Grothendieck Site over the Spectral Tower

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Sheaf Stack over the Dimensional Stratification Site

We define a stratified site $\mathcal{S}$ indexed by dimension $n \in \mathbb{Z}_{\geq 2}$, whose objects are geometric-combinatorial models $\mathcal{I}^{(n)}$ associated to dimension-$n$ F-completions. Morphisms in this site correspond to boundary inclusions, coordinate projections, and degenerations between dimensions:

$$\mathcal{I}^{(n)} \to \mathcal{I}^{(n-1)} \quad \text{via coordinate slice or foliation boundary}.$$

Let $\mathcal{M}_\Delta^{(n)}$ denote the moduli space of $\Delta$-twisted holonomy representations in dimension $n$, with $\mathcal{M}_\Delta^{(2)} = \mathrm{U}(1)^r / \Delta$ the classical representation torus quotient.

We define a sheaf stack of zeta data over $\mathcal{S}$:

$$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks}$$

such that:

  • For each $n$, $\mathscr{Z}(\mathcal{I}^{(n)})$ is a stack (e.g. derived or topological) encoding the family of zeta sheaves over $\mathcal{M}_\Delta^{(n)}$.
  • The stalk of $\mathscr{Z}$ at a point corresponds to a twisted Ihara-type zeta function valued in $\mathbb{C}[[u]]$.
  • Morphisms in $\mathcal{S}$ induce restriction or pullback functors on zeta stacks:

$$\mathscr{Z}(\mathcal{I}^{(n-1)}) \leftarrow \mathscr{Z}(\mathcal{I}^{(n)}),$$

compatible with foliation collapse, singular degeneration, or representation contraction.

  • The delta symmetry group $\Delta$ acts on $\mathscr{Z}$ as a group of natural transformations, preserving the stratified structure.

Sheaf Stack Duality over Foliated Stratification Site Conjecture:

There exists a stacky equivalence

$$\mathscr{Z}(\mathcal{I}^{(n)}) \cong \mathrm{Sh}_\zeta(\mathcal{I}^{(n)})$$

between the zeta sheaf stack over the $n$-dimensional moduli space $\mathcal{M}_\Delta^{(n)}$ and a sheaf of spectral data over the geometric space $\mathcal{I}^{(n)}$.

These equivalences are natural in $n$, delta-equivariant, and compatible with restriction morphisms in $\mathcal{S}$. Thus, the entire tower forms a categorified duality:

$$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks}_{\zeta},$$

encoding the evolution of spectral invariants across dimension.

Derived Categories and Zeta Fourier–Mukai Duality

Let $\mathcal{S}$ be the stratified site indexed by dimension $n \geq 2$, with objects $\mathcal{I}^{(n)}$ and morphisms given by boundary projections and coordinate slicing between F-completions.

For each $n$, we associate:

  • A derived category of sheaves of zeta-type objects:

    $$\mathcal{D}_\zeta^{(n)} := D^b\big(\mathrm{Sh}_\zeta(\mathcal{I}^{(n)})\big),$$

    where $\mathrm{Sh}_\zeta(\mathcal{I}^{(n)})$ is the category of sheaves (or complexes) encoding twisted zeta functions over the stratified geometry $\mathcal{I}^{(n)}$.

  • A derived category of equivariant sheaves over the moduli space:

    $$\mathcal{D}_\Delta^{(n)} := D^b\big(\mathrm{Coh}_\Delta(\mathcal{M}_\Delta^{(n)})\big),$$

    where $\mathrm{Coh}_\Delta$ denotes the category of $\Delta$-equivariant coherent sheaves (or perfect complexes) over the moduli orbifold.

We posit the existence of a Fourier–Mukai-type transform between these categories.

Let $\mathcal{Z}^{(n)} \in D^b\big(\mathrm{Coh}(\mathcal{I}^{(n)} \times \mathcal{M}_\Delta^{(n)})\big)$ denote a zeta kernel object, playing the role of a spectral correspondence. Then define:

$$\Phi_{\mathcal{Z}^{(n)}} : \mathcal{D}_\Delta^{(n)} \longrightarrow \mathcal{D}_\zeta^{(n)}, \quad \mathcal{F} \mapsto Rp_{1*}(\mathcal{Z}^{(n)} \overset{L}{\otimes} p_2^* \mathcal{F}),$$

where $p_1, p_2$ are the projections from $\mathcal{I}^{(n)} \times \mathcal{M}_\Delta^{(n)}$ onto each factor.

Zeta Fourier–Mukai Equivalence

There exists a kernel object $\mathcal{Z}^{(n)}$ such that the Fourier–Mukai transform

$$\Phi_{\mathcal{Z}^{(n)}} : \mathcal{D}_\Delta^{(n)} \xrightarrow{\sim} \mathcal{D}_\zeta^{(n)}$$

is an equivalence of derived categories.

This equivalence respects the stratified structure of $\mathcal{S}$ and is compatible with delta actions, restriction to boundary strata, and deformation of F-completions across dimensions.

Furthermore, for $n > 2$, the entire tower

$$\{ \Phi_{\mathcal{Z}^{(n)}} \}_{n \geq 2}$$

defines a system of zeta transforms across the site $\mathcal{S}$, giving rise to a categorified global Fourier–Mukai theory over the moduli–geometry tower.

Grothendieck Site Structure on the Stratified Tower

We now endow the dimensionally stratified collection $\mathcal{S} = \{ \mathcal{I}^{(n)} \}_{n \geq 2}$ with the structure of a Grothendieck site, allowing us to define stacks, cohomology, and derived functors over this tower.

