Symmetries of the cyclic nerve

We undertake a systematic study of the Hochschild homology, i. e. (the geometric realization of) the cyclic nerve, of (, 1) -categories (and more generally of category-objects in an -category), as a version of factorization homology. In order to do this, we codify (, 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 (, 1) -categories by solidly 1-framed stratified spaces.