Cyclic duality for slice and orbit 2-categories

TL;DR

This paper extends the concept of cyclic duality from the paracyclic category to a broader class of homotopy categories of (2,1)-categories, linking it to geometric and topological structures.

Contribution

It introduces a new framework for duality in homotopy categories of (2,1)-categories, generalizing orbit categories and connecting to geometric examples like submanifolds and circles.

Findings

Generalizes cyclic duality to homotopy categories of (2,1)-categories

Links duality to geometric structures such as submanifolds and $S^1$

Provides a visual interpretation of existing duality results

Abstract

The self-duality of the paracyclic category is extended to a certain class of homotopy categories of (2,1)-categories. These generalise the orbit category of a group and are associated to certain self-dual preorders equipped with a presheaf of groups and a cosieve. Slice 2-categories of equidimensional submanifolds of a compact manifold without boundary form a particular case, and for $S^1$, one recovers cyclic duality. This provides in particular a visualisation of the results of B\"ohm and \c{S}tefan on the topic.