{"id":{"repo_id":"cornell","oai_identifier":"oai:ecommons.cornell.edu:1813/109797"},"canonical_url":"https://search.dev.ndltd.org/etd/cornell/oai:ecommons.cornell.edu:1813/109797","repository":{"repo_id":"cornell","name":"Cornell University","base_url":"https://ecommons.cornell.edu/server/oai/request"},"display":{"title":"Constructing K-theory spectra from algebraic structures with a class of acyclic objects","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_html":"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 <span class=\"etd-inline-math\">(C,C<sup>\\</sup>bot)</span> in an exact category E, together with a subcategory $Z\\subseteq E$ containing <span class=\"etd-inline-math\">C<sup>\\</sup>bot</span>, determines a Waldhausen structure on C in which Z is the class of acyclic objects. This yields a new version of Quillen&#x27;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 <span class=\"etd-inline-math\">S<sub>\\</sub>bullet</span>-construction in the spirit of Waldhausen&#x27;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_has_math":true,"creators":["Sarazola, Maru"],"institution":"Cornell University","degree_name":"Ph. D., Mathematics","degree_level":"Doctor of Philosophy","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":["Aguiar, Marcelo","Holm, Tara"],"year":2021,"date_issued":"2021-05","date_published":"2021-05","updated_at":"2026-07-24T01:49:06Z","subjects":["algebraic K-theory","cotorsion","double categories","exact categories","K-theory","localization"],"languages":["en"],"rights":["Attribution 4.0 International"],"rights_urls":["https://creativecommons.org/licenses/by/4.0/"],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.7298/rw3q-1q83"],"render_values":[{"text":"https://doi.org/10.7298/rw3q-1q83","href":"https://doi.org/10.7298/rw3q-1q83","code":true}]},{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["ProQuest Submission ID: 12461","ProQuest Publication ID: 28416703"],"render_values":[{"text":"ProQuest Submission ID: 12461","href":null,"code":true},{"text":"ProQuest Publication ID: 28416703","href":null,"code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/1813/109797","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Aguiar, Marcelo","Holm, Tara"]},{"key":"dc:creator","label":"Author","values":["Sarazola, Maru"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2021-09-09T17:41:01Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2021-09-09T17:41:01Z"]},{"key":"dc:date.issued","label":"Date","values":["2021-05"]},{"key":"dc:type","label":"Dc Type","values":["dissertation or thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctor of Philosophy"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Constructing K-theory spectra from algebraic structures with a class of acyclic objects"]}]}],"canonical_facts":{"dc:contributor.committeemember":["Aguiar, Marcelo","Holm, Tara"],"dc:creator":["Sarazola, Maru"],"dc:date.accessioned":["2021-09-09T17:41:01Z"],"dc:date.available":["2021-09-09T17:41:01Z"],"dc:date.issued":["2021-05"],"dc:description":["183 pages"],"dc:description.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."],"dc:format.mimetype":["application/pdf"],"dc:identifier.doi":["https://doi.org/10.7298/rw3q-1q83"],"dc:identifier.other":["ProQuest Submission ID: 12461","ProQuest Publication ID: 28416703"],"dc:identifier.uri":["https://hdl.handle.net/1813/109797"],"dc:language.iso":["en"],"dc:rights":["Attribution 4.0 International"],"dc:rights.uri":["https://creativecommons.org/licenses/by/4.0/"],"dc:subject":["algebraic K-theory","cotorsion","double categories","exact categories","K-theory","localization"],"dc:title":["Constructing K-theory spectra from algebraic structures with a class of acyclic objects"],"dc:type":["dissertation or thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctor of Philosophy"],"thesis:degree_name":["Ph. D., Mathematics"],"thesis:institution_name":["Cornell University"]},"updated_at":"2026-07-24T01:49:06Z"}