70,481 research outputs found
Positive model structures for abstract symmetric spectra
We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn’s sense, of a certain positive projective model structure on spectra, where positivity basically means the truncation of the zero level. The localization is by the set of stabilizing morphisms or their truncated version
Symmetric operads in abstract symmetric spectra
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
G-symmetric spectra, semistability and the multiplicative norm
In this paper we develop the basic homotopy theory of G-symmetric spectra
(that is, symmetric spectra with a G-action) for a finite group G, as a model
for equivariant stable homotopy with respect to a G-set universe. This model
lies in between Mandell's equivariant symmetric spectra and the G-orthogonal
spectra of Mandell and May and is Quillen equivalent to the two. We further
discuss equivariant semistability, construct model structures on module,
algebra and commutative algebra categories and describe the homotopical
properties of the multiplicative norm in this context.Comment: Final published versio
A model structure for coloured operads in symmetric spectra
We describe a model structure for coloured operads with values in the
category of symmetric spectra (with the positive model structure), in which
fibrations and weak equivalences are defined at the level of the underlying
collections. This allows us to treat R-module spectra (where R is a cofibrant
ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as
its first term. Using this model structure, we give suficient conditions for
homotopical localizations in the category of symmetric spectra to preserve
module structures.Comment: 16 page
Diagram spaces, diagram spectra, and spectra of units
This article compares the infinite loop spaces associated to symmetric
spectra, orthogonal spectra, and EKMM S-modules. Each of these categories of
structured spectra has a corresponding category of structured spaces that
receives the infinite loop space functor \Omega^\infty. We prove that these
models for spaces are Quillen equivalent and that the infinite loop space
functors \Omega^\infty agree. This comparison is then used to show that two
different constructions of the spectrum of units gl_1 R of a commutative ring
spectrum R agree.Comment: 62 pages. The definition of the functor \mathbb{Q} is changed.
Sections 8, 9, 17 and 18 contain revisions and/or new materia
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