Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric Structures in $\mathbb{R}$-enriched adjunctions

Juan Luis Gastaldi, Samantha Jarvis, Thomas Seiller, John Terilla

TL;DR

The paper develops a canonical metric–polyhedral framework for tropical polytopes by encoding a real matrix M as an \\overline{\\mathbb{R}}-enriched profunctor and studying its Isbell nucleus PNuc(M). Two intertwined geometries emerge: a projective Hilbert-type metric on presheaf and copresheaf realizations, and a canonical polyhedral cell decomposition given by the Isbell inequalities; these are tied via the gap matrix delta^(f,g), whose zero pattern determines cells while its positive entries measure exact distances to boundary events. The main contributions include the Events Theorem linking gap entries to precise distance radii to event loci, the identification of mutually inverse isometries between presheaf and copresheaf realizations under projective Isbell maps, and the construction of order chambers and chamber-indexed towers of formal concept lattices that refine the tropical type decomposition. The framework yields gauge-invariant, canonical structures with potential for reconstruction and persistence-type invariants, enabling navigation and data-analytic techniques on tropical polytopes constructed from \\overline{\\mathbb{R}}-enriched nuclei.

Abstract

A real $m\times n$ matrix $M$ determines tropical row and column polytopes in tropical projective spaces $\mathbb{TP}^{m-1}$ and $\mathbb{TP}^{n-1}$, with canonical polyhedral cell structures that are naturally dual. We reinterpret this picture via Isbell duality: viewing $M$ as an $Rbar$-enriched profunctor $M$: $C^{op}\otimes D\to Rbar$, we study the associated order-reversing Isbell adjunction $M^*\dashv M_*$ and its fixed-point locus, the nucleus $Nuc(M)$. After projectivization, $pnuc(M)$ carries two interacting geometries. On the metric side, $Rbar$-enrichment induces a canonical Hilbert projective--type (max-spread) metric on projective (co)presheaves, and we show that the projective Isbell maps identify the presheaf and copresheaf realizations of $pnuc(M)$ by mutually inverse isometries. On the polyhedral side, in the discrete real setting the Isbell inequalities cut out a canonical polyhedral decomposition of $pnuc(M)$ recovering the usual tropical cell structure. Our main new ingredient is a pointwise invariant of a nucleus point $(f,g)$: the nonnegative $\textit{gap matrix}$ $δ^{(f,g)}(c,d)=M(c,d)-f(c)-g(d)$. Its zero pattern determines the cell containing $(f,g)$, while its positive entries compute exact metric distances to the boundary strata where additional inequalities become tight (Events Theorem). This distance-to-wall principle refines cells into order chambers and supports a constructible tower of complete lattices obtained by thresholding $δ^{(f,g)}$.

Geometric Structures in $\mathbb{R}$-enriched adjunctions

TL;DR

The paper develops a canonical metric–polyhedral framework for tropical polytopes by encoding a real matrix M as an \\overline{\\mathbb{R}}-enriched profunctor and studying its Isbell nucleus PNuc(M). Two intertwined geometries emerge: a projective Hilbert-type metric on presheaf and copresheaf realizations, and a canonical polyhedral cell decomposition given by the Isbell inequalities; these are tied via the gap matrix delta^(f,g), whose zero pattern determines cells while its positive entries measure exact distances to boundary events. The main contributions include the Events Theorem linking gap entries to precise distance radii to event loci, the identification of mutually inverse isometries between presheaf and copresheaf realizations under projective Isbell maps, and the construction of order chambers and chamber-indexed towers of formal concept lattices that refine the tropical type decomposition. The framework yields gauge-invariant, canonical structures with potential for reconstruction and persistence-type invariants, enabling navigation and data-analytic techniques on tropical polytopes constructed from \\overline{\\mathbb{R}}-enriched nuclei.

