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I had some thoughts last month or two about the small nonassociative categories, that is the category over an associative operad (in Set). I know that some people around looked at formal category theory in various framework (especially Mike). Are there any frameworks for nontrivial nonassociative category theory in the sense that a composition of natural transformation is defined and that some weak form of Yoneda lemma is true. I know of some special cases by imitation of some ideas of quasigroup theory; I can defined certain related associative categories (left, right, middle kernel), then certain story about translations in the quasigroup with multiple objects case and dreaming of certain crossed products to combine the two, as well as using some things about isotopy (algebra) in this context. But maybe something can be said from the point of view of formal category theory ? There are nontrivial motivations to the story, e.g. from nonassociative Lie theory which is quite a large subject, starting with the classical work of Mal’cev on analytic loops through the recent work of Sabinin et al. on Sabinin algebras…
Usual category is a monoid with several objects, nonassociative category is a unital magma with several objects. In motivating examples, one usually deals with quasigroups and loops in the sense of algebra. Mal’cev had Lie theory for a class of analytic loops and there are many modern extensions of that theory.
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