94 research outputs found
Universal Toda brackets of ring spectra
We construct and examine the universal Toda bracket of a highly structured
ring spectrum R. This invariant of R is a cohomology class in the Mac Lane
cohomology of the graded ring of homotopy groups of R which carries information
about R and the category of R-module spectra. It determines for example all
triple Toda brackets of R and the first obstruction to realizing a module over
the homotopy groups of R by an R-module spectrum.
For periodic ring spectra, we study the corresponding theory of higher
universal Toda brackets. The real and complex K-theory spectra serve as our
main examples.Comment: 38 pages; a few typos corrected, to appear in Trans. Amer. Math. So
Topological Hochschild homology and the cyclic bar construction in symmetric spectra
The cyclic bar construction in symmetric spectra and B\"okstedt's original
construction are two possible ways to define the topological Hochschild
homology of a symmetric ring spectrum. In this short note we explain how to
correct an error in Shipley's original comparison of these two approaches.Comment: v2: 7 pages; exposition improved, accepted for publication in
Proceedings of the AM
Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories
We prove the theorem stated in the title. More precisely, we show the
stronger statement that every symmetric monoidal left adjoint functor between
presentably symmetric monoidal infinity-categories is represented by a strong
symmetric monoidal left Quillen functor between simplicial, combinatorial and
left proper symmetric monoidal model categories.Comment: v3: 17 pages, references updated and exposition improved, accepted
for publication in Algebraic and Geometric Topolog
Localization sequences for logarithmic topological Hochschild homology
We study the logarithmic topological Hochschild homology of ring spectra with
logarithmic structures and establish localization sequences for this theory.
Our results apply, for example, to connective covers of periodic ring spectra
like real and complex topological K-theory.Comment: v3: 40 pages; minor changes, accepted for publication in
Mathematische Annale
Homotopy invariance of convolution products
The purpose of this paper is to show that various convolution products are
fully homotopical, meaning that they preserve weak equivalences in both
variables without any cofibrancy hypothesis. We establish this property for
diagrams of simplicial sets indexed by the category of finite sets and
injections and for tame -simplicial sets, with the monoid of injective
self-maps of the positive natural numbers. We also show that a certain
convolution product studied by Nikolaus and the first author is fully
homotopical. This implies that every presentably symmetric monoidal
-category can be represented by a symmetric monoidal model category
with a fully homotopical monoidal product.Comment: v2: 31 pages, exposition improve
A strictly commutative model for the cochain algebra of a space
The commutative differential graded algebra of
polynomial forms on a simplicial set is a crucial tool in rational homotopy
theory. In this note, we construct an integral version of
. Our approach uses diagrams of chain complexes indexed by
the category of finite sets and injections to model
differential graded algebras by strictly commutative objects, called
commutative -dgas. We define a functor from
simplicial sets to commutative -dgas and show that it is a
commutative lift of the usual cochain algebra functor. In particular, it gives
rise to a new construction of the dga of cochains.
The functor shares many properties of ,
and can be viewed as a generalization of that works over
arbitrary commutative ground rings. Working over the integers, a theorem by
Mandell implies that determines the homotopy type of
when is a nilpotent space of finite type.Comment: v3: 23 pages; corrected an error in Lemma 3.7, adjusted some model
structures accordingly and made various other small improvement
Virtual vector bundles and graded Thom spectra
We introduce a convenient framework for constructing and analyzing orthogonal
Thom spectra arising from virtual vector bundles. This framework enables us to
set up a theory of orientations and graded Thom isomorphisms with good
multiplicative properties. The theory is applied to the analysis of logarithmic
structures on commutative ring spectra.Comment: v3: 39 pages, minor revision. Accepted for publication in
Mathematische Zeitschrif
Logarithmic topological Hochschild homology of topological K-theory spectra
In this paper we continue our study of logarithmic topological Hochschild
homology. We show that the inclusion of the connective Adams summand into the
p-local complex connective K-theory spectrum, equipped with suitable log
structures, is a formally log THH-etale map, and compute the V(1)-homotopy of
their logarithmic topological Hochschild homology spectra. As an application,
we recover Ausoni's computation of the V(1)-homotopy of the ordinary THH of ku.Comment: v3: 32 pages; slightly revised. Accepted for publication in J. Eur.
Math. Soc. (JEMS
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