  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].