Abstract

A real matrix determines tropical row and column polytopes in tropical projective spaces and , with canonical polyhedral cell structures that are naturally dual. We reinterpret this picture via Isbell duality: viewing as an -enriched profunctor : , we study the associated order-reversing Isbell adjunction and its fixed-point locus, the nucleus . After projectivization, carries two interacting geometries. On the metric side, -enrichment induces a canonical Hilbert projective--type (max-spread) metric on projective (co)presheaves, and we show that the projective Isbell maps identify the presheaf and copresheaf realizations of by mutually inverse isometries. On the polyhedral side, in the discrete real setting the Isbell inequalities cut out a canonical polyhedral decomposition of recovering the usual tropical cell structure. Our main new ingredient is a pointwise invariant of a nucleus point : the nonnegative . Its zero pattern determines the cell containing , while its positive entries compute exact metric distances to the boundary strata where additional inequalities become tight (Events Theorem). This distance-to-wall principle refines cells into order chambers and supports a constructible tower of complete lattices obtained by thresholding .
Paper Structure (32 sections, 27 theorems, 185 equations, 7 figures)

This paper contains 32 sections, 27 theorems, 185 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Relation to existing work
  3. The nucleus of an $\overline{\mathbb{R}}$-profunctor
  4. The extended real numbers $\overline{\mathbb{R}}$
  5. $\overline{\mathbb{R}}$-categories
  6. Presheaves, copresheaves, and Yoneda
  7. Profunctors
  8. Isbell duality and the nucleus
  9. The geometry of the nucleus
  10. Finite index sets and the Isbell transforms
  11. Projective (co)presheaves and the Hilbert--oscillation metric
  12. Projective nuclei and isometries
  13. External gauge transformations
  14. Witness cells
  15. Polyhedral structures of $\mathop{\mathrm{\mathbb{P}\mathrm{Nuc}}}\nolimits(M)$
  16. ...and 17 more sections

Key Result

Proposition 5

The assignments eq:Mupperstar--eq:Mlowerstar define $\overline{\mathbb{R}}$-functors and satisfy meaning that for all $f\in[\mathcal{C}^{\mathrm{op}},\overline{\mathbb{R}}]$ and $g\in[\mathcal{D},\overline{\mathbb{R}}]^{\mathrm{op}}$.

Figures (7)

  • Figure 1: The projective nucleus $\mathop{\mathrm{\mathbb{P}\mathrm{Nuc}}}\nolimits(M)$ for the running example (in an affine chart), with its polyhedral decomposition. Here, the basepoint is marked in green. The other marked points illustrate Theorem \ref{['thm:events']}: a positive gap value at a basepoint computes the distance to the locus where a wall is encountered. The shaded hexagons are metric balls in the max-spread metric.
  • Figure 2: Projective ball and event at radius $1.9$ in $\mathbb{P}\mathcal{C}$ illustrating how Lemma \ref{['lem:events']} and Theorem \ref{['thm:events']} work.
  • Figure 3: The order chambers refining the witness cells.
  • Figure 4: Specialization to a face: if $Q'\le \overline Q$ merges consecutive preorder blocks, then the chamber tower over $Q'$ is obtained from the tower over $Q$ by deleting the intermediate floors. Here $T_{a,b}$ denotes the chosen structure map $L^Q_a\to L^Q_b$ induced by $R^Q_a\subseteq R^Q_b$ (e.g. $T^{\mathrm{ext}}_{a,b}$ or $T^{\mathrm{int}}_{a,b}$), and on a face these maps compose when intermediate relations are deleted. The bottom structure maps are composites of the skipped maps (e.g. $T_{0,2}=T_{1,2}\circ T_{0,1}$).
  • Figure 5: The order chambers refining the witness cells. The point $f_1$ lies directly on an order chamber wall reflecting the tie at $1.6$. Nudging off the wall to the left or right by $0.1$ gives points $f_2$ and $f_3$.
  • ...and 2 more figures

Theorems & Definitions (77)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Definition 7
  • Proposition 8
  • ...and 67 more