We determine what appears to be the bare-bones categorical framework for Poincaré–Birkhoff–Witt (PBW)-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 PBW 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 PBW-type theorems. In particular, we prove a PBW-type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.
We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall-Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann-Holloway-Snashall-Solberg-Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.
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