Traces for factorization homology in dimension 1

David Ayala; John Francis

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arXiv · 2105.01143

Traces for factorization homology in dimension 1

David Ayala·John Francis

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Abstract

We construct a circle-invariant trace from the factorization homology of the circle $ {\sf trace} \colon \int^\alpha_{{\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 To\"en--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 Poincar\'e duality for 1-dimensional factorization homology.

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BibTeX RIS

David Ayala, John Francis. 2021-05-03. Traces for factorization homology in dimension 1. https://arxiv.org/abs/2105.01143

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