Heaps and unpointed stable homotopy theory

TL;DR

This paper explores stability phenomena in unpointed model categories, revealing that homotopy classes form abelian heaps, which are like abelian groups without a fixed zero, differing from the classical pointed case.

Contribution

It introduces the concept of abelian heaps in unpointed homotopy theory and contrasts their properties with classical stable pointed categories.

Findings

Homotopy classes form abelian heaps in unpointed model categories.

These sets can be empty, unlike in pointed stable categories.

Provides a new perspective on stability phenomena in homotopy theory.

Abstract

In this paper, we show how certain ``stability phenomena'' in unpointed model categories provide the sets of homotopy classes with the structure of abelian heaps, i.e. abelian groups without a choice of a zero. In contrast with the classical situation of stable (pointed) model categories, these sets can be empty.