Simplicial Endomorphisms

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.

Key Words:

• Adjunction
• Endomorphisms
• Monads
• Presentation by generators and relations
• Simplicial category
• Temperley–Lieb algebras
• Triples

2000 Mathematics Subject Classification:

• 55U10
• 20M50
• 57M99
• 18G30
• 18C15
• 18A40

ACKNOWLEDGMENTS

I would like to thank Zoran Petrić for reading this article, and an anonymous referee for informing me about Aĭzenštat's article, about which I didn't know when five years ago I wrote the first version (available at: http://arXiv.org/math.GT/0301302). I would like to thank also Vitor Fernandes, who was extremely kind to send me a copy of Aĭzenštat 's article. The writing of the present article was financed by the Ministry of Science of Serbia (Grant 144013).

Notes

Communicated by V. Gould.