Simplicial homotopy theory of algebraic varieties over real closed fields, Part 1

TL;DR

This paper explores the real homotopy type of algebraic varieties over real closed fields, establishing a comparison with étale homotopy types, and introduces foundational results for this new area.

Contribution

It introduces the concept of real homotopy type for algebraic varieties over real closed fields and proves an analogue of Artin-Mazur's theorem relating it to étale homotopy types.

Findings

Defined the real homotopy type for algebraic varieties over real closed fields

Proved an analogue of Artin-Mazur's theorem for real homotopy types

Established foundational results for future research in real algebraic topology

Abstract

We study the homotopy type of the simplicial set of continuous semi-algebraic simplexes of an algebraic variety defined over a real closed field, which we will call the real homotopy type. We prove an analogue of the theorem of Artin-Mazur comparing the real homotopy type with the \'etale homotopy type. This paper is part one of a sequence of papers on this topic.