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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2301.02063 |
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| _version_ | 1866915595572740096 |
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| author | Tetik, Ödül |
| author_facet | Tetik, Ödül |
| contents | Conically smooth spaces (CSSs), introduced by Ayala, Francis and Tanaka, constitute a large class of singular spaces including Whitney-stratified spaces. We reduce the stratified topology of CSSs over depth-$1$ posets to the ordinary topology of linked smooth manifolds, i.e., spans $M\xleftarrowπ L\xrightarrowιN$ of smooth manifolds where $π$ is a fibre bundle and $ι$ is a closed embedding. To that end, we introduce explicit exit path quasi-categories (EPCs) for linked spaces and prove that this induces a fully faithful functor from a quasi-category of linked spaces to the quasi-category of all quasi-categories whose essential image includes the Lurie--MacPherson EPCs of CSSs over depth-$1$ posets. We use linked smooth manifolds to resolve various weaker versions of a conjecture of Ayala--Francis--Rozenblyum in the negative by exhibiting quasi-categories with conservative functors to $\{0<1\}$ satisfying certain finiteness conditions but which are not equivalent to EPCs of CSSs. In a sequel, we develop a tangential theory for linked smooth manifolds and reduce the classification conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02063 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linked spaces and exit paths Tetik, Ödül Algebraic Topology 57N80, 18N60 (Primary) 32S60 (Secondary) Conically smooth spaces (CSSs), introduced by Ayala, Francis and Tanaka, constitute a large class of singular spaces including Whitney-stratified spaces. We reduce the stratified topology of CSSs over depth-$1$ posets to the ordinary topology of linked smooth manifolds, i.e., spans $M\xleftarrowπ L\xrightarrowιN$ of smooth manifolds where $π$ is a fibre bundle and $ι$ is a closed embedding. To that end, we introduce explicit exit path quasi-categories (EPCs) for linked spaces and prove that this induces a fully faithful functor from a quasi-category of linked spaces to the quasi-category of all quasi-categories whose essential image includes the Lurie--MacPherson EPCs of CSSs over depth-$1$ posets. We use linked smooth manifolds to resolve various weaker versions of a conjecture of Ayala--Francis--Rozenblyum in the negative by exhibiting quasi-categories with conservative functors to $\{0<1\}$ satisfying certain finiteness conditions but which are not equivalent to EPCs of CSSs. In a sequel, we develop a tangential theory for linked smooth manifolds and reduce the classification conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds. |
| title | Linked spaces and exit paths |
| topic | Algebraic Topology 57N80, 18N60 (Primary) 32S60 (Secondary) |
| url | https://arxiv.org/abs/2301.02063 |