  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.
