{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/139376"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/139376","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"On the Homotopy Theory of Stratified Spaces","abstract":"This thesis is broken into two parts. In the first part (Chapters 2 to 6) is dedicated to proving a 'homtopy hypothesis' for stratified spaces. Specifically, given a poset P, we show that the ∞-category Strₚ of ∞-categories with a conservative functor to P can be obtained from the ordinary category of P-stratified topological spaces by inverting a class of weak equivalences. For suitably nice P-stratified topological spaces, the corresponding object of Strₚ is the exit-path ∞-category of MacPherson, Treumann, and Lurie. To prove this stratified homotopy hypothesis, we define combinatorial simplicial model structure on the category of simplicial sets over the nerve of 𝑃 whose underlying ∞-category is the ∞-category Strₚ. This model structure on P-stratified simplicial sets allows us to easily compare other theories of P-stratified spaces to ours and deduce that they all embed into ours. The second part (Chapters 7 to 9) explores a number of consequences of this stratified homotopy hypothesis, as well as related results on exit-path ∞-categories and constructible sheaves. This includes an overview of our joint work with Bariwck and Glasman on exit-path categories in algebraic geometry; this work uses as input the perspective on stratified spaces provided by our stratified homotopy hypothesis.","abstract_html":"This thesis is broken into two parts. In the first part (Chapters 2 to 6) is dedicated to proving a 'homtopy hypothesis' for stratified spaces. Specifically, given a poset P, we show that the ∞-category Strₚ of ∞-categories with a conservative functor to P can be obtained from the ordinary category of P-stratified topological spaces by inverting a class of weak equivalences. For suitably nice P-stratified topological spaces, the corresponding object of Strₚ is the exit-path ∞-category of MacPherson, Treumann, and Lurie. To prove this stratified homotopy hypothesis, we define combinatorial simplicial model structure on the category of simplicial sets over the nerve of 𝑃 whose underlying ∞-category is the ∞-category Strₚ. This model structure on P-stratified simplicial sets allows us to easily compare other theories of P-stratified spaces to ours and deduce that they all embed into ours. The second part (Chapters 7 to 9) explores a number of consequences of this stratified homotopy hypothesis, as well as related results on exit-path ∞-categories and constructible sheaves. This includes an overview of our joint work with Bariwck and Glasman on exit-path categories in algebraic geometry; this work uses as input the perspective on stratified spaces provided by our stratified homotopy hypothesis.","abstract_has_math":false,"creators":["Haine, Peter J."],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Barwick, Clark"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-06","date_published":"2021-06","updated_at":"2026-07-22T22:21:44Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"rights_urls":["http://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/139376","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Barwick, Clark"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Department of Mathematics"]},{"key":"dc:creator","label":"Author","values":["Haine, Peter J."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-01-14T15:07:53Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-01-14T15:07:53Z"]},{"key":"dc:date.issued","label":"Date","values":["2021-06"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctoral","Doctor of Philosophy"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright - Educational Use Permitted","Copyright MIT"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/page/InC-EDU/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1721.1/139376"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is broken into two parts. In the first part (Chapters 2 to 6) is dedicated to proving a 'homtopy hypothesis' for stratified spaces. Specifically, given a poset P, we show that the ∞-category Strₚ of ∞-categories with a conservative functor to P can be obtained from the ordinary category of P-stratified topological spaces by inverting a class of weak equivalences. For suitably nice P-stratified topological spaces, the corresponding object of Strₚ is the exit-path ∞-category of MacPherson, Treumann, and Lurie. To prove this stratified homotopy hypothesis, we define combinatorial simplicial model structure on the category of simplicial sets over the nerve of 𝑃 whose underlying ∞-category is the ∞-category Strₚ. This model structure on P-stratified simplicial sets allows us to easily compare other theories of P-stratified spaces to ours and deduce that they all embed into ours. The second part (Chapters 7 to 9) explores a number of consequences of this stratified homotopy hypothesis, as well as related results on exit-path ∞-categories and constructible sheaves. This includes an overview of our joint work with Bariwck and Glasman on exit-path categories in algebraic geometry; this work uses as input the perspective on stratified spaces provided by our stratified homotopy hypothesis."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["On the Homotopy Theory of Stratified Spaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Barwick, Clark"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Haine, Peter J."],"dc:date.accessioned":["2022-01-14T15:07:53Z"],"dc:date.available":["2022-01-14T15:07:53Z"],"dc:date.issued":["2021-06"],"dc:description.abstract":["This thesis is broken into two parts. In the first part (Chapters 2 to 6) is dedicated to proving a 'homtopy hypothesis' for stratified spaces. Specifically, given a poset P, we show that the ∞-category Strₚ of ∞-categories with a conservative functor to P can be obtained from the ordinary category of P-stratified topological spaces by inverting a class of weak equivalences. For suitably nice P-stratified topological spaces, the corresponding object of Strₚ is the exit-path ∞-category of MacPherson, Treumann, and Lurie. To prove this stratified homotopy hypothesis, we define combinatorial simplicial model structure on the category of simplicial sets over the nerve of 𝑃 whose underlying ∞-category is the ∞-category Strₚ. This model structure on P-stratified simplicial sets allows us to easily compare other theories of P-stratified spaces to ours and deduce that they all embed into ours. The second part (Chapters 7 to 9) explores a number of consequences of this stratified homotopy hypothesis, as well as related results on exit-path ∞-categories and constructible sheaves. This includes an overview of our joint work with Bariwck and Glasman on exit-path categories in algebraic geometry; this work uses as input the perspective on stratified spaces provided by our stratified homotopy hypothesis."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/139376"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["On the Homotopy Theory of Stratified Spaces"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:21:44Z"}