dorsal/arxiv
View SchemaGeometric Structures in $\mathbb{R}$-enriched adjunctions
| Authors | Juan Luis Gastaldi, Samantha Jarvis, Thomas Seiller, John Terilla |
|---|---|
| Categories | |
| ArXiv ID | 2601.07900vv1 |
| URL | https://arxiv.org/abs/2601.07900 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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 $\textit{gap matrix}$ $\delta^{(f,g)}(c,d)=M(c,d)-f(c)-g(d)$. 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 $\delta^{(f,g)}$.
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"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 $\\textit{gap matrix}$ $\\delta^{(f,g)}(c,d)=M(c,d)-f(c)-g(d)$. 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 $\\delta^{(f,g)}$.",
"arxiv_id": "2601.07900",
"authors": [
"Juan Luis Gastaldi",
"Samantha Jarvis",
"Thomas Seiller",
"John Terilla"
],
"categories": [
"math.AG",
"math.CT"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Geometric Structures in $\\mathbb{R}$-enriched adjunctions",
"url": "https://arxiv.org/abs/2601.07900",
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