular, the two nontrivial open-adjunction defects form the outer terms of the boundary residual triangle.

4.2. Definition of Naive Spherical MV Packages

Definition 22  

(Spherical selected adjunction). Let

$$a=\left(L_a:{\mathcal{C}}_a\rightleftarrows {\mathcal{D}}_a:R_a\right)$$

be a selected adjunction in an MV package $\mathfrak{A}$. the adjunction a is spherical if its twist and cotwist

$$T_{{\mathcal{C}}_a}\left(a\right):{\mathcal{C}}_a\to {\mathcal{C}}_a,T_{{\mathcal{D}}_a}\left(a\right):{\mathcal{D}}_a\to {\mathcal{D}}_a$$

are equivalences.

Definition 23  

(Compatibility with twist–cotwist equivalences). Let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right).$$

Suppose that every $a\in \mathcal{J}\left(\mathfrak{A}\right)$ is spherical. the recollement compatibility morphisms of $\mathfrak{A}$ are compatible with the twist–cotwist equivalences if the family

$${\left\{T_{{\mathcal{C}}_a}\left(a\right),T_{{\mathcal{D}}_a}\left(a\right)\right\}}_{a\in \mathcal{J}\left(\mathfrak{A}\right)}$$

acts by equivalences on the selected adjunctions and preserves the orthogonality and reconstruction cofiber sequences of Definition 16.

Definition 24  

(Naive spherical MV package). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. A naive spherical MV package in $\mathcal{A}$ is an object

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right)$$

such that:

$$\mathrm{for}\mathrm{every}a\in \mathcal{J}\left(\mathfrak{A}\right),a\mathrm{is}\mathrm{spherical},$$

and the recollement compatibility morphisms of $\mathfrak{A}$ are compatible with the twist–cotwist equivalences in the sense of Definition 23.

Definition 25  

(The category of naive spherical MV packages). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. Define

$$SphMV\left(\mathcal{A}\right)\subset MVAdj\left(\mathcal{A}\right)$$

to be the full subcategory spanned by the naive spherical MV packages.

Remark 21  

(Soundness of the definition). Definition 24 is a structural condition on a given MV package. It does not assert that ordinary recollement packages are spherical. It also does not assert that $SphMV\left(\mathcal{A}\right)$ is the correct nondegenerate target for schober-type gluing. It is a precise naive comparison target whose degeneracy can be tested.

Remark 22  

(Relation with universal sphericalization). Universal sphericalization applies to individual adjunctions. Since an MV package contains several adjunctions sharing the same objects, objectwise sphericalization need not produce an object of $MVAdj\left(\mathcal{A}\right)$. In particular, the colimits defining the sphericalization of one selected adjunction may replace ${\mathcal{C}}_X$ by an object different from the one produced by sphericalizing another selected adjunction. Therefore $SphMV\left(\mathcal{A}\right)$ should not be treated as an automatic output of objectwise sphericalization.

4.3. Degeneracy of Naive Spherical MV Packages

Proposition 3  

(Degeneracy of naive spherical MV packages). Let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right)$$

be an MV adjunction package in a locally stable $(\infty ,2)$-category. Suppose that all four selected adjunctions

$$j_{!}⊣j^{*},j^{*}⊣j_{*},i^{*}⊣i_{*},i_{*}⊣i^{!}$$

are spherical. Then the zero endofunctor of the middle object

$${\mathcal{C}}_X$$

is an equivalence. In particular, in any target where the zero endofunctor is an equivalence only on the zero object, one has

$${\mathcal{C}}_X\simeq 0.$$

Proof. 

By the reconstruction cofiber sequence

$$j_{!}{j}^{*}⟶{id}_{{\mathcal{C}}_X}⟶i_{*}{i}^{*},$$

the cotwist of the adjunction

$$j_{!}⊣j^{*}$$

on ${\mathcal{C}}_X$ is

$$T_{{\mathcal{C}}_X}(j_{!}⊣j^{*})\simeq i_{*}{i}^{*}[-1].$$

Since $j_{!}⊣j^{*}$ is spherical, this cotwist is an equivalence. Hence

$$i_{*}{i}^{*}$$

is an equivalence of ${\mathcal{C}}_X$.

The same reconstruction cofiber sequence also identifies the twist of

$$i^{*}⊣i_{*}$$

on ${\mathcal{C}}_X$ as

$$T_{{\mathcal{C}}_X}(i^{*}⊣i_{*})\simeq j_{!}{j}^{*}\left[1\right].$$

Since $i^{*}⊣i_{*}$ is spherical, this twist is an equivalence. Hence

$$j_{!}{j}^{*}$$

is an equivalence of ${\mathcal{C}}_X$.

Now use the orthogonality relation

$$i^{*}{j}_{!}=0.$$

Composing with $i_{*}$ gives

$$i_{*}{i}^{*}{j}_{!}=0.$$

Since $i_{*}{i}^{*}$ is an equivalence, it follows that

$$j_{!}=0.$$

Therefore

$$j_{!}{j}^{*}=0.$$

But $j_{!}{j}^{*}$ is also an equivalence of ${\mathcal{C}}_X$. Hence the zero endofunctor of ${\mathcal{C}}_X$ is an equivalence. Under the stated nondegeneracy condition on the target, this implies

$${\mathcal{C}}_X\simeq 0.$$

□

Proposition 4  

(Zero-defect obstruction for selected adjunctions). Let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right)$$

be an MV package. Suppose that, for some selected adjunction

$$a=\left(L_a:{\mathcal{C}}_a\rightleftarrows {\mathcal{D}}_a:R_a\right),$$

one has

$$T_{{\mathcal{C}}_a}\left(a\right)\simeq 0.$$

If a is spherical, then the zero endofunctor of ${\mathcal{C}}_a$ is an equivalence. If the zero endofunctor is an equivalence only on the zero object in the target $\mathcal{A}$, then

$${\mathcal{C}}_a\simeq 0.$$

Similarly, if

$$T_{{\mathcal{D}}_a}\left(a\right)\simeq 0$$

and a is spherical, then

$${\mathcal{D}}_a\simeq 0$$

under the same nondegeneracy condition.

Proof. 

If a is spherical, then both

$$T_{{\mathcal{C}}_a}\left(a\right)\mathrm{and}{T}_{{\mathcal{D}}_a}\left(a\right)$$

are equivalences. If one of these endofunctors is zero, then the corresponding zero endofunctor is an equivalence. the final conclusion follows from the stated nondegeneracy condition on $\mathcal{A}$. □

Proposition 5  

(Degeneracy of naive spherical recollement). Let

$$X=U\bigsqcup Z$$

be an open–closed decomposition, and let

$${\mathfrak{A}}_X\in MVAdj\left(\mathrm{St}\right)$$

be the constructible recollement package of Proposition 2. If ${\mathfrak{A}}_X$ is an object of

$$SphMV\left(\mathrm{St}\right),$$

then the zero endofunctor is an equivalence on both

$$D_c^b\left(U\right)\mathrm{and}{D}_c^b\left(Z\right).$$

In particular, if either $D_c^b\left(U\right)$ or $D_c^b\left(Z\right)$ is nonzero, then the ordinary constructible recollement package is not an object of

$$SphMV\left(\mathrm{St}\right).$$

Proof. 

In genuine constructible recollement, the functors $j_{!}$ and $j_{*}$ are fully faithful. Hence

$$j^{*}{j}_{!}\simeq {id}_{D_c^b\left(U\right)},j^{*}{j}_{*}\simeq {id}_{D_c^b\left(U\right)}.$$

For the adjunction

$$j_{!}⊣j^{*},$$

the unit

$${id}_{D_c^b\left(U\right)}⟶j^{*}{j}_{!}$$

is an equivalence. Therefore

$$T_{D_c^b\left(U\right)}(j_{!}⊣j^{*})\simeq 0.$$

If $j_{!}⊣j^{*}$ is spherical, Proposition 4 implies that the zero endofunctor of $D_c^b\left(U\right)$ is an equivalence.

Similarly, for the adjunction

$$j^{*}⊣j_{*},$$

the counit

$$j^{*}{j}_{*}⟶{id}_{D_c^b\left(U\right)}$$

is an equivalence. Hence the corresponding cotwist on $D_c^b\left(U\right)$ is zero. Sphericality again forces the zero endofunctor of $D_c^b\left(U\right)$ to be an equivalence.

The same argument applies to the closed adjunctions. the functor $i_{*}$ is fully faithful, so

$$i^{*}{i}_{*}\simeq {id}_{D_c^b\left(Z\right)},i^{!}{i}_{*}\simeq {id}_{D_c^b\left(Z\right)}.$$

Thus imposing sphericality on

$$i^{*}⊣i_{*}\mathrm{and}{i}_{*}⊣i^{!}$$

forces the zero endofunctor of $D_c^b\left(Z\right)$ to be an equivalence. This proves the claim. □

Example 1  

(A formal naive spherical MV package). Let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right).$$

Suppose that every selected adjunction

$$a\in \mathcal{J}\left(\mathfrak{A}\right)$$

has invertible twist and cotwist, and suppose that the resulting twist–cotwist equivalences preserve the orthogonality morphisms and reconstruction cofiber sequences of Definition 16. Then

$$\mathfrak{A}\in SphMV\left(\mathcal{A}\right).$$

This example is formal. It does not assert that a genuine constructible recollement package has this property.

Remark 23  

(Purpose of the naive target). The purpose of introducing $SphMV\left(\mathcal{A}\right)$ is not to claim that classical MacPherson–Vilonen gluing is already spherical. Rather, the naive definition shows what would be required if one insisted on keeping the original recollement package fixed while imposing sphericality on all selected adjunctions. Proposition 5 shows that this requirement degenerates for genuine constructible recollement.

Remark 24  

(Non-example). A collection of spherical adjunctions

$${\left\{a\right\}}_{a\in \mathcal{J}}$$

with the same object labels

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z$$

does not determine an object of $SphMV\left(\mathcal{A}\right)$ unless the recollement orthogonality and reconstruction data are also specified and are compatible with the twist–cotwist equivalences. Sphericality of the constituent adjunctions alone is insufficient.

Remark 25  

(Coverage limitation). The category $SphMV\left(\mathcal{A}\right)$ is not claimed to exhaust categorical models of perverse schobers on stratified spaces. In light of Proposition 5, it should be regarded as a naive comparison target rather than the final nondegenerate target for spherical gluing. A nondegenerate spherical enhancement of MacPherson–Vilonen gluing must change the target or change coordinates, for example by passing to a spherical gluing envelope or to a nearby/vanishing-cycle model.

5. Conditional Sphericalization Beyond the Raw MV Target

The preceding section shows that the naive spherical MV target is degenerate for raw recollement packages. In particular, imposing sphericality on the selected recollement adjunctions does not produce a nondegenerate spherical form of MacPherson–Vilonen gluing inside the same recollement coordinates. This section records only the conditional template that remains after this no-go result. Universal sphericalization applies to a single adjunction under the hypotheses of Theorem 5; it does not, by itself, preserve a multi-adjunction diagram with shared objects. Therefore no unconditional left adjoint

$$MVAdj\left(\mathcal{A}\right)⟶SphMV\left(\mathcal{A}\right)$$

is asserted.

