The subject of the article is the interpretation of the Leibniz homology of Lie algebras in terms of functor homology.
Leibniz algebras are a generalization of Lie algebras, being defined as vector spaces with a (not necessarily skewsymmetric) bracket which is a derivation of itself. This class of algebraic objects comes with its own homology theory, invented by J.-L. Loday, where the essential change with respect to Lie algebra homology is that the exterior tensor products are replaced by ordinary tensor products.
Functor homology is the study of Tor-functors in the abelian category of functors from some small category to an abelian category. In the present context, the abelian category is the category of vector spaces \({\mathbb K}-\mathrm{Mod}\) and the functors are moreover enriched in \({\mathbb K}-\mathrm{Mod}\). Loday, Pirashvili and Richter remarked that several standard homology theories (like Hochschild and cyclic homology) can be expressed in terms of functor homology. For example, Hochschild homology for commutative algebras is the Tor-functor \(\mathrm{Tor}_*^{\Gamma}(t,{\mathcal L}(A,M))\) in the category of \(\Gamma\)-modules between the coefficient functor \(t\) and the Loday functor \({\mathcal L}(A,M)\) which associates to a commutative associative algebra \(A\) and an \(A\)-module \(M\) the \(\Gamma\)-module \([n]\mapsto M\otimes A^{\otimes n}\). In this way, the combinatorial structure of Hochschild homology is governed by the small category \(\Gamma\) of finite pointed sets.
One difficulty in expressing Lie- or Leibniz homology in terms of functor homology is that expressing the Jacobi- or selfderivation property requires an additive structure. This is why the authors consider categories of functors enriched in \({\mathbb K}-\mathrm{Mod}\).
In independent work, Fresse expressed operad homology as functor homology for symmetric operads. Hoffbeck-Vespa consider instead shuffle operads, leading them to the interpretation of Leibniz homology of Lie algebras in terms of functor homology. As Leibniz algebras cannot be described by a non-symmetric operad, Hoffbeck-Vespa are naturally driven to consider Lie algebras. It would be interesting to better understand how Hoffbeck-Vespa’s result is related to Fresse’ results for the operads Lie and Leib.
The proof of the main theorem relies on the usual axiomatic description of a Tor-functor. The main point is to show that the Tor-functor which Hoffbeck-Vespa define vanishes on projective generators. This is shown combinatorially, using bases of the Lie operad and of the explicit projective generators. These bases then induce a filtration on the Leibniz homology complex and the proof concludes with a spectral sequence argument.