About the contractibility of the walking coinductive equivalence

Abstract

We study the marked simplicial set obtained as the Roberts-Street nerve of the walking coinductive equivalence. We show that it is a non-contractible saturated complicial set for which all of its finite truncations are contractible. When regarding saturated complicial sets as a model for right $(\infty ,\infty )$-categories, it represents a concrete and explicit example of a right $(\infty ,\infty )$-category that is itself non-contractible, but whose reflection to a left $(\infty ,\infty )$-category is contractible.