Left-Garside categories, self-distributivity, and braids

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, Birkhaüser (2000), Chap. IX]. The notion of a Garside monoid emerged at the end of the 1990’s [24, 19] as a development of Garside’s theory of braids [32], and it led to many developments [2, 3, 5, 6, 7, 8, 13, 14, 15, 31, 33, 34, 41, 42, 45, 46, 47, ...]. More recently, Bessis [4], Digne–Michel [27], and Krammer [38] introduced the notion of a Garside category as a further extension, and they used it to capture new, nontrivial examples and improve our understanding of their algebraic structure. The concept of a Garside category is also implicit in [25] and [35], and maybe in the many diagrams of [18]. Here we shall describe and investigate a new example of (left)-Garside category, namely a certain category LD+ associated with the left self-distributivity law (LD) x(yz) = (xy)(xz). The interest in this law originated in the discovery of several nontrivial structures that obey it, in particular in set theory [16, 40] and in low-dimensional topology [36, 30, 44]. A rather extensive theory was developed in the decade 1985-95 [18]. Investigating self-distributivity in the light of Garside categories seems to be a good idea. It turns out that a large part of the theory developed so far can be summarized into one single statement, namely The category LD+ is a left-Garside category, stated as the first part of Theorem 6.1. The interest of the approach should be at least triple. First, it gives an opportunity to restate a number of previously unrelated properties in a new language that is natural and should make them more easily understandable—this is probably not useless. In particular, the connection between self-distributivity and braids is now expressed in the simple statement: There exists a right-lcm preserving surjective functor of LD+ to the Garside category of positive braids, (second part of Theorem 6.1). This result allows one to recover most of the usual algebraic properties of braids as a direct application of the properties of LD+: 1991 Mathematics Subject Classification. 18B40, 20N02, 20F36.