Symmetries of the cyclic nerve

David Ayala; Aaron Mazel-Gee; Nick Rozenblyum

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arXiv · 2405.03897

Symmetries of the cyclic nerve

David Ayala·Aaron Mazel-Gee·Nick Rozenblyum

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Abstract

We undertake a systematic study of the Hochschild homology, i.e. (the geometric realization of) the cyclic nerve, of $(\infty,1)$-categories (and more generally of category-objects in an $\infty$-category), as a version of factorization homology. In order to do this, we codify $(\infty,1)$-categories in terms of quiver representations in them. By examining a universal instance of such Hochschild homology, we explicitly identify its natural symmetries, and construct a non-stable version of the cyclotomic trace map. Along the way we give a unified account of the cyclic, paracyclic, and epicyclic categories. We also prove that this gives a combinatorial description of the $n=1$ case of factorization homology as presented in [AFR18], which parametrizes $(\infty,1)$-categories by solidly 1-framed stratified spaces.

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David Ayala, Aaron Mazel-Gee, Nick Rozenblyum. 2024-05-06. Symmetries of the cyclic nerve. https://arxiv.org/abs/2405.03897

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