On a simplicial monoid whose underlying simplicial set is not a quasi-category

TL;DR

This paper constructs a simplicial monoid whose underlying simplicial set is not a quasi-category, challenging assumptions about the relationship between simplicial monoids and quasi-categories.

Contribution

It provides a counterexample showing that not all simplicial monoids have underlying simplicial sets that are quasi-categories.

Findings

Counterexample of a simplicial monoid with non-quasi-category underlying set

Clarifies the distinction between simplicial monoids and quasi-categories

Highlights limitations in the analogy between groups and monoids

Abstract

It is well known that the underlying simplicial set of any simplicial group is a Kan complex. Roughly speaking, Kan complex is an infinite-dimensional analogue of groupoid, and the relation between groupoids and categories resembles that between groups and monoids. Thus one may ask if the underlying simplicial set of each simplicial monoid is a quasi-category. In this short note, we construct a simplicial monoid whose underlying simplicial set is not a quasi-category.