Salvato in:
| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2601.07900 |
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Sommario:
- The tropical row span and column span of a real matrix are, from the polyhedral point of view, different objects living in different ambient spaces. These polytopes are known to be combinatorially isomorphic as polyhedral complexes; we prove that they are isometric under a Hilbert projective metric. We show that this isometry, along with a considerable amount of additional metric and polyhedral structure, is a direct consequence of a single categorical construction: the Isbell nucleus of the matrix, viewed as a profunctor enriched over the extended reals. The projective nucleus carries two canonical structures inherited from enrichment. The first is a Hilbert projective metric, with respect to which the Isbell conjugate maps are mutually inverse isometries -- this is the Isometry Theorem. The second is a polyhedral cell decomposition cut out by the Isbell inequalities, recovering the type decomposition of tropical convexity. These two structures are linked pointwise by the \emph{gap matrix}. The Events Theorem identifies each positive entry of the gap matrix with the exact projective distance to the locus where the corresponding inequality becomes tight: algebraic slack in the Isbell inequalities equals geometric distance to the cell walls. Thresholding the gap matrix at successive radii produces a constructible sheaf of formal concept lattice towers, extracting discrete algebraic structure from the continuous geometry at each point. In the square case there is generically a unique full-dimensional cell. The Centering Theorem identifies its Chebyshev center -- the point maximally insulated from all cell walls -- and shows that the optimal radius equals the minimum directed cycle mean of an associated digraph, connecting the projective geometry of the nucleus to the classical theory of optimal assignments.