Definition 26 

(Objectwise sphericalization data). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category satisfying the hypotheses of Theorem 5, and let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right).$$

For each selected adjunction $a\in \mathcal{J}\left(\mathfrak{A}\right)$, define

$$S_a\left(a\right):=S\left(a\right),$$

where $S:Adj\left(\mathcal{A}\right)\to Sph\left(\mathcal{A}\right)$ is the universal sphericalization functor. the family

$$S_{obj}\left(\mathfrak{A}\right):={\left\{S_a\left(a\right)\right\}}_{a\in \mathcal{J}\left(\mathfrak{A}\right)}$$

is called the objectwise sphericalization data of $\mathfrak{A}$.

Remark 26 

(Objectwise sphericalization is not diagram-level sphericalization). The family $S_{obj}\left(\mathfrak{A}\right)$ is only a family of spherical adjunctions. It need not define an object of $SphMV\left(\mathcal{A}\right)$, because the colimits defining $S_a\left(a\right)$ may change the source and target objects of the selected adjunctions. Since an MV package shares the three objects

$${\mathcal{C}}_U,{\mathcal{C}}_X,{\mathcal{C}}_Z,$$

sphericalizing the selected adjunctions separately need not preserve a single three-object diagram.

Definition 27 

(Shared-object sphericalization datum). Let

$$\mathfrak{A}\in MVAdj\left(\mathcal{A}\right).$$

A shared-object sphericalization datum for $\mathfrak{A}$ consists of objects

$${\mathcal{C}}_U^S,{\mathcal{C}}_X^S,{\mathcal{C}}_Z^S$$

of $\mathcal{A}$, together with identifications of the source and target objects of the objectwise sphericalizations $S_a\left(a\right)$, $a\in \mathcal{J}\left(\mathfrak{A}\right)$, with the appropriate objects among ${\mathcal{C}}_U^S,{\mathcal{C}}_X^S,{\mathcal{C}}_Z^S$, so that the sphericalized adjunctions have the formal shape

$$j_{!}^S:{\mathcal{C}}_U^S\rightleftarrows {\mathcal{C}}_X^S:j^{S,*},j^{S,*}:{\mathcal{C}}_X^S\rightleftarrows {\mathcal{C}}_U^S:j_{*}^S,$$

and

$$i^{S,*}:{\mathcal{C}}_X^S\rightleftarrows {\mathcal{C}}_Z^S:i_{*}^S,i_{*}^S:{\mathcal{C}}_Z^S\rightleftarrows {\mathcal{C}}_X^S:i^{S,!}.$$

Definition 28 

(Transportability of compatibility morphisms). Let

$$\theta :F\to G$$

be one of the orthogonality or reconstruction morphisms in an MV package $\mathfrak{A}$. We say that θ is transportable through the twist–cotwist system if, for every $m\ge 0$, there is a morphism

$$T^m\left(\theta \right):T^m\left(F\right)\to T^m\left(G\right)$$

compatible with the transition morphisms in the sequential systems defining the objectwise sphericalizations. Equivalently, the family ${\left\{T^m\left(\theta \right)\right\}}_{m\ge 0}$ defines a morphism of the corresponding sequential diagrams.

Remark 27 

(Transportability is additional structure). Transportability is not automatic. the coherent twist–cotwist construction gives an endomorphism of a single adjunction. It does not, by itself, define compatible endomorphisms of all composites appearing in a multi-adjunction recollement diagram.

Theorem 6 

(Conditional diagram-level sphericalization). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category satisfying the hypotheses of Theorem 5. Let

$${MVAdj}^{comp}\left(\mathcal{A}\right)\subseteq MVAdj\left(\mathcal{A}\right)$$

be a subcategory whose objects are MV packages equipped with shared-object sphericalization data and transportable compatibility morphisms. Then objectwise sphericalization assembles, by the chosen additional data, into a functor

$$S_{MV}:{MVAdj}^{comp}\left(\mathcal{A}\right)⟶SphMV\left(\mathcal{A}\right).$$

Proof. 

For each selected adjunction $a\in \mathcal{J}\left(\mathfrak{A}\right)$, universal sphericalization gives a spherical adjunction $S\left(a\right)$. the shared-object datum identifies the source and target objects of these spherical adjunctions with a common triple

$${\mathcal{C}}_U^S,{\mathcal{C}}_X^S,{\mathcal{C}}_Z^S.$$

The transportability hypotheses carry the orthogonality and reconstruction morphisms through the sequential diagrams defining the objectwise sphericalizations. Thus the objectwise sphericalized adjunctions assemble into an object of $SphMV\left(\mathcal{A}\right)$. Functoriality follows from functoriality of universal sphericalization and of the transported compatibility data. □

Remark 28 

(Status of the construction). Theorem 6 is conditional. It does not assert that ${MVAdj}^{comp}\left(\mathcal{A}\right)$ contains any nondegenerate raw recollement package. In light of Proposition 3, the theorem should be read as a formal template for sphericalization after changing target or adding extra structure, not as a sphericalization theorem for ordinary MacPherson–Vilonen recollement.

Corollary 2 

(Conditional mapping-space form). Let

$$\mathfrak{A}\in {MVAdj}^{comp}\left(\mathcal{A}\right)\mathrm{and}\mathfrak{B}\in SphMV\left(\mathcal{A}\right).$$

Assume that the objectwise universal properties of Theorem 5 are compatible with the transported finite MV relation diagram. Then there is a natural equivalence

$${Map}_{SphMV\left(\mathcal{A}\right)}\left(S_{MV}\left(\mathfrak{A}\right),\mathfrak{B}\right)\simeq {Map}_{{MVAdj}^{comp}\left(\mathcal{A}\right)}\left(\mathfrak{A},\iota \mathfrak{B}\right).$$

Remark 29 

(Universal target). The equivalence in Corollary 2 is conditional on the chosen compatibility subcategory and on compatibility of the objectwise universal properties with the finite MV relation diagram. It is not an unconditional universal property on all of

$$MVAdj\left(\mathcal{A}\right).$$

6. Recollement Defects and MV Boundary Residuals

This section gives the precise defect calculation behind the MacPherson–Vilonen boundary residual. For an open–closed decomposition $X=U\bigsqcup Z$, the recollement triangles identify the difference between lower and upper open reconstruction with closed-stratum terms [1,2,3,4]. We first recall the elementary residuals

$${Res}_Z^{+}\left(M\right),{Res}_Z^{-}\left(M\right),$$

and the boundary residual

$${BRes}_Z\left(M\right).$$

We then compute the twist–cotwist defects of the four recollement adjunctions. the main point is that ${BRes}_Z\left(M\right)$ is not a vague shadow of sphericalization: it is an extension of the two nontrivial open-adjunction defect terms.

6.1. Lower and Upper Reconstruction Residuals

Definition 29  

(Open–closed residuals). Let

$$X=U\bigsqcup Z$$

be an open–closed decomposition of a stratified space, with inclusions

$$j:U\hookrightarrow X,i:Z\hookrightarrow X.$$

Let

$$M\in D_c^b\left(X\right).$$

The lower reconstruction residual of M along Z is

$${Res}_Z^{+}\left(M\right):=Cone\left(j_{!}{j}^{*}M\to M\right).$$

The upper reconstruction residual of M along Z is

$${Res}_Z^{-}\left(M\right):=Cone\left(M\to j_{*}{j}^{*}M\right)[-1].$$

Proposition 6  

(Identification of elementary residuals). For every

$$M\in D_c^b\left(X\right),$$

there are canonical isomorphisms

$${Res}_Z^{+}\left(M\right)\simeq i_{*}{i}^{*}M,{Res}_Z^{-}\left(M\right)\simeq i_{*}{i}^{!}M.$$

Proof. 

The first recollement triangle is

$$j_{!}{j}^{*}M⟶M⟶i_{*}{i}^{*}M\stackrel{+1}{⟶}.$$

By the definition of the cone in a stable category,

$$Cone\left(j_{!}{j}^{*}M\to M\right)\simeq i_{*}{i}^{*}M.$$

This gives

$${Res}_Z^{+}\left(M\right)\simeq i_{*}{i}^{*}M.$$

The second recollement triangle is

$$i_{*}{i}^{!}M⟶M⟶j_{*}{j}^{*}M\stackrel{+1}{⟶}.$$

Thus

$$Cone\left(M\to j_{*}{j}^{*}M\right)\simeq i_{*}{i}^{!}M\left[1\right].$$

Shifting by $[-1]$ gives

$$Cone\left(M\to j_{*}{j}^{*}M\right)[-1]\simeq i_{*}{i}^{!}M.$$

Therefore

$${Res}_Z^{-}\left(M\right)\simeq i_{*}{i}^{!}M.$$

□

Remark 30  

(Elementary residuals). The objects

$$i_{*}{i}^{*}M,i_{*}{i}^{!}M$$

are supported on Z. They measure the failure of M to coincide with the lower and upper open reconstructions

$$j_{!}{j}^{*}M,j_{*}{j}^{*}M.$$

6.2. The Boundary Residual

Definition 30  

(Boundary residual). Let

$$M\in D_c^b\left(X\right).$$

The boundary residual of M along Z is

$${BRes}_Z\left(M\right):=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right).$$

Proposition 7  

(Boundary residual triangle). For every

$$M\in D_c^b\left(X\right),$$

there is a functorial distinguished triangle

$$i_{*}{i}^{*}M⟶{BRes}_Z\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

Proof. 

Consider the composable morphisms

$$j_{!}{j}^{*}M⟶M⟶j_{*}{j}^{*}M.$$

Apply the octahedral axiom to this composable pair. the resulting distinguished triangle is

$$Cone\left(j_{!}{j}^{*}M\to M\right)⟶Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right)⟶Cone\left(M\to j_{*}{j}^{*}M\right)\stackrel{+1}{⟶}.$$

By Definition 30,

$$Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right)={BRes}_Z\left(M\right).$$

By Proposition 6,

$$Cone\left(j_{!}{j}^{*}M\to M\right)\simeq i_{*}{i}^{*}M.$$

The second recollement triangle gives

$$Cone\left(M\to j_{*}{j}^{*}M\right)\simeq i_{*}{i}^{!}M\left[1\right].$$

Substitution gives

$$i_{*}{i}^{*}M⟶{BRes}_Z\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

Functoriality follows from functoriality of the recollement triangles and of the octahedral construction. □

Corollary 3   

(Support of the boundary residual). The object

$${BRes}_Z\left(M\right)$$

is supported on Z.

Proof. 

