Category enriched over Lie algebras Recently, I was wondering if the category $\mathbf{Lie_k}$ of Lie $k$-algebras is enriched over itself (as the category of abelian groups is). The answer is no, it is not difficult to see it. My question is : can you give me an easy example of a category enriched over $\mathbf{Lie_k}$ where the Lie bracket is not trivial ? Moreover, in additive categories the additive structure is unique, can we have an analogue with the bracket ?
Another (soft) question is : are these examples interesting in a sense ? Do we have litterature on it ?
 A: To enrich categories over $\mathbf{Lie}_k$, it is virtually necessary to enrich $\mathbf{Lie}_k$ over itself. Indeed, essentially all enriched category theory involves enrichment in a closed monoidal category $\mathcal{V}$, and since $\mathbf{Lie}_k$ is appropriately nice (technically, it is finitely presentable,) it suffices to give either the internal hom or the tensor product of the closed monoidal structure. If the internal hom is to respect the usual underlying set functor, then its unit must be the free Lie algebra on one generator, which determines the entire internal hom up to unique isomorphism, since every Lie algebra is a colimit of copies of the free Lie algebra on one generator. Concretely, such an internal hom must try to give the set of Lie algebra homomorphisms the "levelwise" Lie algebra structure, which, as you already observed, is not a Lie algebra at all. This is the generic situation with most such attempts: Lie algebras, like rings, commutative rings, groups, abelian groups, monoids, commutative monoids, magmas, pointed sets, etc, etc, are models of an algebraic theory, and the "pointwise" self-enrichment only exists when the algebraic theory is commutative, as for abelian groups, commutative monoids, and pointed sets, but not the other examples. That's why you've never heard, for instance, of a category enriched in $k$-algebras.
