On the Homotopy Theory of Stratified Spaces
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...
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| Médium: | Diplomová práce |
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Massachusetts Institute of Technology
2022
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| On-line přístup: | https://hdl.handle.net/1721.1/139376 https://orcid.org/0000-0002-6662-2035 |
| _version_ | 1826204735492849664 |
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| author | Haine, Peter J. |
| author2 | Barwick, Clark |
| author_facet | Barwick, Clark Haine, Peter J. |
| author_sort | Haine, Peter J. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:59:51Z |
| format | Thesis |
| id | mit-1721.1/139376 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T12:59:51Z |
| publishDate | 2022 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1393762022-01-15T03:29:48Z On the Homotopy Theory of Stratified Spaces Haine, Peter J. Barwick, Clark Massachusetts Institute of Technology. Department of Mathematics 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. Ph.D. 2022-01-14T15:07:53Z 2022-01-14T15:07:53Z 2021-06 2021-05-25T12:46:58.338Z Thesis https://hdl.handle.net/1721.1/139376 https://orcid.org/0000-0002-6662-2035 In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Haine, Peter J. On the Homotopy Theory of Stratified Spaces |
| title | On the Homotopy Theory of Stratified Spaces |
| title_full | On the Homotopy Theory of Stratified Spaces |
| title_fullStr | On the Homotopy Theory of Stratified Spaces |
| title_full_unstemmed | On the Homotopy Theory of Stratified Spaces |
| title_short | On the Homotopy Theory of Stratified Spaces |
| title_sort | on the homotopy theory of stratified spaces |
| url | https://hdl.handle.net/1721.1/139376 https://orcid.org/0000-0002-6662-2035 |
| work_keys_str_mv | AT hainepeterj onthehomotopytheoryofstratifiedspaces |