G-Global Algebraic K-Theory

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Author
Lenz, Tobias
Type of Scholarly Publication
Dissertation
Date of Exam
04.11.2021
Date of Publication
09.11.2021
Advisor
Schwede, Stefan
Co-Referee
Nikolaus, Thomas
Involved Institutions
Rheinische Friedrich-Wilhelms-Universität Bonn
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Abstract
We develop the foundations of G-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the G-global algebraic K-theory of small symmetric monoidal categories with G-action, unifying G-equivariant algebraic K-theory, as considered for example by Shimakawa, and Schwede's global algebraic K-theory.
As an application of the theory, we prove that the G-global algebraic K-theory functor exhibits the category of small symmetric monoidal categories with G-action as a model of connective G-global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic K-theory.
Subjects
Algebraische Topologie, Äquivariante Homotopietheorie, Algebraische K-Theorie, Ultrakommutativität, Unendliche Schleifenräume, Symmetrisch monoidale Kategorien, Gamma-Räume, algebraic topology, equivariant homotopy theory, algebraic K-theory, global homotopy theory, ultra-commutativity, infinite loop spaces, symmetric monoidal categories, Gamma spaces
Classification (DDC)
510 Mathematik
Lenz, Tobias: G-Global Algebraic K-Theory. - Bonn, 2021. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-64017
@phdthesis{handle:20.500.11811/9401,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-64017,
author = {{Tobias Lenz}},
title = {G-Global Algebraic K-Theory},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2021,
month = nov,
note = {We develop the foundations of G-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the G-global algebraic K-theory of small symmetric monoidal categories with G-action, unifying G-equivariant algebraic K-theory, as considered for example by Shimakawa, and Schwede's global algebraic K-theory.
As an application of the theory, we prove that the G-global algebraic K-theory functor exhibits the category of small symmetric monoidal categories with G-action as a model of connective G-global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic K-theory.},
url = {https://hdl.handle.net/20.500.11811/9401}
}

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