Signs in objective linear algebra, exemplified with exterior powers and determinants

TL;DR

This paper introduces an objective linear algebra framework using a cardinality functor capable of negative values, providing a new perspective on signs via homotopies and orientations, with applications to exterior powers and determinants.

Contribution

It develops a novel objective linear algebra theory incorporating signs through homotopies and orientations, with explicit applications to exterior powers and determinants.

Findings

Objective linear algebra with negative cardinality values

Signs interpreted as homotopies and orientation ratios

Objective treatment of exterior powers and determinants

Abstract

We develop objective linear algebra in a new setting with a cardinality functor that can take negative values. The signs arise as little homotopies, as ratios between orientations. To illustrate the workings of the theory we give an objective treatment of exterior powers and determinants.