Correspondences and stable homotopy theory

TL;DR

This paper introduces a general framework for constructing correspondences and spectral categories from symmetric ring objects, applying it to recover stable homotopy theories in algebraic and motivic contexts.

Contribution

It provides a novel method to derive stable homotopy theories from symmetric ring objects and modules, unifying algebraic and motivic homotopy theories.

Findings

Recovered stable homotopy theory of spectra from modules over a commutative symmetric ring spectrum.

Reconstructed stable motivic homotopy theory from spectral modules over spectral categories.

Established a general method for producing spectral categories from symmetric ring objects.

Abstract

A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra $SH$ is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory $SH(k)$ from spectral modules over associated spectral categories.