This paper sets up the foundations for derived algebraic geometry,
Goerss--Hopkins obstruction theory, and the construction of commutative ring
spectra in the abstract setting of operadic algebras in symmetric spectra in an
(essentially) arbitrary model category.
We show that one can do derived algebraic geometry a la To\"en--Vezzosi in an
abstract category of spectra. We also answer in the affirmative a question of
Goerss and Hopkins by showing that the obstruction theory for operadic algebras
in spectra can be done in the generality of spectra in an (essentially)
arbitrary model category. We construct strictly commutative simplicial ring
spectra representing a given cohomology theory and illustrate this with a
strictly commutative motivic ring spectrum representing higher order products
on Deligne cohomology.
These results are obtained by first establishing Smith's stable positive
model structure for abstract spectra and then showing that this category of
spectra possesses excellent model-theoretic properties: we show that all
colored symmetric operads in symmetric spectra valued in a symmetric monoidal
model category are admissible, i.e., algebras over such operads carry a model
structure. This generalizes the known model structures on commutative ring
spectra and E-infinity ring spectra in simplicial sets or motivic spaces. We
also show that any weak equivalence of operads in spectra gives rise to a
Quillen equivalence of their categories of algebras. For example, this extends
the familiar strictification of E-infinity rings to commutative rings in a
broad class of spectra, including motivic spectra. We finally show that
operadic algebras in Quillen equivalent categories of spectra are again Quillen
equivalent.Comment: 34 pages. Comments and questions are very welcome. v2: Identical to
the journal version except for formatting and styl