The triangle of Proposition 7 has first and third terms supported on Z. Since the full subcategory of $D_c^b\left(X\right)$ consisting of objects supported on Z is triangulated, the middle term

$${BRes}_Z\left(M\right)$$

is also supported on Z. □

Remark 31  

(Boundary residual as total residual). The object

$${BRes}_Z\left(M\right)$$

compares the two open reconstructions

$$j_{!}{j}^{*}M\mathrm{and}{j}_{*}{j}^{*}M.$$

The triangle of Proposition 7 expresses this total boundary residual as an extension of the lower and upper closed residual terms.

6.3. Twist–Cotwist Defects of Recollement Adjunctions

Definition 31  

(Adjunction defect). Let

$$L:\mathcal{C}\rightleftarrows \mathcal{D}:R$$

be an adjunction in a locally stable $(\infty ,2)$-category. Its adjunction defect package is the pair

$$\left(cof\left({id}_{\mathcal{C}}\to RL\right),fib\left(LR\to {id}_{\mathcal{D}}\right)\right).$$

Equivalently, this is the twist–cotwist pair

$$\left(T_{\mathcal{C}}(L⊣R),T_{\mathcal{D}}(L⊣R)\right)$$

of Definition 4.

Definition 32  

(Recollement notation for defect computation). For the open–closed decomposition

$$X=U\bigsqcup Z,$$

write

$${\mathcal{C}}_U:=D_c^b\left(U\right),{\mathcal{C}}_X:=D_c^b\left(X\right),{\mathcal{C}}_Z:=D_c^b\left(Z\right).$$

All twist–cotwist functors below are computed in the corresponding endofunctor categories.

Proposition 8  

(Twist–cotwist defects of recollement). With notation as in Definition 32, the four recollement adjunctions have the following defects:

$$T_{{\mathcal{C}}_U}(j_{!}⊣j^{*})\simeq 0,T_{{\mathcal{C}}_X}(j_{!}⊣j^{*})\simeq i_{*}{i}^{*}(-)[-1],$$

$$T_{{\mathcal{C}}_X}(j^{*}⊣j_{*})\simeq i_{*}{i}^{!}(-)\left[1\right],T_{{\mathcal{C}}_U}(j^{*}⊣j_{*})\simeq 0,$$

$$T_{{\mathcal{C}}_X}(i^{*}⊣i_{*})\simeq j_{!}{j}^{*}(-)\left[1\right],T_{{\mathcal{C}}_Z}(i^{*}⊣i_{*})\simeq 0,$$

and

$$T_{{\mathcal{C}}_Z}(i_{*}⊣i^{!})\simeq 0,T_{{\mathcal{C}}_X}(i_{*}⊣i^{!})\simeq j_{*}{j}^{*}(-)[-1].$$

Proof. 

We compute each pair from the unit, counit, and recollement triangles.

For the adjunction

$$j_{!}⊣j^{*},$$

the unit is

$${id}_{{\mathcal{C}}_U}⟶j^{*}{j}_{!}.$$

Since $j_{!}$ is fully faithful in recollement,

$$j^{*}{j}_{!}\simeq {id}_{{\mathcal{C}}_U}.$$

Hence

$$T_{{\mathcal{C}}_U}(j_{!}⊣j^{*})\simeq 0.$$

The counit is

$$j_{!}{j}^{*}⟶{id}_{{\mathcal{C}}_X}.$$

The first recollement triangle gives

$$j_{!}{j}^{*}⟶{id}_{{\mathcal{C}}_X}⟶i_{*}{i}^{*}\stackrel{+1}{⟶}.$$

Thus

$$fib\left(j_{!}{j}^{*}\to {id}_{{\mathcal{C}}_X}\right)\simeq i_{*}{i}^{*}[-1],$$

and therefore

$$T_{{\mathcal{C}}_X}(j_{!}⊣j^{*})\simeq i_{*}{i}^{*}(-)[-1].$$

For the adjunction

$$j^{*}⊣j_{*},$$

the unit is

$${id}_{{\mathcal{C}}_X}⟶j_{*}{j}^{*}.$$

The second recollement triangle gives

$$i_{*}{i}^{!}⟶{id}_{{\mathcal{C}}_X}⟶j_{*}{j}^{*}\stackrel{+1}{⟶}.$$

Hence

$$cof\left({id}_{{\mathcal{C}}_X}\to j_{*}{j}^{*}\right)\simeq i_{*}{i}^{!}\left[1\right],$$

so

$$T_{{\mathcal{C}}_X}(j^{*}⊣j_{*})\simeq i_{*}{i}^{!}(-)\left[1\right].$$

The counit is

$$j^{*}{j}_{*}⟶{id}_{{\mathcal{C}}_U}.$$

Since $j_{*}$ is fully faithful,

$$j^{*}{j}_{*}\simeq {id}_{{\mathcal{C}}_U}.$$

Therefore

$$T_{{\mathcal{C}}_U}(j^{*}⊣j_{*})\simeq 0.$$

For the adjunction

$$i^{*}⊣i_{*},$$

the unit is

$${id}_{{\mathcal{C}}_X}⟶i_{*}{i}^{*}.$$

Using the first recollement triangle again,

$$j_{!}{j}^{*}⟶{id}_{{\mathcal{C}}_X}⟶i_{*}{i}^{*}\stackrel{+1}{⟶},$$

we get

$$cof\left({id}_{{\mathcal{C}}_X}\to i_{*}{i}^{*}\right)\simeq j_{!}{j}^{*}\left[1\right].$$

Hence

$$T_{{\mathcal{C}}_X}(i^{*}⊣i_{*})\simeq j_{!}{j}^{*}(-)\left[1\right].$$

The counit

$$i^{*}{i}_{*}⟶{id}_{{\mathcal{C}}_Z}$$

is an equivalence because $i_{*}$ is fully faithful. Therefore

$$T_{{\mathcal{C}}_Z}(i^{*}⊣i_{*})\simeq 0.$$

For the adjunction

$$i_{*}⊣i^{!},$$

the unit

$${id}_{{\mathcal{C}}_Z}⟶i^{!}{i}_{*}$$

is an equivalence because $i_{*}$ is fully faithful. Hence

$$T_{{\mathcal{C}}_Z}(i_{*}⊣i^{!})\simeq 0.$$

The counit is

$$i_{*}{i}^{!}⟶{id}_{{\mathcal{C}}_X}.$$

The second recollement triangle gives

$$i_{*}{i}^{!}⟶{id}_{{\mathcal{C}}_X}⟶j_{*}{j}^{*}\stackrel{+1}{⟶}.$$

Thus

$$fib\left(i_{*}{i}^{!}\to {id}_{{\mathcal{C}}_X}\right)\simeq j_{*}{j}^{*}[-1],$$

and therefore

$$T_{{\mathcal{C}}_X}(i_{*}⊣i^{!})\simeq j_{*}{j}^{*}(-)[-1].$$

This proves all stated identifications. □

Remark 32  

(Not a twist of one recollement adjunction). The boundary residual

$${BRes}_Z\left(M\right)=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right)$$

is not the twist or cotwist of a single recollement adjunction. Instead, it is an extension of the two nontrivial open-adjunction defect terms computed in Proposition 8.

Proposition 9  

(Boundary residual from open-adjunction defects). For every

$$M\in D_c^b\left(X\right),$$

there is a functorial distinguished triangle

$$T_{{\mathcal{C}}_X}(j_{!}⊣j^{*})\left(M\right)\left[1\right]⟶{BRes}_Z\left(M\right)⟶T_{{\mathcal{C}}_X}(j^{*}⊣j_{*})\left(M\right)\stackrel{+1}{⟶}.$$

Equivalently, this is the boundary residual triangle

$$i_{*}{i}^{*}M⟶{BRes}_Z\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

Proof. 

By Proposition 8,

$$T_{{\mathcal{C}}_X}(j_{!}⊣j^{*})\simeq i_{*}{i}^{*}(-)[-1],$$

so

$$T_{{\mathcal{C}}_X}(j_{!}⊣j^{*})\left(M\right)\left[1\right]\simeq i_{*}{i}^{*}M.$$

Similarly,

$$T_{{\mathcal{C}}_X}(j^{*}⊣j_{*})\simeq i_{*}{i}^{!}(-)\left[1\right],$$

so

$$T_{{\mathcal{C}}_X}(j^{*}⊣j_{*})\left(M\right)\simeq i_{*}{i}^{!}M\left[1\right].$$

Substituting these identifications into the boundary residual triangle of Proposition 7 gives the displayed triangle. □

Remark 33  

(Soundness and coverage). Proposition 9 is a precise recollement-defect statement. It does not assert that every object supported on Z arises as

$${BRes}_Z\left(M\right),$$

nor that every schober-theoretic defect is realized by a constructible recollement residual. It also does not require a nondegenerate sphericalization of the full recollement package.

6.4. Perverse-Heart Formulation

Definition 33  

(Intermediate extension). Let

$$A\in Perv\left(U\right).$$

The intermediate extension of A to X is

$$j_{!*}A:=im\left(j_{!}A\to j_{*}A\right)$$

computed in the abelian category

$$Perv\left(X\right).$$

Proposition 10  

(Residual-minimality of intermediate extension). Let

$$A\in Perv\left(U\right).$$

The object $j_{!*}A$ is characterized by the following properties:

$$j^{*}{j}_{!*}A\simeq A,$$

$$j_{!*}A\mathrm{has}\mathrm{no}\mathrm{nonzero}\mathrm{subobject}\mathrm{supported}\mathrm{on}Z,$$

and

$$j_{!*}A\mathrm{has}\mathrm{no}\mathrm{nonzero}\mathrm{quotient}\mathrm{supported}\mathrm{on}Z.$$

Equivalently, $j_{!*}A$ is the residual-minimal extension of A between $j_{!}A$ and $j_{*}A$.

Proof. 

By definition,

$$j_{!*}A=im(j_{!}A\to j_{*}A).$$

Applying $j^{*}$ gives

$$j^{*}{j}_{!}A\simeq A,j^{*}{j}_{*}A\simeq A.$$

Since $j^{*}$ is exact for the perverse t-structure, the image satisfies

$$j^{*}{j}_{!*}A\simeq A.$$

Let

$$K\subset j_{!*}A$$

be a subobject supported on Z. Then

$$j^{*}K=0.$$

The inclusion

$$K\hookrightarrow j_{!*}A\hookrightarrow j_{*}A$$

gives a subobject of $j_{*}A$ supported on Z. By the standard recollement property of the upper extension $j_{*}$, the object $j_{*}A$ has no nonzero subobject supported on Z. Hence

$$K=0.$$

Let Q

be a quotient of $j_{!*}A$ supported on Z. the quotient map

$$j_{!*}A\to Q$$

composes with the surjection

$$j_{!}A\to j_{!*}A$$

to give a quotient of $j_{!}A$ supported on Z. By the standard recollement property of the lower extension $j_{!}$, the object $j_{!}A$ has no nonzero quotient supported on Z. Hence

$$Q=0.$$

Thus $j_{!*}A$ is an extension of A with neither closed-supported subobject nor closed-supported quotient. Conversely, any extension of A with these two properties is identified with the image of

$$j_{!}A\to j_{*}A$$

by the standard perverse recollement argument [1,3,4]. Therefore it is isomorphic to $j_{!*}A$. □

Remark 34  

(Abelian residual formulation). The object $j_{!*}A$ is the abelian-heart analogue of residual minimization. the lower and upper extensions

$$j_{!}A,j_{*}A$$

are the two extremal open reconstructions, and $j_{!*}A$ is obtained by removing the closed-supported quotient residual from $j_{!}A$ and the closed-supported subobject residual from $j_{*}A$.

