4,058 research outputs found
Behavior of Quillen (co)homology with respect to adjunctions
This paper aims to answer the following question: Given an adjunction between
two categories, how is Quillen (co)homology in one category related to that in
the other? We identify the induced comparison diagram, giving necessary and
sufficient conditions for it to arise, and describe the various comparison
maps. Examples are given. Along the way, we clarify some categorical
assumptions underlying Quillen (co)homology: cocomplete categories with a set
of small projective generators provide a convenient setup.Comment: Minor corrections. To appear in Homology, Homotopy and Application
Completed power operations for Morava E-theory
We construct and study an algebraic theory which closely approximates the
theory of power operations for Morava E-theory, extending previous work of
Charles Rezk in a way that takes completions into account. These algebraic
structures are made explicit in the case of K-theory. Methodologically, we
emphasize the utility of flat modules in this context, and prove a general
version of Lazard's flatness criterion for module spectra over associative ring
spectra.Comment: Version 3: Minor corrections. Journal version, up to small cosmetic
change
Moduli spaces of 2-stage Postnikov systems
Using the obstruction theory of Blanc-Dwyer-Goerss, we compute the moduli
space of realizations of 2-stage Pi-algebras concentrated in dimensions 1 and n
or in dimensions n and n+1. The main technical tools are Postnikov truncation
and connected covers of Pi-algebras, and their effect on Quillen cohomology.Comment: Version 3: Added conventions in section 1.3. Minor change
Eilenberg–MacLane mapping algebras and higher distributivity up to homotopy
Primary cohomology operations, i.e., elements of the Steenrod algebra, are
given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps
(before taking homotopy classes) form the topological version of the Steenrod
algebra. Composition of such maps is strictly linear in one variable and linear
up to coherent homotopy in the other variable. To describe this structure, we
introduce a hierarchy of higher distributivity laws, and prove that the
topological Steenrod algebra satisfies all of them. We show that the higher
distributivity laws are homotopy invariant in a suitable sense. As an
application of -distributivity, we provide a new construction of a
derivation of degree of the mod Steenrod algebra.Comment: v3: Minor changes. Final versio
2-track algebras and the Adams spectral sequence
In previous work of the first author and Jibladze, the -term of the
Adams spectral sequence was described as a secondary derived functor, defined
via secondary chain complexes in a groupoid-enriched category. This led to
computations of the -term using the algebra of secondary cohomology
operations. In work with Blanc, an analogous description was provided for all
higher terms . In this paper, we introduce -track algebras and tertiary
chain complexes, and we show that the -term of the Adams spectral sequence
is a tertiary Ext group in this sense. This extends the work with Jibladze,
while specializing the work with Blanc in a way that should be more amenable to
computations.Comment: v2: Added Appendix A on models for homotopy 2-types. To appear in the
Journal of Homotopy and Related Structure
Higher Toda brackets and the Adams spectral sequence in triangulated categories
The Adams spectral sequence is available in any triangulated category
equipped with a projective or injective class. Higher Toda brackets can also be
defined in a triangulated category, as observed by B. Shipley based on J.
Cohen's approach for spectra. We provide a family of definitions of higher Toda
brackets, show that they are equivalent to Shipley's, and show that they are
self-dual. Our main result is that the Adams differential in any Adams
spectral sequence can be expressed as an -fold Toda bracket and as an
order cohomology operation. We also show how the result
simplifies under a sparseness assumption, discuss several examples, and give an
elementary proof of a result of Heller, which implies that the three-fold Toda
brackets in principle determine the higher Toda brackets.Comment: v2: Added Section 7, about an application to computing maps between
modules over certain ring spectra. Minor improvements elsewhere. v3: Minor
updates throughout; closely matches published versio
The DG-category of secondary cohomology operations
We study track categories (i.e., groupoid-enriched categories) endowed with
additive structure similar to that of a 1-truncated DG-category, except that
composition is not assumed right linear. We show that if such a track category
is right linear up to suitably coherent correction tracks, then it is weakly
equivalent to a 1-truncated DG-category. This generalizes work of the first
author on the strictification of secondary cohomology operations. As an
application, we show that the secondary integral Steenrod algebra is
strictifiable.Comment: v3: Minor revision
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