Hochschild homology, lax codescent, and duplicial structure

TL;DR

This paper explores the structure of duplicial objects, generalizing cyclic objects, through a 2-categorical perspective on Hochschild homology, with applications to nerves of categories and bicategories.

Contribution

It provides a new conceptual framework for duplicial objects using decalage comonads and 2-categorical Hochschild homology, extending previous constructions.

Findings

Describes duplicial objects via decalage comonads

Provides a 2-categorical account of Hochschild homology

Analyzes duplicial structures on nerves of categories and bicategories

Abstract

We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Bohm and Stefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.