2026-09-20T12:47:05Zhttps://ecommons.cornell.edu/server/oai/requestoai:ecommons.cornell.edu:1813/1097972026-05-15T19:50:16Zcom_1813_35col_1813_47
Sarazola, Maru
Zakharevich, Inna I.
Aguiar, Marcelo
Holm, Tara
2021-09-09T17:41:01Z
2021-09-09T17:41:01Z
2021-05
ProQuest Submission ID: 12461
ProQuest Publication ID: 28416703
https://hdl.handle.net/1813/109797
https://doi.org/10.7298/rw3q-1q83
15049531
183 pages
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 $(C,C^\bot)$ in an exact category E, together with a subcategory $Z\subseteq E$ containing $C^\bot$, 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 $A\subseteq B$ 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 $S_\bullet$-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.
en
Attribution 4.0 International
https://creativecommons.org/licenses/by/4.0/
algebraic K-theory
cotorsion
double categories
exact categories
K-theory
localization
Constructing K-theory spectra from algebraic structures with a class of acyclic objects
dissertation or thesis
https://newcatalog.library.cornell.edu/catalog/15049531
application/pdf
Mathematics
Cornell University
Doctor of Philosophy
Ph. D., Mathematics
https://hdl.handle.net/1813/59810
Publication
Cornell Theses and Dissertations
Sarazola, Maru
5893a6ea-7af3-41d7-abc6-04bcd26ab5df
open.access
open.access
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="0b518018-d166-49b0-bdc8-f6a2e16d3e3c">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Language>en</Language>
<Title>Constructing K-theory spectra from algebraic structures with a class of acyclic objects</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2021-05</PublicationDate>
<DOI>https://doi.org/10.7298/rw3q-1q83</DOI>
<Authors>
<Author>
<DisplayName>Sarazola, Maru</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<OrgUnit />
</Publisher>
</Publishers>
<License>https://creativecommons.org/licenses/by/4.0/</License>
<Keyword>algebraic K-theory</Keyword>
<Keyword>cotorsion</Keyword>
<Keyword>double categories</Keyword>
<Keyword>exact categories</Keyword>
<Keyword>K-theory</Keyword>
<Keyword>localization</Keyword>
<Abstract>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 $(C,C^\bot)$ in an exact category E, together with a subcategory $Z\subseteq E$ containing $C^\bot$, 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 $A\subseteq B$ 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 $S_\bullet$-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.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>