7. Iterated Stratifications and Residual Towers

This section iterates the open–closed residual construction of Section 6 over a finite stratification. A filtration by closed unions of strata reduces stratified reconstruction to successive recollement steps, each governed by the adjoint triples and support triangles of constructible sheaf theory [1,2,3,4]. At each step, the boundary residual is the cone comparing the lower and upper open reconstructions. the resulting residual tower is unconditional: it uses only recollement triangles and the octahedral axiom. Any spherical enhancement of such a tower is conditional on the additional shared-object and transportability hypotheses isolated in Section 5; no unconditional spherical completion of ordinary constructible recollement is asserted here.

7.1. Filtrations by Closed Unions of Strata

Definition 34  

(Stratified filtration). Let X be a stratified space. A finite stratified filtration of X is a chain of closed unions of strata

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

For $0\le k\le n$, define

$$U_k:=X_k\setminus X_{k-1}.$$

Let

$$j_k:U_k\hookrightarrow X_k,i_k:X_{k-1}\hookrightarrow X_k$$

be the open and closed inclusions.

Remark 35  

(Open–closed step). For each $0\le k\le n$, the pair

$$X_k=U_k\bigsqcup X_{k-1}$$

is an open–closed decomposition. Hence the categories

$$D_c^b\left(U_k\right),D_c^b\left(X_k\right),D_c^b\left(X_{k-1}\right)$$

satisfy the recollement formalism of Definition 10.

Definition 35  

(Stagewise restriction). Let

$$M\in D_c^b\left(X\right).$$

For $0\le k\le n$, define

$$M_k{:=M|}_{X_k}.$$

Thus

$$M_k\in D_c^b\left(X_k\right).$$

7.2. Iterated MV Packages

Definition 36  

(Stagewise MV package). Let $X_{•}$ be a stratified filtration as in Definition 34. For each $0\le k\le n$, the open–closed pair

$$X_k=U_k\bigsqcup X_{k-1}$$

defines an MV adjunction package

$${\mathfrak{A}}_{MV}(X_k;X_{k-1})\in MVAdj\left(\mathrm{St}\right)$$

by assigning

$${\mathcal{C}}_U:=D_c^b\left(U_k\right),{\mathcal{C}}_X:=D_c^b\left(X_k\right),{\mathcal{C}}_Z:=D_c^b\left(X_{k-1}\right),$$

with structural functors

$$j_{k!},j_k^{*},j_{k*},i_k^{*},i_{k*},i_k^{!}.$$

Definition 37  

(Iterated MV package). The iterated MV adjunction package associated to $X_{•}$ is the finite family

$${\mathfrak{A}}_{MV}\left(X_{•}\right):={\left\{{\mathfrak{A}}_{MV}(X_k;X_{k-1})\right\}}_{k=0}^n,$$

together with the identifications along the shared categories

$$D_c^b\left(X_{k-1}\right)$$

appearing as the closed category at stage k and as the middle category at the preceding stage whenever $k\ge 1$.

Remark 36  

(Diagrammatic meaning). The object ${\mathfrak{A}}_{MV}\left(X_{•}\right)$ is not a single recollement diagram. It is a finite linked system of recollement diagrams. Each stage records one open–closed reconstruction step, and the identifications along

$$D_c^b\left(X_{k-1}\right)$$

encode the iteration.

Definition 38  

(Iterated package in a locally stable target). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category. An iterated MV package in $\mathcal{A}$ indexed by $X_{•}$ is a finite family

$${\left\{{\mathfrak{A}}_k\right\}}_{k=0}^n,{\mathfrak{A}}_k\in MVAdj\left(\mathcal{A}\right),$$

together with identifications of the closed object of ${\mathfrak{A}}_k$ with the middle object of ${\mathfrak{A}}_{k-1}$, whenever $k\ge 1$.

Remark 37  

(Soundness of iteration). The construction in Definition 38 assumes only the specified finite filtration and the corresponding recollement data at each stage. It does not assert that every constructible object is determined uniquely by the residual tower without the gluing morphisms. the reconstruction and coverage questions are separate from the existence of the iterated package.

7.3. Residual Towers

Definition 39  

(Stagewise boundary residual). Let $X_{•}$ be a stratified filtration and let

$$M\in D_c^b\left(X\right).$$

For $0\le k\le n$, let

$$M_k{:=M|}_{X_k}.$$

The k-th boundary residual of M is

$${BRes}_k\left(M\right):=Cone\left(j_{k!}{j}_k^{*}{M}_k\to j_{k*}{j}_k^{*}{M}_k\right)\in D_c^b\left(X_k\right).$$

Definition 40  

(Residual tower). The residual tower of M with respect to $X_{•}$ is the finite family

$${\mathfrak{R}}_{X_{•}}\left(M\right):={\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n.$$

Proposition 11  

(Stagewise residual triangle). For every

$$M\in D_c^b\left(X\right)$$

and every $0\le k\le n$, there is a functorial distinguished triangle

$$i_{k*}{i}_k^{*}{M}_k⟶{BRes}_k\left(M\right)⟶i_{k*}{i}_k^{!}{M}_k\left[1\right]\stackrel{+1}{⟶}.$$

Proof. 

Fix k. the open–closed decomposition

$$X_k=U_k\bigsqcup X_{k-1}$$

has inclusions

$$j_k:U_k\hookrightarrow X_k,i_k:X_{k-1}\hookrightarrow X_k.$$

Apply Proposition 7 to the object

$$M_k\in D_c^b\left(X_k\right).$$

The boundary residual in that proposition is

$$Cone\left(j_{k!}{j}_k^{*}{M}_k\to j_{k*}{j}_k^{*}{M}_k\right),$$

which is ${BRes}_k\left(M\right)$ by Definition 39. the resulting distinguished triangle is

$$i_{k*}{i}_k^{*}{M}_k⟶{BRes}_k\left(M\right)⟶i_{k*}{i}_k^{!}{M}_k\left[1\right]\stackrel{+1}{⟶}.$$

Functoriality follows from the functoriality in Proposition 7. □

Corollary 4 

(Support of stagewise residuals). For every $0\le k\le n$, the object

$${BRes}_k\left(M\right)$$

is supported on $X_{k-1}\subset X_k$.

Proof. 

By Proposition 11, the object

$${BRes}_k\left(M\right)$$

lies in a distinguished triangle whose other two terms are

$$i_{k*}{i}_k^{*}{M}_k,i_{k*}{i}_k^{!}{M}_k\left[1\right].$$

Both are in the essential image of

$$i_{k*}:D_c^b\left(X_{k-1}\right)\to D_c^b\left(X_k\right).$$

The essential image of $i_{k*}$ is a triangulated subcategory. Hence

$${BRes}_k\left(M\right)$$

is also supported on $X_{k-1}$. □

Remark 38  

(Interpretation of the tower). The family

$${\mathfrak{R}}_{X_{•}}\left(M\right)={\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n$$

records, stage by stage, the difference between lower and upper reconstruction from the open stratum $U_k$ inside $X_k$. the tower depends on the chosen filtration $X_{•}$.

Remark 39  

(No reconstruction without gluing data). The residual tower alone does not reconstruct M. Reconstruction also requires the underlying open restrictions

$$j_k^{*}{M}_k$$

and the gluing morphisms appearing in the corresponding MV packages. Thus

$${\mathfrak{R}}_{X_{•}}\left(M\right)$$

is a residual invariant of the stratified reconstruction process, not a complete invariant of M.

7.4. Conditional Spherical Enhancement of Residual Towers

Definition 41  

(Conditionally spherical iterated package). Let $\mathcal{A}$ be a locally stable $(\infty ,2)$-category satisfying the hypotheses of Theorem 5. Let

$${\mathfrak{A}}_{MV}\left(X_{•}\right)$$

be an iterated MV package in $\mathcal{A}$. A conditionally spherical enhancement of ${\mathfrak{A}}_{MV}\left(X_{•}\right)$ is a choice, at every stage k, of the additional data required in Definition 27 and Definition 28, compatible with the identifications of shared categories in Definition 37.

Proposition 12  

(Conditional stagewise spherical enhancement). Assume $\mathcal{A}$ satisfies the hypotheses of Theorem 5. Let

$${\mathfrak{A}}_{MV}\left(X_{•}\right)$$

be an iterated MV package in $\mathcal{A}$. If

$${\mathfrak{A}}_{MV}\left(X_{•}\right)$$

is equipped with a conditionally spherical enhancement in the sense of Definition 41, then the objectwise sphericalizations of its stagewise selected adjunctions assemble into a compatible family of spherical MV packages.

Proof. 

For each $0\le k\le n$, the stage

$${\mathfrak{A}}_{MV}(X_k;X_{k-1})$$

is an object of $MVAdj\left(\mathcal{A}\right)$. By Theorem 5, each selected adjunction at that stage has an objectwise sphericalization. the conditionally spherical enhancement supplies the shared-object identifications and transportability of compatibility morphisms required by Theorem 6. Hence each stage gives a spherical MV package in the conditional sense of Section 5. the compatibility of these data with the identifications of shared categories in Definition 37 allows the stagewise spherical packages to assemble into a compatible finite family. □

Corollary 5  

(Conditional universal property of the iterated enhancement). Let

$${\mathfrak{A}}_{MV}\left(X_{•}\right)$$

be an iterated MV package equipped with a conditionally spherical enhancement, and let

$${\mathfrak{B}}_{•}$$

be an iterated spherical MV package indexed by the same filtration $X_{•}$. Assume the objectwise universal properties are compatible with the finite diagram of transported MV relations at every stage. Then the corresponding mapping spaces satisfy the finite-limit equivalence

$$Map\left(S_{MV}\left({\mathfrak{A}}_{MV}\left(X_{•}\right)\right),{\mathfrak{B}}_{•}\right)\simeq Map\left({\mathfrak{A}}_{MV}\left(X_{•}\right),\iota {\mathfrak{B}}_{•}\right),$$

where the mapping spaces are computed in the corresponding conditional diagram categories.

Proof. 

