Traces for factorization homology in dimension 1
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913606367444992 |
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| author | Ayala, David Francis, John |
| author_facet | Ayala, David Francis, John |
| contents | We construct a circle-invariant trace from the factorization homology of the circle $ {\sf trace} \colon \int^α_{{\mathbb S}^1} \\underline{\sf End}(V) \longrightarrow \uno $ associated to a dualizable object $V\in {\boldsymbol{\mathfrak X}}$ in a symmetric monoidal $\infty$-category. This proves a conjecture of Toën--Vezzosi on existence of circle-invariant traces. Underlying our construction is a calculation of the factorization homology over the circle of the walking adjunction in terms of the paracyclic category of Getzler--Jones: $ \int_{{\mathbb S}^1} {\sf Adj} ~\simeq~ {\bDelta_{\circlearrowleft}}^{\triangleleft\!\triangleright} ~. $ This calculation exhibits a form of Poincaré duality for 1-dimensional factorization homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_01143 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Traces for factorization homology in dimension 1 Ayala, David Francis, John Algebraic Topology Category Theory K-Theory and Homology Quantum Algebra We construct a circle-invariant trace from the factorization homology of the circle $ {\sf trace} \colon \int^α_{{\mathbb S}^1} \\underline{\sf End}(V) \longrightarrow \uno $ associated to a dualizable object $V\in {\boldsymbol{\mathfrak X}}$ in a symmetric monoidal $\infty$-category. This proves a conjecture of Toën--Vezzosi on existence of circle-invariant traces. Underlying our construction is a calculation of the factorization homology over the circle of the walking adjunction in terms of the paracyclic category of Getzler--Jones: $ \int_{{\mathbb S}^1} {\sf Adj} ~\simeq~ {\bDelta_{\circlearrowleft}}^{\triangleleft\!\triangleright} ~. $ This calculation exhibits a form of Poincaré duality for 1-dimensional factorization homology. |
| title | Traces for factorization homology in dimension 1 |
| topic | Algebraic Topology Category Theory K-Theory and Homology Quantum Algebra |
| url | https://arxiv.org/abs/2105.01143 |