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

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 $d_r$ in any Adams spectral sequence can be expressed as an $(r+1)$-fold Toda bracket and as an $r^{\text{th}}$ 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

La charge de l’orignal épormyable : une dramaticité sauvage

La charge de l’orignal épormyable est un objet curieux. Il convoque une série anarchique d’architextes consacrés et met ainsi en échec l’homogénéité dramatique. Le drame, composé des figures clés d’architextes élémentaires, reproduit le dialogisme le plus traditionnel et déborde enfin le cadre de la représentation par sa dimension expressionniste. C’est pour ébranler le système des conventions dramatiques que Gauvreau ne respecte pas le pacte théâtral.La charge de l’orignal épormyable is a strange piece. It summons a anarchical set of established architexts and consequently undermine the dramatic homogeneity. The play, having the key stances of elementary architexts for main characters, reproduces the traditional dialogism and finally break out of the stage bounds with an expressionnist angle. Gauvreau differs from the dramatic pact to disturb his conventionnal system