For each stage k, Corollary 2 gives the conditional equivalence

$${Map}_{SphMV\left(\mathcal{A}\right)}\left(S_{MV}\left({\mathfrak{A}}_{MV}(X_k;X_{k-1})\right),{\mathfrak{B}}_k\right)\simeq {Map}_{{MVAdj}^{comp}\left(\mathcal{A}\right)}\left({\mathfrak{A}}_{MV}(X_k;X_{k-1}),\iota {\mathfrak{B}}_k\right).$$

The iterated package is a finite diagram obtained by imposing compatibility along the shared stage categories. Taking the finite limit of the stagewise mapping-space equivalences over this compatibility diagram gives the displayed equivalence. □

Remark 40  

(Conditional nature). The residual tower

$${\mathfrak{R}}_{X_{•}}\left(M\right)$$

exists unconditionally in the constructible derived category. A spherical enhancement of the iterated MV package is conditional on additional shared-object and transportability data. It is not a formal consequence of ordinary recollement and is not asserted for arbitrary constructible stratifications.

Remark 41  

(Possible comparison problems). The residual tower construction produces an invariant of M relative to a fixed filtration $X_{•}$. It does not assert invariance under refinement of the filtration, compatibility with all realization functors, or preservation of compact, constructible, or motivic subcategories under any spherical enhancement. These are separate comparison problems.

8. Filtered and Graded Interpretation

This section records the filtered–graded interpretation of the residual tower. Universal sphericalization has a filtered–graded model: the walking spherical adjunction is represented by the associated-graded adjunction between filtered and graded spectra [7,12]. This fact explains why spherical gluing theories naturally interact with filtrations. In the present paper, however, the residual towers are defined before and independently of sphericalization: they are obtained from successive open–closed recollement triangles and the octahedral axiom [1,2,3]. Thus the filtered–graded language below is interpretive. It describes

$${\left\{{BRes}_k\left(M\right)\right\}}_k$$

as boundary associated-graded data for stratified reconstruction, not as a splitting of M, not as a theorem of unconditional spherical completion, and not as a motivic or Hodge-theoretic lift.

8.1. The Universal Filtered/Graded Model

Definition 42  

(Filtered and graded spectra). Let $\mathrm{Sp}$ denote the stable ∞-category of spectra. Define

$$f\mathrm{Sp}:=Fun(\mathbf{Z},\mathrm{Sp}),$$

the stable ∞-category of filtered spectra, and define

$$g\mathrm{Sp}:=Fun({\mathbf{Z}}^{\simeq },\mathrm{Sp}),$$

the stable ∞-category of graded spectra. For

$$X={(X_m,{\phi }_m)}_{m\in \mathbf{Z}}\in f\mathrm{Sp},$$

where

$${\phi }_m:X_m\to X_{m+1},$$

the associated graded object is

$$gr{\left(X\right)}_m:=cof\left({\phi }_{m-1}\right)$$

for every $m\in \mathbf{Z}$.

Definition 43  

(Trivial filtration). For

$$Y={\left(Y_m\right)}_{m\in \mathbf{Z}}\in g\mathrm{Sp},$$

let

$$triv\left(Y\right)\in f\mathrm{Sp}$$

denote the filtered spectrum whose m-th filtered piece is $Y_m$ and whose transition maps are zero:

$$Y_m⟶Y_{m+1}.$$

Definition 44  

(Walking adjunctions). Let

$$Y(+)$$

denote the positive walking adjunction, and let

$$Y(-)$$

denote the negative walking adjunction, in the sense of the universal sphericalization formalism. Thus $S\left(Y\right(+\left)\right)$ and $S\left(Y\right(-\left)\right)$ denote their universal sphericalizations.

Theorem 7  

(Universal filtered–graded spherical adjunction). There is an equivalence of spherical adjunctions

$$S\left(Y(+)\right)\simeq \left(gr:f\mathrm{Sp}\rightleftarrows g\mathrm{Sp}:triv\right).$$

Dually, there is an equivalence

$$S\left(Y(-)\right)\simeq \left(triv[-1]\langle 1\rangle :g\mathrm{Sp}\rightleftarrows f\mathrm{Sp}:gr\right).$$

Proof. 

This is the filtered–graded description of the walking spherical adjunction proved in [7]. the proof uses Dold–Kan type equivalences for stable ∞-categories and the compatibility of those equivalences with the adjunction data [7,12]. □

Remark 42  

(Use in this paper). Theorem 7 is not used to define constructible residuals. Those residuals were defined in Section 6 from recollement and the octahedral axiom. the theorem is used only as a conceptual guide: it explains why filtrations and associated-graded data are natural in spherical contexts. The residual tower itself remains an ordinary constructible recollement construction.

8.2. Stratified Objects as Filtered Objects

Definition 45  

(Stratified filtration of an object). Let

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

be a filtration by closed unions of strata. Let

$$M\in D_c^b\left(X\right).$$

For $0\le k\le n$, set

$$M_k{:=M|}_{X_k}.$$

The family

$$M_{•}:=(M_0,M_1,\dots ,M_n)$$

is the stratified filtration of M associated to $X_{•}$.

Definition 46  

(Stagewise open piece). For $0\le k\le n$, let

$$U_k:=X_k\setminus X_{k-1},$$

and let

$$j_k:U_k\hookrightarrow X_k,i_k:X_{k-1}\hookrightarrow X_k$$

be the complementary inclusions. the k-th open stratum restriction of M is

$$j_k^{*}{M}_k\in D_c^b\left(U_k\right).$$

Definition 47  

(Boundary associated-graded term). For $M\in D_c^b\left(X\right)$ and $0\le k\le n$, define

$${gr}_k^{\partial }\left(M\right):={BRes}_k\left(M\right)=Cone\left(j_{k!}{j}_k^{*}{M}_k\to j_{k*}{j}_k^{*}{M}_k\right).$$

The finite family

$${gr}_{X_{•}}^{\partial }\left(M\right):={\left\{{gr}_k^{\partial }\left(M\right)\right\}}_{k=0}^n$$

is the boundary associated-graded residual family of M with respect to $X_{•}$.

Proposition 13  

(Stagewise boundary graded triangle). For every $M\in D_c^b\left(X\right)$ and every $0\le k\le n$, there is a functorial distinguished triangle

$$i_{k*}{i}_k^{*}{M}_k⟶{gr}_k^{\partial }\left(M\right)⟶i_{k*}{i}_k^{!}{M}_k\left[1\right]\stackrel{+1}{⟶}.$$

Proof. 

By Definition 47,

$${gr}_k^{\partial }\left(M\right)={BRes}_k\left(M\right).$$

The assertion is therefore Proposition 11. □

Remark 43  

(Boundary graded data). The family

$${gr}_{X_{•}}^{\partial }\left(M\right)$$

is not an associated graded object of M in the sense of a chosen splitting of $M_{•}$. It is the boundary analogue of associated-graded data: it records the successive defects between lower and upper reconstruction from $U_k$ at each stage

$$X_k=U_k\bigsqcup X_{k-1}.$$

8.3. Residual Towers as Boundary Associated-Graded Data

Definition 48  

(Boundary graded residual tower). Let

$$X_{•}:⌀=X_{-1}\subset X_0\subset \dots \subset X_n=X$$

be a filtration by closed unions of strata, and let

$$M\in D_c^b\left(X\right).$$

The boundary graded residual tower of M is

$${gr}_{X_{•}}^{\partial }\left(M\right)={\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n.$$

Proposition 14  

(Residual tower as boundary associated-graded data). Let

$$M\in D_c^b\left(X\right)$$

and let $X_{•}$ be a filtration by closed unions of strata. the residual tower

$${\mathfrak{R}}_{X_{•}}\left(M\right)={\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n$$

is equal to the boundary graded residual tower

$${gr}_{X_{•}}^{\partial }\left(M\right).$$

Each term is the ordinary constructible boundary term associated to the corresponding open–closed recon- struction step

$$X_k=U_k\bigsqcup X_{k-1}.$$

Proof. 

By Definition 40,

$${\mathfrak{R}}_{X_{•}}\left(M\right)={\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n.$$

By Definition 48,

$${gr}_{X_{•}}^{\partial }\left(M\right)={\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n.$$

Hence the two finite families are equal. For each k, the term

$${BRes}_k\left(M\right)$$

is the cone

$$Cone\left(j_{k!}{j}_k^{*}{M}_k\to j_{k*}{j}_k^{*}{M}_k\right),$$

and so it is exactly the boundary term comparing lower and upper open reconstruction at the stage

$$X_k=U_k\bigsqcup X_{k-1}.$$

□

Remark 44  

(No arbitrary graded splitting). Proposition 14 does not choose a splitting of the stratified filtration $M_{•}$. It also does not identifyM

with a direct sum of residual terms. the residual tower records boundary defects of successive recollement steps; reconstruction still requires the open restrictions and the gluing morphisms.

Remark 45  

(Conditional relation with sphericalization). The filtered–graded model of Theorem 7 suggests why spherical gluing theories naturally interact with filtrations and associated graded objects. In this paper, however, the residual tower is constructed without invoking sphericalization. Any comparison with a spherical filtered–graded MV package requires the additional shared-object and transportability hypotheses of Section 5. Likewise, compatibility with Hodge, motivic, or realization-theoretic refinements requires separately constructed realization categories and functors. No arbitrary Hodge projector, unconstructed motivic lift, or target-class-dependent splitting is used in this section.

9. Duality and Functoriality

Verdier duality is one of the fundamental symmetries of constructible and perverse sheaf theory, exchanging lower and upper extension functors while preserving the recollement formalism and the self-duality of the perverse t-structure [1,2,4]. Since the residual constructions introduced in Section 6 are defined from recollement triangles and cones, it is natural to record their duality behavior. the lower and upper residuals are exchanged by Verdier duality. the boundary residual is not strictly self-dual; it is exchanged with the boundary residual of the Verdier dual object up to a shift. We also record a cautious refinement statement for filtrations. No invariance theorem under refinement is claimed.

9.1. Verdier Duality Exchanges Lower and Upper Residuals

Proposition 15  

(Duality of MV residuals). Let

$$X=U\bigsqcup Z$$

be an open–closed decomposition of a stratified space, and let

$$M\in D_c^b\left(X\right).$$

Then there are canonical isomorphisms

$$\mathbb{D}{Res}_Z^{+}\left(M\right)\simeq {Res}_Z^{-}\left(\mathbb{D}M\right),$$

and

$$\mathbb{D}{Res}_Z^{-}\left(M\right)\simeq {Res}_Z^{+}\left(\mathbb{D}M\right).$$

Moreover,

$$\mathbb{D}{BRes}_Z\left(M\right)\simeq {BRes}_Z\left(\mathbb{D}M\right)[-1].$$

Proof. 

Verdier duality is an exact contravariant equivalence

$$\mathbb{D}:D_c^b\left(X\right)⟶D_c^b\left(X\right),$$

and satisfies the standard identities

$$\mathbb{D}{j}_{!}\simeq j_{*}\mathbb{D},\mathbb{D}{j}_{*}\simeq j_{!}\mathbb{D},$$

together with

$$\mathbb{D}{i}^{*}\simeq i^{!}\mathbb{D},\mathbb{D}{i}^{!}\simeq i^{*}\mathbb{D}$$

[1,2].

