scholarly journals MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN RETRACTIVE SPACES

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.

scholarly journals The cotangent complex and Thom spectra

2020 ◽  
Vol 90 (2) ◽  
pp. 229-252
Author(s):  
Nima Rasekh ◽  
Bruno Stonek
Keyword(s):  
Deformation Theory ◽  
Commutative Rings ◽  
Smash Product ◽  
Ring Spectrum ◽  
Central Object ◽  
Calculus Of Functors ◽  
Cotangent Complex ◽  
Thom Spectra ◽  
Ring Spectra ◽  
Formula Group

AbstractThe cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $$E_\infty $$ E ∞ -ring spectra in various ways. In this work we first establish, in the context of $$\infty $$ ∞ -categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of $$E_\infty $$ E ∞ -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an $$E_\infty $$ E ∞ -ring spectrum and $$\mathrm {Pic}(R)$$ Pic ( R ) denote its Picard $$E_\infty $$ E ∞ -group. Let Mf denote the Thom $$E_\infty $$ E ∞ -R-algebra of a map of $$E_\infty $$ E ∞ -groups $$f:G\rightarrow \mathrm {Pic}(R)$$ f : G → Pic ( R ) ; examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of $$R\rightarrow Mf$$ R → M f is equivalent to the smash product of Mf and the connective spectrum associated to G.

scholarly journals Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

2014 ◽  
Vol 7 (4) ◽  
pp. 1077-1117 ◽  
Author(s):  
Matthew Ando ◽  
Andrew J. Blumberg ◽  
David Gepner ◽  
Michael J. Hopkins ◽  
Charles Rezk
Keyword(s):  
Loop Space ◽  
Space Theory ◽  
Thom Spectra ◽  
Ring Spectra ◽  
Infinite Loop ◽  
Infinite Loop Space

scholarly journals Stable homotopical algebra and Γ-spaces

1999 ◽  
Vol 126 (2) ◽  
pp. 329-356 ◽  
Author(s):  
STEFAN SCHWEDE
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Develop Model ◽  
Smash Product ◽  
Spectral Sequences ◽  
Homotopical Algebra ◽  
Stable Homotopy Theory ◽  
Set Up

In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

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