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dc.contributor.advisorHopkins, Michael
dc.contributor.authorBarton, Reid William
dc.date.accessioned2019-12-11T10:06:35Z
dc.date.created2019-11
dc.date.issued2019-09-10
dc.date.submitted2019
dc.identifier.citationBarton, Reid William. 2019. A Model 2-Category of Enriched Combinatorial Premodel Categories. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.
dc.identifier.urihttp://nrs.harvard.edu/urn-3:HUL.InstRepos:42013127*
dc.description.abstractQuillen equivalences induce equivalences of homotopy theories and therefore form a natural choice for the "weak equivalences" between model categories. In [21], Hovey asked whether the 2-category Mod of model categories has a "model 2-category structure" with these weak equivalences. We give an example showing that Mod does not have pullbacks, so cannot be a model 2-category. We can try to repair this lack of limits by generalizing the notion of model category. The lack of limits in Mod is due to the two-out-of-three axiom, so we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a 2-category CPM with excellent algebraic properties: CPM has all limits and colimits and is equipped with a tensor product (representing Quillen bifunctors) which is adjoint to an internal Hom. The homotopy theory of a model category depends in an essential way on the weak equivalences, so it does not extend directly to premodel categories. We build a substitute homotopy theory under an additional axiom on the premodel category, which holds automatically for a premodel category enriched in a monoidal model category V. The 2-category of combinatorial V-premodel categories VCPM is simply the 2-category of modules over the monoid object V, so VCPM inherits the algebraic structure of CPM. We construct a model 2-category structure on VCPM for V a tractable symmetric monoidal model category, by adapting Szumiło's construction of a fibration category of cofibration categories [35]. For set-theoretic reasons, constructing factorizations for this model 2-category structure requires a technical variant of the small object argument which relies on an analysis of the rank of combinatoriality of a premodel category.
dc.description.sponsorshipMathematics
dc.format.mimetypeapplication/pdf
dc.language.isoen
dash.licenseLAA
dc.subjectmodel categories
dc.titleA Model 2-Category of Enriched Combinatorial Premodel Categories
dc.typeThesis or Dissertation
dash.depositing.authorBarton, Reid William
dc.date.available2019-12-11T10:06:35Z
thesis.degree.date2019
thesis.degree.grantorGraduate School of Arts & Sciences
thesis.degree.grantorGraduate School of Arts & Sciences
thesis.degree.levelDoctoral
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
thesis.degree.nameDoctor of Philosophy
dc.contributor.committeeMemberLurie, Jacob
dc.contributor.committeeMemberMiller, Haynes
dc.type.materialtext
thesis.degree.departmentMathematics
thesis.degree.departmentMathematics
dash.identifier.vireo
dc.identifier.orcid0000-0002-8365-7168
dash.author.emailrwbarton@alum.mit.edu


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