Applying $\mathbb{D}$ to

$${Res}_Z^{+}\left(M\right)=Cone\left(j_{!}{j}^{*}M\to M\right)$$

gives

$$\mathbb{D}{Res}_Z^{+}\left(M\right)\simeq Cone\left(\mathbb{D}M\to j_{*}{j}^{*}\mathbb{D}M\right)[-1],$$

which is precisely

$${Res}_Z^{-}\left(\mathbb{D}M\right).$$

Similarly, applying $\mathbb{D}$ to

$${Res}_Z^{-}\left(M\right)=Cone\left(M\to j_{*}{j}^{*}M\right)[-1]$$

gives

$$\mathbb{D}{Res}_Z^{-}\left(M\right)\simeq Cone\left(j_{!}{j}^{*}\mathbb{D}M\to \mathbb{D}M\right),$$

which is

$${Res}_Z^{+}\left(\mathbb{D}M\right).$$

For the boundary residual, use the defining cone

$${BRes}_Z\left(M\right)=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right).$$

Applying $\mathbb{D}$ gives

$$\mathbb{D}{BRes}_Z\left(M\right)\simeq Cone\left(j_{!}{j}^{*}\mathbb{D}M\to j_{*}{j}^{*}\mathbb{D}M\right)[-1],$$

because duality reverses the arrow

$$j_{!}{j}^{*}M⟶j_{*}{j}^{*}M$$

to the corresponding arrow

$$j_{!}{j}^{*}\mathbb{D}M⟶j_{*}{j}^{*}\mathbb{D}M$$

after the usual cone shift for a contravariant exact functor. Therefore

$$\mathbb{D}{BRes}_Z\left(M\right)\simeq {BRes}_Z\left(\mathbb{D}M\right)[-1].$$

□

Corollary 6 

(Shifted self-duality of the boundary residual). If

$$M\simeq \mathbb{D}M,$$

then

$$\mathbb{D}{BRes}_Z\left(M\right)\simeq {BRes}_Z\left(M\right)[-1].$$

Proof. 

Substitute

$$M\simeq \mathbb{D}M$$

into Proposition 15. □

Remark 46  

(Residual symmetry). The lower and upper residuals are exchanged by Verdier duality. the boundary residual is preserved only up to the shift

$$[-1].$$

Thus the boundary residual is naturally shift-skew under Verdier duality.

9.2. Compatibility with Refinement of Stratifications

Definition 49  

(Refinement of filtrations). Let

$$X_{•}:⌀=X_{-1}\subset X_0\subset \dots \subset X_n=X$$

and

$$X_{•}^{\prime }:⌀=X_{-1}^{\prime }\subset X_0^{\prime }\subset \dots \subset X_m^{\prime }=X$$

be filtrations by closed unions of strata. We say that

$$X_{•}^{\prime }$$

refines

$$X_{•}$$

if every closed subset

$$X_k$$

occurs among the closed subsets

$$X_{\ell }^{\prime }.$$

Definition 50  

(Common stages). Let $X_{•}^{\prime }$ refine $X_{•}$. For each $0\le k\le n$, let

$$\rho \left(k\right)$$

be the unique integer such that

$$X_{\rho \left(k\right)}^{\prime }=X_k.$$

The integers $\rho \left(k\right)$ are called the common-stage indices of the refinement.

Definition 51  

(Restriction of a refined residual tower to common stages). Let $X_{•}^{\prime }$ refine $X_{•}$, and let

$$M\in D_c^b\left(X\right).$$

The restriction of the refined residual tower

$${\mathfrak{R}}_{X_{•}^{\prime }}\left(M\right)$$

to the common stages is the finite family

$${\left\{{BRes}_{\rho \left(k\right)}^{\prime }\left(M\right)\right\}}_{k=0}^n,$$

where

$${BRes}_{\ell }^{\prime }\left(M\right):=Cone\left(j_{\ell !}^{\prime }{j}_{\ell }^{\prime *}{M}_{\ell }^{\prime }\to j_{\ell *}^{\prime }{j}_{\ell }^{\prime *}{M}_{\ell }^{\prime }\right)$$

is the boundary residual associated to the filtration $X_{•}^{\prime }$, and

$$M_{\ell }^{\prime }{:=M|}_{X_{\ell }^{\prime }}.$$

Remark 47  

(Refinement changes the residual terms). A refinement generally changes the open pieces

$$U_k=X_k\setminus X_{k-1}$$

by replacing one open–closed step with several smaller steps. Consequently, the residual tower for $X_{•}^{\prime }$ is not canonically the same object as the residual tower for $X_{•}$. Any comparison between the two requires additional choices or an explicit comparison diagram of open–closed decompositions.

Definition 52  

(Residual comparison datum). Let $X_{•}^{\prime }$ refine $X_{•}$. A residual comparison datum from

$${\mathfrak{R}}_{X_{•}}\left(M\right)$$

to

$${\mathfrak{R}}_{X_{•}^{\prime }}\left(M\right)$$

is a choice, for every $0\le k\le n$, of a morphism

$${BRes}_k\left(M\right)⟶{BRes}_{\rho \left(k\right)}^{\prime }\left(M\right)$$

compatible with the corresponding maps of open–closed recollement diagrams at the common stage

$$X_k=X_{\rho \left(k\right)}^{\prime }.$$

Proposition 16  

(Conditional functoriality under refinement). Let

$$X_{•}^{\prime }$$

refine

$$X_{•}.$$

Let

$$M\in D_c^b\left(X\right).$$

Suppose a residual comparison datum is given in the sense of Definition 52. Then this datum determines a morphism from the residual tower of M with respect to $X_{•}$ to the restriction of the residual tower of M with respect to $X_{•}^{\prime }$ along the common stages:

$${\mathfrak{R}}_{X_{•}}\left(M\right)⟶{\left\{{BRes}_{\rho \left(k\right)}^{\prime }\left(M\right)\right\}}_{k=0}^n.$$

Proof. 

By Definition 52, for each $0\le k\le n$ we are given a morphism

$${BRes}_k\left(M\right)⟶{BRes}_{\rho \left(k\right)}^{\prime }\left(M\right).$$

The residual tower

$${\mathfrak{R}}_{X_{•}}\left(M\right)$$

is the finite family

$${\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n.$$

The restricted refined tower is the finite family

$${\left\{{BRes}_{\rho \left(k\right)}^{\prime }\left(M\right)\right\}}_{k=0}^n.$$

Collecting the chosen componentwise morphisms gives the displayed morphism of finite families. □

Remark 48  

(Soundness versus invariance). Proposition 16 is deliberately conditional. It does not assert that every refinement produces canonical comparison morphisms without choices. It does not assert that the resulting residual towers are canonically isomorphic. It does not assert that any spherical enhancement is independent of the chosen filtration.

Remark 49  

(Future comparison problems). The comparison problem for residual towers under refinement is a separate question. Natural directions include barycentric subdivision, Whitney refinements, Verdier specialization, realization functors, and singularity-specific stratifications. the present paper uses only the residual tower associated to a fixed filtration.

10. Examples

This section records elementary instances of the residual formalism. the examples are not classification results. They show how the definitions above specialize to familiar situations in constructible and perverse sheaf theory: a single closed stratum, the intermediate extension in the perverse heart, finite stratifications, and the basic one-point stratification $X={\mathbb{A}}^1$, $Z=\left\{0\right\}$. the last example is included to make the degeneracy mechanism concrete: in ordinary recollement, the identities

$$j^{*}{j}_{!}\simeq id,j^{*}{j}_{*}\simeq id$$

force zero twist or cotwist terms on the open category. Thus the raw recollement coordinates are not the coordinates in which a nondegenerate spherical structure is visible [1,2,3,4,8].

10.1. A Single Closed Stratum

Definition 53  

(Single-stratum decomposition). Let X be a stratified space. Let

$$Z\subset X$$

be a single closed stratum, and set

$$U:=X\setminus Z.$$

Let

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

be the inclusions.

Proposition 17  

(Residuals for one closed stratum). Let $Z\subset X$ be a single closed stratum as in Definition 53. For every

$$M\in D_c^b\left(X\right),$$

there are canonical isomorphisms

$${Res}_Z^{+}\left(M\right)\simeq i_{*}{i}^{*}M,{Res}_Z^{-}\left(M\right)\simeq i_{*}{i}^{!}M.$$

In particular,

$${Res}_Z^{+}\left(M\right),{Res}_Z^{-}\left(M\right)$$

are supported on Z.

Proof. 

The isomorphisms are the identifications of Proposition 6. Since $i_{*}$ has essential image the full triangulated subcategory of $D_c^b\left(X\right)$ consisting of objects supported on Z, both

$$i_{*}{i}^{*}M\mathrm{and}{i}_{*}{i}^{!}M$$

are supported on Z. Hence the two residuals are supported on Z. □

Proposition 18  

(Boundary residual for one closed stratum). Let $Z\subset X$ be a single closed stratum. For every

$$M\in D_c^b\left(X\right),$$

the boundary residual

$${BRes}_Z\left(M\right)=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right)$$

is supported on Z, and it fits into a functorial distinguished triangle

$$i_{*}{i}^{*}M⟶{BRes}_Z\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

Proof. 

The distinguished triangle is Proposition 7. Its first and third terms lie in the essential image of

$$i_{*}:D_c^b\left(Z\right)\to D_c^b\left(X\right).$$

This essential image is a triangulated subcategory. Therefore the middle term

$${BRes}_Z\left(M\right)$$

also lies in the essential image of $i_{*}$, and hence is supported on Z. □

Remark 50  

(Interpretation). For a single closed stratum, the boundary residual measures the gap between the two open reconstructions

$$j_{!}{j}^{*}M\mathrm{and}{j}_{*}{j}^{*}M.$$

The residual is entirely closed-stratum supported. It is therefore a local boundary object for the open–closed pair $X=U\bigsqcup Z$.

10.2. Perverse Sheaves and Intermediate Extension

Definition 54  

(Perverse open datum). Let $X=U\bigsqcup Z$ be an open–closed decomposition, and let

$$A\in Perv\left(U\right).$$

The lower and upper perverse extensions of A are

$$j_{!}A\in Perv\left(X\right),j_{*}A\in Perv\left(X\right),$$

whenever $j_{!}$ and $j_{*}$ are taken in the perverse recollement sense.

Proposition 19  

(Intermediate extension as residual-minimal extension). Let

$$A\in Perv\left(U\right).$$

Then

$$j_{!*}A=im\left(j_{!}A\to j_{*}A\right)$$

in $Perv\left(X\right)$. Moreover $j_{!*}A$ is characterized by

$$j^{*}{j}_{!*}A\simeq A,$$

and by the two vanishing conditions:

$$j_{!*}A\mathrm{has}\mathrm{no}\mathrm{nonzero}\mathrm{subobject}\mathrm{supported}\mathrm{on}Z,$$

$$j_{!*}A\mathrm{has}\mathrm{no}\mathrm{nonzero}\mathrm{quotient}\mathrm{supported}\mathrm{on}Z.$$

Proof. 

