Entropy and Holography through Adjunctions: A Bicategorical Perspective on Landauer's Principle

Abstract

We develop a bicategorical formulation of entropy and Landauer's principle in which deterministic monotone maps between logical and thermodynamic state spaces are embedded into a broader theory of open, many-to-many feasibility interfaces. Entropy-ordered state spaces form the objects of a locally posetal bicategory, Boolean profunctors encode feasible implementations, and refinements define 2-morphisms. Although this bicategory admits genuinely non-representable interfaces, every exact Landauer adjunction lies in its representable sector and is generated by a unique monotone implementation map. Such an adjunction also induces an idempotent boundary monad, together with a dual idempotent bulk comonad. Our main result identifies the Eilenberg-Moore object of the boundary monad with a universal boundary-stable quotient determined by the bulk-mediated round-trip relation. This provides a canonical reconstruction principle: the quotient captures exactly those boundary distinctions that remain observable after implementation in the bulk and subsequent readout. In the representable case, two boundary states are identified precisely when they have the same image under the implementation map. Therefore, the result turns irreversible information loss into an explicit universal object and provides a compositional framework for studying reconstruction, coarse-graining, and bulk-boundary visibility in open thermodynamic systems. Finally, we outline a potential cost-enriched extension in which interfaces carry dissipation costs and composition minimizes accumulated dissipation over admissible intermediate realizations, opening a route from Boolean feasibility to quantitative thermodynamic implementation.

56 pages