Symmetries of the cyclic nerve

TL;DR

This paper systematically studies the Hochschild homology of $( abla,1)$-categories, revealing their symmetries and constructing a non-stable cyclotomic trace map, unifying cyclic, paracyclic, and epicyclic categories.

Contribution

It provides a detailed analysis of Hochschild homology for $( abla,1)$-categories, identifying symmetries and offering a unified framework for cyclic categories.

Findings

Explicit identification of symmetries in Hochschild homology

Construction of a non-stable cyclotomic trace map

Unified account of cyclic, paracyclic, and epicyclic categories

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.