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

Joachim Kock, Jesper Michael Møller

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.

Signs in objective linear algebra, exemplified with 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.
Paper Structure (29 sections, 27 theorems, 137 equations)

This paper contains 29 sections, 27 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions
  4. Related work
  5. Outlook
  6. Organisation of the paper
  7. Linear algebra
  8. Objective linear algebra without signs
  9. Parities and the 'ground field'
  10. Group actions and homotopy sums
  11. Orientations
  12. Linear maps
  13. Monoidal structures
  14. The negative of a span
  15. Scalars
  16. ...and 14 more sections

Key Result

Lemma 2.3.4

Every $X{\to}\mathbb{P}$ is (uniquely) the homotopy sum of copies of $\mathsf{e}:1{\to}\mathbb{P}$.

Theorems & Definitions (41)

  • Lemma 2.3.4
  • Proposition 2.3.5
  • Lemma 2.4.1
  • proof
  • Lemma 2.5.6
  • Lemma 2.5.7
  • proof
  • Lemma 2.8.5
  • Proposition 2.8.6
  • Lemma 2.9.3
  • ...and 31 more