  Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant
homotopy theoretic construction of the Brown-Peterson spectrum at a prime p.
They discussed May's notions of nuclear complexes and of cores of spaces,
spectra, and commutative S-algebras. Their most striking conclusions, due to Hu
and Kriz, were negative: cores are not unique up to equivalence, and BP is not
a core of MU considered as a commutative S-algebra, although it is a core of MU
considered as a p-local spectrum. We investigate these ideas further, obtaining
much more positive conclusions. We show that nuclear complexes have several
non-obviously equivalent characterizations. Up to equivalence, they are
precisely the irreducible complexes, the minimal atomic complexes, and the
Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz
dimension, which we call complexes with no mod p detectable homotopy. Unlike
the notion of a nuclear complex, these other notions are all invariant under
equivalence. This simple and conceptual criterion for a complex to be minimal
atomic allows us to prove that many familiar spectra, such as ko, eo_2, and BoP
at the prime 2, all BP<n> at any prime p, and the indecomposable wedge summands
of the suspension spectra of $CP^\infty$ and $HP^\infty$ at any prime p are
minimal atomic.
