Possible connections between whiskered categories and groupoids, many object Leibniz algebras, automorphism structures and local-to-global questions

We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered $R$-categories, thus answering questions of what might be `commutative versions’ of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theory here does not satisfy the Leibniz identity. We also discuss potential applications and extensions, for example to resolutions of monoids.

💡 Research Summary

The paper introduces a novel enrichment of ordinary categories and groupoids by attaching to every object a pair of endomorphisms, called “whiskers”. Formally, a whiskered category consists of a usual small category together with, for each object (A), two distinguished morphisms (\lambda_A) and (\rho_A) (the left and right whiskers) that act on any arrow whose source or target is (A). When the underlying category is a groupoid, the authors call the structure a whiskered groupoid.

The central technical contribution is a generalized commutator theory for whiskered groupoids. In a plain groupoid the commutator of two composable arrows (g) and (h) is (g^{-1}h^{-1}gh). In the whiskered setting the authors replace the identity maps in this expression by the whisker maps, defining
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