Constructing K-theory spectra from algebraic structures with a class of acyclic objects

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This thesis studies different ways to construct categories admitting an algebraic K-theory spectrum, focusing on categories that contain some flavor of underlying algebraic structure as well as relevant homotopical information. In Part I, published as [20], we show that under certain technical conditions, a cotorsion pair (C,C) in an exact category E, together with a subcategory ZE containing C, determines a Waldhausen structure on C in which Z is the class of acyclic objects. This yields a new version of Quillen's Localization Theorem, relating the K-theory of exact categories AB to that of a cofiber. The novel approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, A need not be a Serre subcategory, which results in new examples. In Part II, joint work with Brandon Shapiro, we upgrade the K-theory of (A)CGW categories due to Campbell and Zakharevich by defining a new type of structures, called FCGWA categories, that incorporate the data of weak equivalences. FCGWA categories admit an S-construction in the spirit of Waldhausen's, which produces a K-theory spectrum, and satisfies analogues of the Additivity and Fibration Theorems. Weak equivalences are determined by choosing a subcategory of acyclic objects satisfying minimal conditions, which results in a Localization Theorem that generalizes previous versions in the literature. Our main example is chain complexes of sets with quasi-isomorphisms; these satisfy a Gillet--Waldhausen Theorem, yielding an equivalent presentation of the K-theory of finite sets.

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183 pages


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algebraic K-theory; cotorsion; double categories; exact categories; K-theory; localization


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Union Local


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Zakharevich, Inna I.

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Aguiar, Marcelo
Holm, Tara

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Ph. D., Mathematics

Degree Level

Doctor of Philosophy

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Government Document




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Attribution 4.0 International


dissertation or thesis

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