Stratified F-completion Site $\mathcal{S}$

Let $\mathcal{S}$ be the category whose objects are compact stratified spaces $\mathcal{I}^{(n)}$ arising as coordinate slices of an $n$-dimensional F-completion, and whose morphisms are:

  • Coordinate projection maps $\pi_{n,m} : \mathcal{I}^{(n)} \to \mathcal{I}^{(m)}$ for $m < n$;
  • Singular degeneration maps collapsing cone points or foliation leaves;
  • Foliation-preserving inclusions along boundary strata.

We define a Grothendieck topology on $\mathcal{S}$ by declaring a family $\{ f_i: \mathcal{I}^{(n_i)} \to \mathcal{I}^{(n)} \}_{i \in I}$ to be a covering if the images of the $f_i$ jointly cover all regular and singular strata of $\mathcal{I}^{(n)}$.

This topology captures local charts in the stratified sense (e.g., charts homeomorphic to open subsets of $\mathbb{R}^k / G$ for some finite $G$), respecting delta symmetry and foliation structure.

Sheaf and Stack Theory over $\mathcal{S}$

Given this site structure, we define:

  • A zeta sheaf stack

    $$\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \longrightarrow \mathbf{Stacks},$$

    assigning to each $\mathcal{I}^{(n)}$ a stack of zeta-invariant sheaves encoding holonomy-twisted spectral data.

    

  • A derived category of global sections

    $$\mathcal{D}_{\text{global}} := D^b(\Gamma(\mathcal{S}, \mathscr{Z})),$$

    representing globally defined complexes of zeta sheaves over the stratified tower.

  • For each pair of strata $\mathcal{I}^{(n)}$, $\mathcal{I}^{(m)}$, a space of derived morphisms:

    $$\operatorname{Ext}^k_{\mathcal{S}}(\mathscr{Z}(\mathcal{I}^{(m)}), \mathscr{Z}(\mathcal{I}^{(n)})) := \mathrm{Ext}^k_{\mathbf{Stacks}}(i_! \mathscr{Z}(\mathcal{I}^{(m)}), i_* \mathscr{Z}(\mathcal{I}^{(n)})),$$

    where $i: \mathcal{I}^{(m)} \hookrightarrow \mathcal{S}$ is the inclusion functor.

Zeta Stack Cohomology Conjecture:

The cohomology of the zeta sheaf stack over $\mathcal{S}$ encodes the stratified spectral flow of zeta invariants across dimensions. In particular:

$$H^k(\mathcal{S}, \mathscr{Z}) \cong \bigoplus_{n \geq 2} \operatorname{Ext}^k_{\mathcal{S}}(\mathscr{Z}(\mathcal{I}^{(n)}), \mathscr{Z}_{\mathrm{tot}}),$$

where $\mathscr{Z}_{\mathrm{tot}}$ is the terminal zeta object capturing global delta-equivariant sections.

Furthermore, these Ext pairings organize the zeta flows into a spectral sequence, with:

$$E_1^{n,k} = H^k(\mathcal{I}^{(n)}, \mathscr{Z}) \Rightarrow H^{n+k}(\mathcal{S}, \mathscr{Z}),$$

indicating how local spectral data lifts to global geometric information over the tower.

Motivic Zeta Function of the Stratified Site

We define a global motivic zeta function that captures the cumulative structure of zeta sheaf data across all strata $\mathcal{I}^{(n)}$ of the stratified site $\mathcal{S}$.

Motivic Zeta Function

Let $\mathscr{Z} : \mathcal{S}^{\mathrm{op}} \to \mathbf{Stacks}$ be the zeta sheaf stack over the Grothendieck site $\mathcal{S}$.

The motivic zeta function of the tower is defined as the formal power series:

$$\zeta_{\mathcal{S}}(t) := \sum_{n=2}^\infty \chi\big(\mathcal{D}_\zeta^{(n)}\big) \cdot t^n,$$

where:

  • $\mathcal{D}_\zeta^{(n)} := D^b(\mathrm{Sh}_\zeta(\mathcal{I}^{(n)}))$ is the derived category of zeta sheaves over $\mathcal{I}^{(n)}$,
  • $\chi(-)$ denotes the categorical Euler characteristic, or alternatively, a suitable trace invariant such as:

    $$\chi(\mathcal{D}_\zeta^{(n)}) := \sum_k (-1)^k \dim \operatorname{Ext}^k_{\mathcal{S}}(\mathscr{Z}(\mathcal{I}^{(n)}), \mathscr{Z}_{\mathrm{tot}}).$$

Motivic Zeta Cohomology Conjecture:

The global motivic zeta function $\zeta_{\mathcal{S}}(t)$ admits the interpretation:

$$\zeta_{\mathcal{S}}(t) = \operatorname{Tr}\left( (-1)^F \,|\, \mathbb{H}^\bullet(\mathcal{S}, \mathscr{Z}) \right),$$

where $F$ is the cohomological degree functor and $\mathbb{H}^\bullet(\mathcal{S}, \mathscr{Z})$ denotes the hypercohomology of the sheaf stack.

Moreover, if each $\mathcal{I}^{(n)}$ corresponds to a representation stratum of a classifying stack $B\pi_1(\Gamma_n)$, then $\zeta_{\mathcal{S}}(t)$ may admit a rational expression analogous to a motivic zeta function in the sense of Denef–Loeser:

$$\zeta_{\mathcal{S}}(t) = \prod_{n=2}^\infty \left( \frac{1}{1 - \mathbb{L}^{w_n} t^n} \right)^{\chi_n},$$

where:

  • $\mathbb{L}$ is the class of the affine line in the Grothendieck ring of stacks,
  • $w_n$ measures the weight of zeta data on level $n$,
  • $\chi_n = \chi(\mathcal{I}^{(n)})$ or its delta-equivariant refinement.

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