Algebraic models of non-connected spaces and homotopy theory ofL∞algebras

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of ∞ \infty -categories C \mathcal {C} and classify such Goodwillie towers in terms of the derivatives of the identity functor of C \mathcal {C} . As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p p -local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching’s spectral version of the Lie operad. This is a close analogue of Quillen’s results on rational homotopy.

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Rational homotopy theory for non-simply connected spaces

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-99-02463-0 ◽

1999 ◽

Vol 352 (4) ◽

pp. 1493-1525 ◽

Cited By ~ 8

Author(s):

Antonio Gómez-Tato ◽

Stephen Halperin ◽

Daniel Tanré

Keyword(s):

Homotopy Theory ◽

Rational Homotopy Theory ◽

Rational Homotopy ◽

Simply Connected ◽

Connected Spaces

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Equivariant Topology and Derived Algebra

10.1017/9781108942874 ◽

2021 ◽

Keyword(s):

Homotopy Theory ◽

Algebraic Models ◽

Research Papers ◽

Model Categories ◽

Homotopy Limits ◽

Equivariant Topology ◽

Mackey Functors ◽

Chromatic Homotopy ◽

Equivariant Spectra ◽

Bousfield Localization

This volume contains eight research papers inspired by the 2019 'Equivariant Topology and Derived Algebra' conference, held at the Norwegian University of Science and Technology, Trondheim in honour of Professor J. P. C. Greenlees' 60th birthday. These papers, written by experts in the field, are intended to introduce complex topics from equivariant topology and derived algebra while also presenting novel research. As such this book is suitable for new researchers in the area and provides an excellent reference for established researchers. The inter-connected topics of the volume include: algebraic models for rational equivariant spectra; dualities and fracture theorems in chromatic homotopy theory; duality and stratification in tensor triangulated geometry; Mackey functors, Tambara functors and connections to axiomatic representation theory; homotopy limits and monoidal Bousfield localization of model categories.

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MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN RETRACTIVE SPACES

Forum of Mathematics Sigma ◽

10.1017/fms.2020.11 ◽

2020 ◽

Vol 8 ◽

Author(s):

FABIAN HEBESTREIT ◽

STEFFEN SAGAVE ◽

CHRISTIAN SCHLICHTKRULL

Keyword(s):

Homotopy Theory ◽

Companion Paper ◽

Smash Product ◽

Algebraic Models ◽

Thom Spectra ◽

Ring Spectra ◽

Symmetric Spectra ◽

Point Set

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set-level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding $\infty$ -categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted $K$ -theory.

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