This is Proposition 10. the equality

$$j_{!*}A=im\left(j_{!}A\to j_{*}A\right)$$

is the standard definition of the intermediate extension in the perverse heart. the absence of a nonzero closed-supported subobject follows from the embedding into $j_{*}A$. the absence of a nonzero closed-supported quotient follows from the quotient of $j_{!}A$. Conversely, any extension of A to X satisfying these two conditions identifies with the image of

$$j_{!}A\to j_{*}A$$

by the standard perverse recollement argument [1,3,4]. □

Definition 55  

(Perverse boundary residual). For

$$A\in Perv\left(U\right),$$

define the perverse boundary residual by

$${BRes}_Z^{♥}\left(A\right):=p{H}^{0}Cone\left(j_{!}A\to j_{*}A\right).$$

Remark 51  

(Abelian residual formulation). The object

$${BRes}_Z^{♥}\left(A\right)$$

is the perverse-heart form of the boundary residual. the intermediate extension

$$j_{!*}A$$

is residual-minimal: it removes closed-supported quotients from $j_{!}A$ and closed-supported subobjects from $j_{*}A$.

10.3. Finite Stratifications

Definition 56  

(Finite filtration example). Let

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

be a finite filtration by closed unions of strata. For $0\le k\le n$, set

$$U_k:=X_k\setminus X_{k-1},$$

and let

$$j_k:U_k\hookrightarrow X_k,i_k:X_{k-1}\hookrightarrow X_k$$

be the complementary inclusions.

Proposition 20  

(Residual tower for a finite stratification). Let

$$M\in D_c^b\left(X\right).$$

For $0\le k\le n$, set

$$M_k{:=M|}_{X_k}$$

and define

$${BRes}_k\left(M\right):=Cone\left(j_{k!}{j}_k^{*}{M}_k\to j_{k*}{j}_k^{*}{M}_k\right).$$

Then each stage fits into a functorial distinguished triangle

$$i_{k*}{i}_k^{*}{M}_k⟶{BRes}_k\left(M\right)⟶i_{k*}{i}_k^{!}{M}_k\left[1\right]\stackrel{+1}{⟶}.$$

Proof. 

This is Proposition 11, applied to each open–closed decomposition

$$X_k=U_k\bigsqcup X_{k-1}.$$

□

Remark 52  

(Successive closed-stratum corrections). The tower

$${\left\{{BRes}_k\left(M\right)\right\}}_{k=0}^n$$

records the successive closed-stratum corrections produced by reconstructing M over

$$X_0,X_1,\dots ,X_n=X.$$

It does not determine M without the corresponding open restrictions

$$j_k^{*}{M}_k$$

and the gluing morphisms in the iterated MV package.

Remark 53  

(Conditional spherical enhancement). A spherical enhancement of the iterated MV package associated to the filtration is not automatic. It requires the additional shared-object and transportability data specified in Section 5. Thus the residual tower exists unconditionally, while any spherical completion of the tower is conditional.

10.4. The One-Point Stratification of the Affine Line

Definition 57  

(The basic one-point example). Let

$$X={\mathbb{A}}^1,Z=\left\{0\right\},U={\mathbb{A}}^1\setminus \left\{0\right\}\simeq {\mathbb{G}}_m.$$

Let

$$j:{\mathbb{G}}_m\hookrightarrow {\mathbb{A}}^1,i:\left\{0\right\}\hookrightarrow {\mathbb{A}}^1$$

be the inclusions.

Proposition 21  

(Raw recollement is not spherical in the one-point example). For the open–closed decomposition

$${\mathbb{A}}^1={\mathbb{G}}_m\bigsqcup \left\{0\right\},$$

the ordinary recollement adjunctions satisfy

$$j^{*}{j}_{!}\simeq {id}_{D_c^b\left({\mathbb{G}}_m\right)},j^{*}{j}_{*}\simeq {id}_{D_c^b\left({\mathbb{G}}_m\right)}.$$

Consequently,

$$T_{D_c^b\left({\mathbb{G}}_m\right)}(j_{!}⊣j^{*})\simeq 0,T_{D_c^b\left({\mathbb{G}}_m\right)}(j^{*}⊣j_{*})\simeq 0.$$

Similarly,

$$i^{*}{i}_{*}\simeq {id}_{D_c^b\left(\left\{0\right\}\right)},i^{!}{i}_{*}\simeq {id}_{D_c^b\left(\left\{0\right\}\right)},$$

and hence the corresponding closed-category defects vanish:

$$T_{D_c^b\left(\left\{0\right\}\right)}(i^{*}⊣i_{*})\simeq 0,T_{D_c^b\left(\left\{0\right\}\right)}(i_{*}⊣i^{!})\simeq 0.$$

Thus the raw constructible recollement package for

$${\mathbb{A}}^1={\mathbb{G}}_m\bigsqcup \left\{0\right\}$$

is not a nondegenerate naive spherical MV package.

Proof. 

The identities

$$j^{*}{j}_{!}\simeq {id}_{D_c^b\left({\mathbb{G}}_m\right)},j^{*}{j}_{*}\simeq {id}_{D_c^b\left({\mathbb{G}}_m\right)}$$

are the full-faithfulness identities for the lower and upper open extensions in recollement. The identities

$$i^{*}{i}_{*}\simeq {id}_{D_c^b\left(\left\{0\right\}\right)},i^{!}{i}_{*}\simeq {id}_{D_c^b\left(\left\{0\right\}\right)}$$

are the full-faithfulness identities for the closed embedding. Applying the definitions of twist and cotwist gives the four displayed zero defect terms. Since the categories

$$D_c^b\left({\mathbb{G}}_m\right)\mathrm{and}{D}_c^b\left(\left\{0\right\}\right)$$

are nonzero, the zero endofunctor is not an equivalence on either category. Hence the raw recollement package is not nondegenerately spherical. □

Proposition 22  

(Boundary residual in the one-point example). Let

$$M\in D_c^b\left({\mathbb{A}}^1\right).$$

For the decomposition

$${\mathbb{A}}^1={\mathbb{G}}_m\bigsqcup \left\{0\right\},$$

the boundary residual is

$${BRes}_{\left\{0\right\}}\left(M\right)=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right),$$

and it fits into a functorial distinguished triangle

$$i_{*}{i}^{*}M⟶{BRes}_{\left\{0\right\}}\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

Equivalently,

$${BRes}_{\left\{0\right\}}\left(M\right)$$

is an extension of the two nontrivial open-adjunction defect terms

$$T_{D_c^b\left({\mathbb{A}}^1\right)}(j_{!}⊣j^{*})\left(M\right)\left[1\right]\mathrm{and}{T}_{D_c^b\left({\mathbb{A}}^1\right)}(j^{*}⊣j_{*})\left(M\right).$$

Proof. 

This is Proposition 9 applied to

$$X={\mathbb{A}}^1,Z=\left\{0\right\},U={\mathbb{G}}_m.$$

□

Remark 54  

(Meaning of the example). The one-point stratification of ${\mathbb{A}}^1$ displays both sides of the paper’s thesis. On the one hand, the raw recollement adjunctions are not nondegenerately spherical because full-faithfulness forces zero defects on the open and closed categories. On the other hand, the boundary residual remains nontrivial and is governed by the two nonzero defect terms on the middle category $D_c^b\left({\mathbb{A}}^1\right)$. This is why the passage from recollement to spherical gluing should be understood as a change of coordinates, not as naive sphericality of the raw $j_{!},j^{*},j_{*}$ recollement package.

11. Motivic and Schober-Theoretic Outlook

In this section we record two conditional extensions of the construction. First, the residual formalism is not intrinsically restricted to ordinary constructible sheaves: it uses only open–closed recollement triangles and the octahedral axiom, hence it has analogues in any six-functor setting where such recollement is available [1,2,9,10]. Second, the relation with spherical gluing and schobers is more delicate. Universal sphericalization applies to individual adjunctions under cocompleteness and local-stability hypotheses [7]. Raw recollement, however, is not itself a nondegenerate spherical package, because full-faithfulness forces several twist–cotwist defects to vanish. Thus the motivic and schober-theoretic statements below are formulated as conditional outlooks, not as unconditional extensions [6,8].

11.1. Motivic Residuals

Definition 58  

(Motivic six-functor recollement context). Let X be a scheme or analytic space equipped with a finite stratification. Let

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

be an open–closed decomposition by unions of strata. A motivic six-functor recollement context on X consists of stable categories

$$\mathcal{M}\left(U\right),\mathcal{M}\left(X\right),\mathcal{M}\left(Z\right),$$

together with functors

$$j_{!},j^{*},j_{*},i^{*},i_{*},i^{!},$$

such that

$$j_{!}⊣j^{*}⊣j_{*},i^{*}⊣i_{*}⊣i^{!},$$

and such that the usual open–closed recollement triangles hold in $\mathcal{M}\left(X\right)$.

Definition 59  

(Motivic lower and upper residuals). Let

$$M\in \mathcal{M}\left(X\right)$$

in a motivic six-functor recollement context. Define

$${Res}_{Z,\mathrm{mot}}^{+}\left(M\right):=Cone\left(j_{!}{j}^{*}M\to M\right),$$

and

$${Res}_{Z,\mathrm{mot}}^{-}\left(M\right):=Cone\left(M\to j_{*}{j}^{*}M\right)[-1].$$

Definition 60  

(Motivic boundary residual). Let

$$M\in \mathcal{M}\left(X\right).$$

Define

$${BRes}_{Z,\mathrm{mot}}\left(M\right):=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right).$$

Proposition 23  

(Motivic residual triangle). Assume the setting of Definition 58. For every

$$M\in \mathcal{M}\left(X\right),$$

there are canonical isomorphisms

$${Res}_{Z,\mathrm{mot}}^{+}\left(M\right)\simeq i_{*}{i}^{*}M,{Res}_{Z,\mathrm{mot}}^{-}\left(M\right)\simeq i_{*}{i}^{!}M,$$

and a functorial distinguished triangle

$$i_{*}{i}^{*}M⟶{BRes}_{Z,\mathrm{mot}}\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

Proof. 

The proof is formal. the open–closed recollement triangles in $\mathcal{M}\left(X\right)$ are

$$j_{!}{j}^{*}M⟶M⟶i_{*}{i}^{*}M\stackrel{+1}{⟶},$$

and

$$i_{*}{i}^{!}M⟶M⟶j_{*}{j}^{*}M\stackrel{+1}{⟶}.$$

The first triangle gives

$$Cone\left(j_{!}{j}^{*}M\to M\right)\simeq i_{*}{i}^{*}M.$$

The second triangle gives

$$Cone\left(M\to j_{*}{j}^{*}M\right)[-1]\simeq i_{*}{i}^{!}M.$$

This proves the two residual identifications.

