Stable homotopy theory of higher categories

TL;DR

This paper extends stable homotopy theory principles to higher categories, showing that inverting endomorphism categories yields a stable homotopy theory of higher categories with new phenomena.

Contribution

It introduces a higher categorical stable homotopy theory framework, including a Brown representability theorem and applications to derived categories of rigs.

Findings

Classical stable homotopy theory is recovered by inverting morphisms.

Stabilization is achieved via spectrum objects in higher categories.

Categorical homology theories support long exact sequences and higher homological algebra.

Abstract

Stable homotopy theory is governed by the principle that after inverting loop spaces, homotopy types become the representing objects for homology theories. We show that this principle extends to higher category theory: inverting endomorphism categories leads to a stable homotopy theory of higher categories, in which higher categories play the role of spaces and categorical spectra represent homology theories of higher categories. Classical stable homotopy theory is recovered by inverting morphisms. While several fundamental features of classical stable homotopy theory persist in this setting, new phenomena arise from categorical dimension. In particular, stabilization is realized by spectrum objects, and the passage from unstable to stable homotopy theory is controlled within a stable range. Our main result is a categorical Brown representability theorem classifying categorical homology theories by categorical spectra. As a consequence, categorical homology theories give rise to long exact sequences and support a higher-categorical homological algebra. As an application, we construct the derived category of a rig, extending homological algebra beyond additive contexts.