A Spectral "schematization" of Homotopy Types

TL;DR

This paper develops a formalism for schematization of homotopy types using symmetric spectra, generalizing topological spaces and defining new homotopy groups via mapping spaces in symmetric spectra.

Contribution

It introduces a novel approach to schematization by formalizing it through symmetric spectra and mapping spaces, extending Grothendieck's original ideas.

Findings

Provides a new formalism for schematization of homotopy types.

Defines homotopy groups of schematized types via mapping spaces in symmetric spectra.

Generalizes topological spaces to symmetric spectra for homotopy analysis.

Abstract

Toen has interpreted the schematization problem as originally imagined by Grothendieck in "Pursuing Stacks" in such a way that solution(s) to this problem could be given. As he pointed out, there are many solutions available, and he gave two constructions solving this problem. What we do in the present work is reconsider what Grothendieck initially had in mind and develop a formalism that provides a concept of "schematization" and corresponding homotopy groups of "schematized" homotopy types. In our view this can be realized if we generalize topological spaces to symmetric spectra, and the mapping spaces Map(S,A) in the category of symmetric sequences play the role of homotopy groups, S the sphere spectrum, A a symmetric spectrum.