Apply the octahedral axiom to the composable morphisms

$$j_{!}{j}^{*}M⟶M⟶j_{*}{j}^{*}M.$$

The resulting distinguished triangle is

$$Cone\left(j_{!}{j}^{*}M\to M\right)⟶Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right)⟶Cone\left(M\to j_{*}{j}^{*}M\right)\stackrel{+1}{⟶}.$$

Substituting the two residual identifications gives

$$i_{*}{i}^{*}M⟶{BRes}_{Z,\mathrm{mot}}\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}.$$

□

Remark 55  

(Formal nature of motivic residuals). Proposition 23 uses only recollement triangles and the octahedral axiom. It does not require a motivic realization functor, a motivic lift of sphericalization, or a comparison with Hodge or ℓ-adic realizations.

Definition 61  

(Realization compatibility for residuals). Let

$$Real:\mathcal{M}\left(X\right)\to D_c^b\left(X\right)$$

be an exact realization functor commuting with

$$j_{!},j^{*},j_{*},i^{*},i_{*},i^{!}.$$

We say that Real is residual-compatible for the pair $(U,Z)$ if, for every $M\in \mathcal{M}\left(X\right)$, the natural morphism

$$Real\left({BRes}_{Z,\mathrm{mot}}\left(M\right)\right)⟶{BRes}_Z\left(Real\left(M\right)\right)$$

is an isomorphism.

Proposition 24  

(Realization compatibility under six-functor compatibility). Let

$$Real:\mathcal{M}\left(X\right)\to D_c^b\left(X\right)$$

be exact and commute with

$$j_{!},j^{*},j_{*}.$$

Then Real is residual-compatible for the boundary residual:

$$Real\left({BRes}_{Z,\mathrm{mot}}\left(M\right)\right)\simeq {BRes}_Z\left(Real\left(M\right)\right).$$

If Real also commutes with $i^{*},i_{*},i^{!}$, then it carries the motivic residual triangle of Proposition 23 to the constructible residual triangle of Proposition 7.

Proof. 

By definition,

$${BRes}_{Z,\mathrm{mot}}\left(M\right)=Cone\left(j_{!}{j}^{*}M\to j_{*}{j}^{*}M\right).$$

Since Real is exact, it commutes with cones. Since it commutes with $j_{!},j^{*},j_{*}$, there are canonical isomorphisms

$$Real\left(j_{!}{j}^{*}M\right)\simeq j_{!}{j}^{*}Real\left(M\right),$$

and

$$Real\left(j_{*}{j}^{*}M\right)\simeq j_{*}{j}^{*}Real\left(M\right).$$

Therefore

$$Real\left({BRes}_{Z,\mathrm{mot}}\left(M\right)\right)\simeq Cone\left(j_{!}{j}^{*}Real\left(M\right)\to j_{*}{j}^{*}Real\left(M\right)\right),$$

which is

$${BRes}_Z\left(Real\left(M\right)\right).$$

If Real also commutes with $i^{*},i_{*},i^{!}$, then applying Real to

$$i_{*}{i}^{*}M⟶{BRes}_{Z,\mathrm{mot}}\left(M\right)⟶i_{*}{i}^{!}M\left[1\right]\stackrel{+1}{⟶}$$

gives

$$i_{*}{i}^{*}Real\left(M\right)⟶{BRes}_Z\left(Real\left(M\right)\right)⟶i_{*}{i}^{!}Real\left(M\right)\left[1\right]\stackrel{+1}{⟶}.$$

This is the constructible residual triangle. □

Remark 56  

(Motivic caution). The existence of motivic residuals is formal once recollement is available. the existence of a realization-compatible motivic theory depends on the particular motivic category and realization functor. No unconstructed motivic lift is assumed in this paper [9,10].

11.2. Conditional Motivic Sphericalization

Definition 62  

(Motivic MV adjunction package). Let $\mathcal{M}(-)$ be a motivic six-functor theory satisfying open–closed recollement. For an open–closed decomposition

$$X=U\bigsqcup Z,$$

the motivic MV adjunction package is

$${\mathfrak{A}}_{MV}^{\mathrm{mot}}(X;Z):=\left(\mathcal{M}\left(U\right),\mathcal{M}\left(X\right),\mathcal{M}\left(Z\right);j_{!},j^{*},j_{*},i^{*},i_{*},i^{!}\right).$$

Proposition 25  

(Conditional motivic sphericalization). Let $\mathcal{M}(-)$ be a motivic six-functor theory whose categories assemble into a sufficiently cocomplete locally stable $(\infty ,2)$-category. Suppose further that the subcategory under consideration is closed under the sequential colimits appearing in universal sphericalization. Finally, suppose that the motivic MV package is equipped with the shared-object sphericalization data and transportable compatibility morphisms required in Section 5. Then the conditional compatible sphericalization construction applies to the motivic recollement package

$${\mathfrak{A}}_{MV}^{\mathrm{mot}}(X;Z).$$

In particular, under these hypotheses there is a conditional spherical package

$$S_{MV}\left({\mathfrak{A}}_{MV}^{\mathrm{mot}}(X;Z)\right).$$

Proof. 

Under the stated hypotheses, the selected motivic recollement adjunctions form objects of

$$Adj\left(\mathcal{A}\right)$$

for a sufficiently cocomplete locally stable $(\infty ,2)$-category $\mathcal{A}$. By Theorem 5, each selected adjunction admits universal sphericalization. the closure hypothesis ensures that the sequential colimits used in the construction remain inside the chosen subcategory. the shared-object sphericalization data and transportability hypotheses ensure that the objectwise sphericalizations assemble into a single diagram-level package. Therefore Theorem 6 applies to

$${\mathfrak{A}}_{MV}^{\mathrm{mot}}(X;Z).$$

□

Remark 57  

(Constructibility and geometric origin). The hypotheses of Proposition 25 are not automatic for constructible, compact, or geometric-origin motivic subcategories. Universal sphericalization uses sequential colimits, and such colimits may leave a small or compact subcategory. Preservation of constructibility or geometric origin is therefore a separate condition, not a formal consequence of the present paper.

Remark 58  

(No unconditional Ruimy–Tubach claim). The categories of Ruimy–Tubach provide a natural motivic setting in which one may ask for integral residuals, realization compatibility, and six-functor recollement. the present paper does not claim that these categories automatically satisfy all hypotheses of Proposition 25. Such a claim requires verifying local stability, required cocompleteness, closure under sphericalization colimits, shared-object compatibility, transportability of compatibility morphisms, and compatibility with realization functors [9,10].

11.3. Toward Spherical MV Gluing and Schobers

Definition 63  

(Conditional local spherical MV completion). Let

$$X=U\bigsqcup Z$$

be an open–closed decomposition, and let

$${\mathfrak{A}}_{MV}(X;Z)$$

be the corresponding MV adjunction package in a locally stable $(\infty ,2)$-category. A conditional local spherical MV completion of the pair $(X,Z)$ is the object

$$S_{MV}\left({\mathfrak{A}}_{MV}(X;Z)\right)$$

whenever the hypotheses of Theorem 6 are satisfied for

$${\mathfrak{A}}_{MV}(X;Z).$$

Proposition 26  

(Conditional local universal property). Let

$${\mathfrak{A}}_{MV}(X;Z)$$

be as in Definition 63, and assume that it belongs to the compatibility subcategory

$${MVAdj}^{comp}\left(\mathcal{A}\right)$$

of Theorem 6. Let

$$\mathfrak{B}\in SphMV\left(\mathcal{A}\right).$$

Assume moreover that the objectwise universal properties are compatible with the finite diagram of transported MV relations. Then there is a natural equivalence

$${Map}_{SphMV\left(\mathcal{A}\right)}\left(S_{MV}\left({\mathfrak{A}}_{MV}(X;Z)\right),\mathfrak{B}\right)\simeq {Map}_{{MVAdj}^{comp}\left(\mathcal{A}\right)}\left({\mathfrak{A}}_{MV}(X;Z),\iota \mathfrak{B}\right).$$

Proof. 

This is Corollary 2 applied to the MV adjunction package

$${\mathfrak{A}}_{MV}(X;Z)$$

inside the compatibility subcategory

$${MVAdj}^{comp}\left(\mathcal{A}\right).$$

□

Remark 59  

(Relation with schobers). Spherical functors and spherical adjunctions provide local models in the theory of perverse schobers [5,6,8,13]. The recollement calculations of this paper explain why one should not expect the raw MacPherson–Vilonen package itself to be spherical. the full-faithfulness identities in ordinary recollement force zero twist or cotwist terms. Schober-type spherical gluing is therefore better understood as a different local coordinate system, closer to nearby and vanishing-cycle data, rather than as naive sphericality of the raw

$$j_{!},j^{*},j_{*}$$

recollement coordinates.

Remark 60  

(Nearby and vanishing-cycle coordinates). For a disk with one possible singular point, the schober model is expressed in terms of a spherical functor, while classical perverse-sheaf gluing may be described using nearby and vanishing cycles. This suggests that the correct comparison problem is not to sphericalize the four raw recollement adjunctions

$$j_{!}⊣j^{*},j^{*}⊣j_{*},i^{*}⊣i_{*},i_{*}⊣i^{!},$$

inside the same recollement target. Rather, the natural problem is to compare the residual defect calculus of recollement with a nearby/vanishing-cycle or Beilinson-type gluing description [4,8].

Remark 61  

(Descent and global gluing). A global theory of schobers on stratified spaces requires descent, compatibility across multiple strata, and coherence under refinement of stratifications. the construction in this paper supplies the local recollement-defect calculation and the conditional form of a diagram-level sphericalization step. It does not assert that all descent data are automatically satisfied.

Remark 62  

(Synthesis). The results of the paper may be summarized by the following chain of unconditional constructible constructions:

$$\begin{array}{cc}\hfill \mathrm{open}--\mathrm{closed}\mathrm{recollement}& ⇝\mathrm{MV}\mathrm{adjunction}\mathrm{package},\hfill \\ \hfill \mathrm{MV}\mathrm{adjunction}\mathrm{package}& ⇝\mathrm{twist}--\cot \mathrm{wist}\mathrm{defect}\mathrm{calculus},\hfill \\ \hfill \mathrm{defect}\mathrm{calculus}& ⇝\mathrm{boundary}\mathrm{residual}\mathrm{triangle}.\hfill \end{array}$$

At the constructible level, the boundary residual is

$${BRes}_Z(-)=Cone\left(j_{!}{j}^{*}(-)\to j_{*}{j}^{*}(-)\right),$$

and the main defect-theoretic statement is the triangle

$$T_{{\mathcal{C}}_X}(j_{!}⊣j^{*})(-)\left[1\right]⟶{BRes}_Z(-)⟶T_{{\mathcal{C}}_X}(j^{*}⊣j_{*})(-)\stackrel{+1}{⟶}.$$

At the iterated stratified level, these boundary residuals form the tower

$${\left\{{BRes}_k(-)\right\}}_k.$$

A spherical completion of this data is conditional on the hypotheses of universal sphericalization, shared-object compatibility, and transportability. the residual tower itself is a formal consequence of recollement and the octahedral axiom.