Reconstructing the spectrum of F_1 from the stable homotopy category

TL;DR

This paper applies a reconstruction theorem from algebraic geometry to the stable homotopy category, revealing its topological structure and classifying certain subcategories, thus connecting homotopy theory with geometric intuition.

Contribution

It demonstrates how to recover the topological space from the stable homotopy category and classifies key subcategories, bridging algebraic geometry and homotopy theory.

Findings

Recovered the one point topological space from the stable homotopy category.

Classified filtering subsets of principal thick subcategories.

Extended classification results to p-local versions.

Abstract

The finite stable homotopy category S_0 has been suggested as a candidate for a category of perfect complexes over the monoid scheme Spec F_1. We apply a reconstruction theorem from algebraic geometry to S_0, and show that one recovers the one point topological space. We also classify filtering subsets of the set of principal thick subcategories of S_0, and of its p-local versions. This is motivated by a result saying that the analogous classification for the category of perfect complexes over an affine scheme provides topological information.