81 research outputs found
The algebra of secondary homotopy operations in ring spectra
The primary algebraic model of a ring spectrum is the ring of homotopy
groups. We introduce the secondary model which has the structure of a secondary
analogue of a ring. This new algebraic model determines Massey products and
cup-one squares. As an application we obtain new derivations of the homotopy
ring.Comment: We have changed the title and we have included new computations and
applications, 55 page
Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence
We describe the dualization of the algebra of secondary cohomology operations
in terms of generators extending the Milnor dual of the Steenrod algebra. In
this way we obtain explicit formulae for the computation of the E_3-term of the
Adams spectral sequence converging to the stable homotopy groups of spheres.Comment: 101 page
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
Comparing cohomology obstructions
We show that three different kinds of cohomology - Baues-Wirsching
cohomology, the (S,O)-cohomology of Dwyer-Kan, and the Andre-Quillen cohomology
of a Pi-algebra - are isomorphic, under certain assumptions.
This is then used to identify the cohomological obstructions in three general
approaches to realizability problems: the track category version of
Baues-Wirsching, the diagram rectifications of Dwyer-Kan-Smith, and the
Pi-Algebra realization of Dwyer-Kan-Stover. Our main tool in this
identification is the notion of a mapping algebra: a simplicially enriched
version of an algebra over a theory
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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