Simplicial Endomorphisms

TL;DR

This paper explores the monoids of simplicial endomorphisms, linking them to Temperley-Lieb algebras and adjoint functor situations, providing a new presentation and completeness proof.

Contribution

It offers a new, self-contained presentation and completeness proof for the monoids of simplicial endomorphisms, expanding understanding of their algebraic structure.

Findings

Established the connection between simplicial endomorphisms and Temperley-Lieb algebras.

Provided a new presentation with a completeness proof.

Surveyed and reworked previous results on these monoids.

Abstract

The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphisms are linked to arbitrary adjoint situations. This link is established through diagrams of the kind found in Temperley-Lieb algebras. Results about these matters, which were previously prefigured up to a point, are here surveyed and reworked. A presentation of monoids of simplicial endomorphisms by generators and relations has been given a long time ago. Here a closely related presentation is given, with completeness proved in a new and self-contained manner.