The broadly applied notions of Lie bialgebras, Manin triples, classical r-matrices and 𝒪 {\mathcal{O}} -operators of Lie algebras owe their importance to the close relationships among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, this paper introduces, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras.

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Courant 代数胚与李 2-代数之间的关系; CLWX (CLWX 是 "Courant-刘张炬-Weinstein-徐平" 的缩 写) 2-代数胚与李 3-代数之间的关系; (非交换) omni-(n)-李代数与李 2-代数之间的关系; 多辛几何 以及高阶 Courant 代数胚的迷向对合子丛与李 n-代数之间的关系.