We determine what appears to be the bare-bones categorical framework for Poincare-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a Poincare-Birkhoff-Witt property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new Poincare-Birkhoff-Witt type theorems. In particular, we prove a PBW type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.

中文翻译：

---

内函子和 Poincaré–Birkhoff–Witt 定理

我们确定了关于各种代数结构的通用包络代数的 Poincare-Birkhoff-Witt 类型定理的基本分类框架。我们的语言是内函子的语言；我们确定 monad 的自然转换只有在该转换使其 codomain 成为其域上的自由权利模块时才具有 Poincare-Birkhoff-Witt 性质。我们以许多应用程序结束，以展示这种统一方法如何证明各种新旧 Poincare-Birkhoff-Witt 类型定理。特别是，我们证明了 pre-Lie 代数的普遍包络树状代数的 PBW 类型结果，回答了 Loday 的一个问题。