Symmetries of the cyclic nerve

TL;DR

The work develops a comprehensive, quiver-based framework for Hochschild homology of $(\infty,1)$-categories, revealing a canonical ${\mathbb W^{\mathrm{op}}}$-module symmetry and a non-stable cyclotomic trace. It unifies cyclic, paracyclic, and epicyclic categories via a universal construction built from quivers, and shows how Hochschild homology naturally fits into factorization homology at level $n=1$, aligning with the AFR2 program. The authors construct a universal ambient category ${\boldsymbol{\mathcal{M}}}$ to encode endomorphisms and Hochschild objects, give explicit descriptions of the morphisms and excision sites, and formulate a combinatorial model for factorization homology that matches the geometric theory. The results provide a robust, coordinate-free way to study traces and cyclotomic phenomena across enriched $(\infty,1)$-categories, with a concrete bridge to topological field-theoretic constructions. Overall, the paper advances a unifying algebraic-combinatorial apparatus for higher-categorical Hochschild invariants and their diffeomorphism-invariant symmetries, with direct implications for K-theory traces and cyclotomic invariants.

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.

Figures (3)

• Figure 1: Just as one can present spectra by probing them either by all finite pointed spaces or merely by spheres, one can present $(\infty,1)$-categories by probing them either by all quivers or merely by nonempty linear quivers.
• Figure 2: A graph $\Gamma$ and its exit path category.
• Figure 3: An object $M$ with an excision site and a picture of $M_2$.

Theorems & Definitions (161)

• Theorem A: \ref{['t73']}\ref{['theorem A']}
• Theorem B: \ref{['t73']}\ref{['theorem B']}
• Theorem C: \ref{['t44']}
• Theorem D: \ref{['t2']}
• Remark 4.1
• Theorem E
• Remark 5.1
• Remark 5.2
• Definition 1.1.1
• Remark 1.1.2