Constanze Roitzheim

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.

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories.

Derived A(infinity)-algebras in an operadic context

Derived A1 -algebras were developed recently by Sagave. Their advantage over classical A1 -algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A1 -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, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad A1 as a resolution of the operad As encoding associative algebras. We further show that Sagaves definition of morphisms agrees with the infinity- morphisms of dA1 -algebras arising from operadic machinery. We also study the operadic homology of derived A1 -algebras. 16E45, 18D50; 18G55, 18G10

Uniqueness of A-infinity structures and Hochschild cohomology

Working over a commutative ground ring, we establish a Hochschild cohomology criterion for uniqueness of derived A-infinity algebra structures in the sense of Sagave. We deduce a Hochschild cohomology criterion for intrinsic formality of a differential graded algebra. This generalizes a classical result of Kadeishvili for the case of a graded algebra over a field.

Bousfield Localisations along Quillen Bifunctors

Consider a Quillen adjunction of two variables between combinatorial model categories from 𝓒×𝓓

Rational equivariant rigidity

\mathcal {C}\times \mathcal {D}

Towers and Fibered Products of Model Structures

to 𝓔

A case of monoidal uniqueness of algebraic models

\mathcal {E}

On the algebraic classification of K-local spectra

, a set 𝒮

Rigidity and exotic models for the K -local stable homotopy category

\mathcal {S}