Partial Linearity and Partial Natural Mal’tsevness

We introduced in [7] a notion of Mal’tsevness relative to a specific class Σ of split epimorphisms. We investigate here the induced relative notion of natural Mal’tsevness, with a special attention to the example of quandles. Introduction In [7] we introduced a notion of Mal’tsevness which is only relative to a class Σ of split epimorphisms (stable under pullback and containing the isomorphisms), and we investigated what is remaining of the properties of the global Mal’tsev context (A. Carboni and all [11],[12]), after a first work about partial pointed protomodularity [10]. The Mal’tsev context contains, in particular, the naturally Mal’tsev one introduced by P.T. Johnstone [15] which corresponds to the “additive heart” of the theory. So, we shall investigate here what is remaining of the properties of the global naturally Mal’tsev context inside the relative frame. The generic example for the partial Mal’tsev context in [8] was the category of quandles, an algebraic structure independently introduced in [16] and [18] for Knot theorists, since it formalized the Reidemeister moves on oriented link diagrams, see also [13]; so here will be a special attention to the notion of autonomous quandle which retains the partial naturally Mal’tsev part of the theory. In [6], the author specified that the non-pointed additive context was actually structured by a subtle hierarchy of notions. It is not unexpected that, in the relative context, the previous subtleties grow up in complexity: for instance, there will appear examples of Mal’tsev (or Σ-Mal’tsev) categories which become naturally Mal’tsev for a certain subclass Σ′. This gives rise to the beginning of a kind of cartography for the linear and additive parts in Categorical Algebra. Notice that our notion of partial Mal’tsevness is different from the relative Mal’tsevness studied in [14]. This article is organized along the following lines: Section 1 is devoted to some recalls and to the definition of the partial naturally Mal’tsev context while Section 2 is devoted to what remains of the results of from [15] and [4] in this relative context. Section 3 investigates the particular case of the point-congruous Received by the editors 2015-06-26 and, in final form, 2016-05-20. Transmitted by Clemens Berger. Published on 2016-05-28. 2010 Mathematics Subject Classification: 18C99, 18A10, 08B05, 57M27.