A calculus of types in Isbell nuclei

2026-06-02

AI Summary

This work unifies the orthogonality-closure type construction from linear logic and realizability theory with the kernel construction arising in enriched Isbell duality. Assuming only an associative execution operation and a real-valued metric—without requiring compatibility—we demonstrate that types defined via orthogonality precisely coincide with fixed points of the Isbell kernel, thereby establishing their equivalence in a minimal setting for the first time. This correspondence naturally yields an associative type product endowed with a bicomodule structure and an intrinsic non-commutative Lambek calculus, along with a bidirectional generation mechanism linking types and quantitative relations. Key contributions include a coherence theorem for the tripartite construction, an explicit formula for the product in finite-dimensional cases, and the closure of derived types within the original system.

Abstract

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.

Problem

• Isbell nucleus
• linear logic
• orthogonality
• Lambek calculus
• type construction

Innovation

• Isbell nucleus
• orthogonality
• Lambek calculus
• realisability
• enriched duality