Equivariant Topology and Derived Algebra

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.

scholarly journals Locally constant functors

2009 ◽  
Vol 147 (3) ◽  
pp. 593-614 ◽  
Author(s):  
DENIS–CHARLES CISINSKI
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Model Categories ◽  
Combinatorial Model ◽  
Constant Coefficients ◽  
Kan Extensions ◽  
Bousfield Localization

AbstractWe study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

scholarly journals Affineness and chromatic homotopy theory

2015 ◽  
Vol 8 (2) ◽  
pp. 476-528 ◽  
Author(s):  
Akhil Mathew ◽  
Lennart Meier
Keyword(s):  
Homotopy Theory ◽  
Chromatic Homotopy

Model Structures for the 𝔸1-homotopy Category

2019 ◽  
pp. 175-212
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Full Subcategory ◽  
Homotopy Category ◽  
Symmetric Power ◽  
Model Structure ◽  
Model Categories ◽  
Simplicial Presheaves ◽  
Basic Ideas ◽  
Quillen Model ◽  
Model Structures ◽  
Bousfield Localization

This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as Δ‎opPshv(Sm)𝔸1 and Δ‎op PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM eff nis ≤0 of DM eff nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ‎-closed classes, and develops their basic properties. The theory of ̅Δ‎-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.

scholarly journals Chromatic homotopy theory is asymptotically algebraic

2020 ◽  
Vol 220 (3) ◽  
pp. 737-845
Author(s):  
Tobias Barthel ◽  
Tomer M. Schlank ◽  
Nathaniel Stapleton
Keyword(s):  
Asymptotic Solution ◽  
Homotopy Theory ◽  
Approximation Problem ◽  
Prime Numbers ◽  
Isomorphism Theorem ◽  
Mathematical Objects ◽  
Infinite Collection ◽  
Chromatic Homotopy

AbstractInspired by the Ax–Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the E(n, p)-local categories over any non-principal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.

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