Categories of Kirchhoff relations

TL;DR

This paper characterizes Kirchhoff relations within affine Lagrangian relations over finite fields, linking them to electrical circuit components and error detection matrices, and explores their subcategories including deterministic and lossless relations.

Contribution

It introduces a matrix-based characterization of Kirchhoff relations and identifies key subcategories, connecting circuit theory with algebraic and categorical structures.

Findings

Kirchhoff relations are characterized by parity-check matrices.

Deterministic Kirchhoff relations are generated by resistor circuits.

Lossless relations form a fundamental hyper-categorical structure.

Abstract

It is known that the category of affine Lagrangian relations, AffLagRel_F, over a field, F, of integers modulo a prime p (with p > 2) is isomorphic to the category of stabilizer quantum circuits for p-dits. Furthermore, it is known that electrical circuits (generalized for the field F) occur as a natural subcategory of AffLagRel_F. The purpose of this paper is to provide a characterization of the relations in this subcategory -- and in important subcategories thereof -- in terms of parity-check and generator matrices as used in error detection. In particular, we introduce the subcategory consisting of Kirchhoff relations to be (affinely) those Lagrangian relations that conserve total momentum or equivalently satisfy Kirchhoff's current law. Maps in this subcategory can be generated by electrical components (generalized for the field F): namely resistors, current dividers, and current and voltage sources. The "source" electrical components deliver the affine nature of the maps while current dividers add an interesting quasi-stochastic aspect. We characterize these Kirchhoff relations in terms of parity-check matrices and in addition, characterizes two important subcategories: the deterministic Kirchhoff relations and the lossless relations. The category of deterministic Kirchhoff relations as electrical circuits are generated by resistors circuits. Lossless relations, which are deterministic Kirchhoff, provide exactly the basic hyper-categorical structure of these settings.