The Weyl functor - Introduction to Absolute Arithmetic

TL;DR

This paper explores the concept of absolute arithmetic by connecting Tits' interpretation of symmetric groups as Chevalley groups over a hypothetical field with only one element, developing combinatorial geometry and linear algebra in this context.

Contribution

It introduces the Weyl functor and foundational principles of absolute arithmetic, bridging classical algebraic structures with a novel conceptual framework.

Findings

Development of combinatorial geometry over the 'field' with one element

Formulation of linear algebra concepts in absolute arithmetic

Establishment of the 'absolute mantra' as a core principle

Abstract

Starting from an ancient observation of Tits concerning the interpretation of symmetric groups as Chevalley groups over a (non-existing) field having only one element, we describe combinatorial geometry over this field, as well as Linear Algebra. We arrive at an "absolute mantra" which is one of the basic principles of the present book.