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