Left-Garside categories, self-distributivity, and braids

TL;DR

This paper introduces the concepts of left-Garside categories and locally left-Garside monoids, linking self-distributivity laws to braid theory and proposing a new approach to the Embedding Conjecture.

Contribution

It develops the theory of left-Garside categories and connects self-distributivity with braid categories, offering a new framework for braid-related conjectures.

Findings

A category associated with self-distributivity is a left-Garside category.

This category projects onto the standard Garside category of braids.

The approach provides a program for the Embedding Conjecture.

Abstract

In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This approach leads to a realistic program for establishing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhauser (2000), Chap. IX].