arXiv subjects

Samantha Jarvis

Publications and source records attributed to Samantha Jarvis.

4 recordsLinked to original sources

arXiv2026

A calculus of types in Isbell nuclei

We identify two constructions from different mathematical traditions. In linear logic and realisability, logical types are generated rather than fixed in advance: one begins with a universe of realisers equipped with execution, uses orthogonality to test their interactions, and takes types to be the biorthogonally closed subsets. In enriched Isbell duality, a quantitative relation induces an adjunction whose fixed points form a category, its nucleus. These constructions proceed by different means; we show that, in the present setting, they produce the same objects. The shared datum is minimal: an associative product, called execution, and a real-valued measurement, with no compatibility assumed between them. The failure of the measurement to be additive is at once the relation defining orthogonality and the quantitative relation whose Isbell nucleus we form, and the types cut out by orthogonality are exactly the fixed points of the associated adjunction. The identification pays off in both directions. The most natural product of types fails to be associative; repairing this failure forces a different notion of type, sensitive to both sides of a composite, on which the induced product is associative and, when execution has units, carries two residuals. What emerges is a noncommutative Lambek calculus, derived directly from execution and orthogonality rather than imposed. In the reverse direction, each such type, read on the categorical side, generates a quantitative relation of its own, and with it a derived adjunction and a further generation of types; these derived types are again types of the original situation, computed by the residuals of the Lambek calculus. We also prove a coherence theorem for the threefold arrangements of this construction and, in the finite-dimensional case, give explicit formulas for the product.

cs.LO↗

arXiv2026

Projective metric geometry of tropical nuclei: gap matrices, event loci, and order chambers

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.

math.AG↗

arXiv2025

Spectral properties of graphs associated to the Basilica group

We provide the foundation of the spectral analysis of the Laplacian on the orbital Schreier graphs of the Basilica group, the iterated monodromy group of the quadratic polynomial $z^2-1$. This group is an important example in the class of self-similar amenable but not elementary amenable finite automata groups studied by Grigorchuk, Żuk, \v Sunić, Bartholdi, Virág, Nekrashevych, Kaimanovich, Nagnibeda et al. We prove that the spectrum of the Laplacian has infinitely many gaps and that the support of the KNS Spectral Measure is a Cantor set. Moreover, on a generic blowup, the spectrum coincides with this Cantor set, and is pure point with localized eigenfunctions and eigenvalues located at the endpoints of the gaps.

math.GR↗

arXiv2023

Monoidal Structures on Generalized Polynomial Categories

Recently, there has been renewed interest in the theory and applications of de Paiva's dialectica categories and their relationship to the category of polynomial functors. Both fall under the theory of generalized polynomial categories, which are free coproduct completions of free product completions of (monoidal) categories. Here we extend known monoidal structures on polynomial functors and dialectica categories to generalized polynomial categories. We highlight one such monoidal structure, an asymmetric operation generalizing composition of polynomial functors, and show that comonoids with respect to this structure correspond to categories enriched over a related free coproduct completion. Applications include modeling compositional bounds on dynamical systems.

math.CT↗