The Geometry of the Delta Groupoid

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This post serves to marry the last three posts into one cohesive framework. The connection to foliational completions is also made clear.


Enriched Graphs and Delta Groupoid


Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph, embedded in $\mathbb{R}^3$. Define an enrichment of $\Gamma$ to be a triple:

$$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$

where:

- $\phi: E \to \mathcal{C}$ is a coloring or stratification into a finite set of colors $\mathcal{C}$,

- $\mathcal{L}$ is a local system (functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$), assumed flat,

- $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation.


Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments.


Define the Delta groupoid $\mathcal{D}_\Gamma$ as follows:

- Objects: enrichments $\mathcal{X} \in \mathcal{E}_\Gamma$.

- Morphisms: invertible maps

  $$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')$$

  satisfying:

- The underlying graph $\Gamma$ is fixed:

$$Z_\Gamma(u) = Z_{\delta(\Gamma)}(u)$$

- The coloring transforms via pullback:

$$\phi' = \delta^* \phi$$

- The local system transforms functorially:

$$\mathcal{L}' = \delta^* \mathcal{L}, \quad \rho' = \rho \circ \delta_*$$

- Composition: $(\delta_2 \circ \delta_1)(\mathcal{X}) = \delta_2(\delta_1(\mathcal{X}))$.

- Each object has an identity, and morphisms are invertible.

Morphisms $\delta$ preserve $Z_{\Gamma}(u)$ but generally do not preserve $\phi$, $\mathcal{L}$, or $\rho$. Thus, $\delta$ encodes invariance of cycle structure and variance of enrichment data.

Given $\mathcal{X}, \mathcal{X}' \in \mathcal{E}_\Gamma$ with $\delta(\mathcal{X}) = \mathcal{X}'$, an important question is:

$$H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \mathcal{L}') \quad \text{?}$$


Suppose:

- $\delta$ is a geometric Delta morphism (e.g., rotation about $z=0$),

- $\Gamma \subset \mathbb{R}^3$ lies in $z=0$ plane,

- $\delta$ acts as an automorphism of $\Gamma$ and $\mathcal{L}$ preserving flatness.

Then:

$$\rho \simeq \rho \circ \delta_*, \quad H^1(\Gamma, \mathcal{L}) \cong H^1(\Gamma, \delta^*\mathcal{L}).$$


Such $\delta$ form a distinguished subgroupoid preserving the full enrichment.


Foliational Completions and the Interface Structure.


Let $X$ be a smooth, open, geodesically convex $n$-dimensional manifold with regular polytope boundary, and $V = \{v_i\}$ a finite set of vertices.


Define the block:

$$\mathcal{B} := X \cup \partial X$$

A foliational completion (or $\mathcal{F}$-completion) is given by foliations $\mathcal{F}_{v_{ij}}$ for each distinct pair $(v_i, v_j)$, satisfying:

$$\lim_{p \to \partial X} L_\alpha(p) = \{v_i, v_j\}$$

for leaves $L_\alpha$ of $\mathcal{F}_{v_{ij}}$.

The $\mathcal{F}$-completion is:

$$CX_V := \bigcup_{(v_i, v_j) \in V \times V} \mathcal{F}_{v_{ij}}$$


The Interface Structure $\mathcal{I}$


Inside $CX_V$ in dimension $n=3$, define $\mathcal{I}$ as the union of four symmetrical maximal surfaces of revolution inside the unit cube, each having constant Gaussian curvature.

Slicing $\mathcal{I}$ with planes $x_1, x_2, x_3 = 1/2$ yields two overlapping oval curves per slice, with intersections forming pieces of the graph $\Gamma$.

Thus, $\mathcal{I}$ geometrically generates $\Gamma$, tying together the foliational and graph-based frameworks.


Summary


The enriched Delta groupoid $\mathcal{D}_\Gamma$ and the foliational Interface $\mathcal{I}$ are intertwined:

- $\mathcal{I}$ provides a natural geometric source of $\Gamma$,

- $\mathcal{D}_\Gamma$ captures symmetries, preserving cycle structure but transforming local systems and cohomology.

Together, they describe a system where coarse spectral data remains invariant while finer enrichment data evolves under geometric transformations.

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