arXiv subjects

Constanze Roitzheim

Publications and source records attributed to Constanze Roitzheim.

25 records · Page 2Linked to original sources

arXiv2012

Derived A-infinity algebras in an operadic context

Derived A-infinity algebras were developed recently by Sagave. Their advantage over classical A-infinity algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A-infinity algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas. This generalises the established result describing the operad A-infinity as a resolution of the operad As encoding associative algebras. We further show Sagave's definition of morphisms agrees with the infinity-morphisms of dA-infinity algebras arising from operadic machinery. We also study the operadic homology of derived A-infinity algebras.

math.AT↗

arXiv2012

Stable left and right Bousfield localisations

We study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.

math.AT↗

arXiv2012

Rational Equivariant Rigidity

We prove that if G is the circle group or a profinite group, then the all of the homotopical information of the category of rational G-spectra is captured by triangulated structure of the rational G-equivariant stable homotopy category. That is, for G profinite or S1, the rational G-equivariant stable homotopy category is rigid. For the case of profinite groups this rigidity comes from an intrinsic formality statement, so we carefully relate the notion of intrinsic formality of a differential graded algebra to rigidity.

math.AT↗

arXiv2011

Monoidality of Franke's Exotic Model

We discuss the monoidal structure on Franke's algebraic model for the $K_{(p)}$-local stable homotopy category at odd primes and show that its Picard group is isomorphic to the integers.

math.AT↗

arXiv2011

Local Framings

Framings provide a way to construct Quillen functors from simplicial sets to any given model category. A more structured set-up studies stable frames giving Quillen functors from spectra to stable model categories. We will investigate how this is compatible with Bousfield localisation to gain insight into the deeper structure of the stable homotopy category. We further show how these techniques relate to rigidity questions and how they can be used to study algebraic model categories.

math.AT↗

arXiv2010

Uniqueness of $A_\infty$-structures and Hochschild cohomology

This paper investigates if a differential graded algebra can have more than one $A_\infty$-structure extending the given differential graded algebra structure. We give a sufficient condition for uniqueness of such an $A_\infty$-structure up to quasi-isomorphism using Hochschild cohomology. We then extend this condition to Sagave's notion of derived $A_\infty$-algebras after introducing a notion of Hochschild cohomology that applies to this.

math.AT↗

arXiv2007

On the algebraic classification of K-local spectra

In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K_(p)-local stable homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist's point of view.

math.AT↗