Cellular objects and Shelah's singular compactness theorem

TL;DR

This paper presents a unified structural formulation of Shelah's singular compactness theorem, extending its applicability to cellular structures and connecting it with recent developments in abstract homotopy theory.

Contribution

It introduces a new functorial and cellular formulation of singular compactness that generalizes existing results across various mathematical structures.

Findings

Unified structural statement of singular compactness

Cellular structures exhibit a relative notion of freeness

Connections to abstract homotopy theory

Abstract

The best-known version of Shelah's celebrated singular cardinal compactness theorem states that if the cardinality of an abelian group is singular, and all its subgroups of lesser cardinality are free, then the group itself is free. The proof can be adapted to cover a number of analogous situations in the setting of non-abelian groups, modules, graph colorings, set transversals etc. We give a single, structural statement of singular compactness that covers all examples in the literature that we are aware of. A case of this formulation, singular compactness for cellular structures, is of special interest; it expresses a relative notion of freeness. The proof of our functorial formulation is motivated by a paper of Hodges, based on a talk of Shelah. The cellular formulation is new, and related to recent work in abstract homotopy theory.