Cellular objects and Shelah's singular compactness theorem

Abstract

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.

A Proof of the Countable Telescope Conjecture for Module Categories

“…This meta-mathematical approach will lead to several generalizations of Shelah's compactness principle, as pointed out and formalised by Beke and Rosicky in [5]. Towards the end of our project, we will present a theorem stated by Saroch and Stovicek in [20] that can be regarded from this perspective.…”

“…it restricts to each ι δ ). Then, following a classical construction (see [5] for some more context), we can define the new term as the pushout of the iso ι and of the non-split inclusion ⊆: K ⊗ H (λ) ↪− → N ⊗ H (λ) , i.e. M α := P O(ι, ⊆).…”

“…Then, some set-theoretical prerequisites are introduced. In particular, we present the notion of filters, as well as a brief overview of the model theoretical approach, and we mention a 'revisited' approach to freeness conditions, following the work of Beke and Rosicky in [5]. Again, everything is made explicit, although the model theoretical digression on elementary cogenerators or Ziegler's spectrum are barely sketched, since each of them itself would require an independent review.…”