From pre-Lie bialgebras to phase spaces of Lie algebras: a categorical correspondence

This paper establishes a categorical framework for phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical s-matrices, and relative Rota-Baxter operators by introducing the concept of coherent homomorphisms. Starting with endo pre-Lie algebras (pre-Lie algebras equipped with endomorphisms), we extend classical constructions to this enhanced setting, which leads to the notion of coherent endomorphisms for each class of structures. Through polarization, these endomorphisms naturally generalize to coherent homomorphisms, establishing well-defined categories of these algebraic objects. Furthermore, mappings between categories are elevated to functors or equivalences, formalizing interconnections among the constructions. Finally, exploiting the categorical correspondence between s-matrices and relative Rota-Baxter operators, we develop cohomology and deformation of s-matrices, thereby bridging algebraic structures with category-theoretic methods.