Miscellaneous structures associated to algebras over operads

In this chapter, we recall the definition of miscellaneous structures associated to algebras over operads: enveloping operads, which model comma categories of algebras over operads; enveloping algebras, which are associative algebras formed from the structure of enveloping operads; representations, which are nothing but modules over enveloping algebras; and modules of Kähler differentials, which occur in the definition of the homology of algebras over operads. In §10, we address enveloping objects and modules of Kähler differentials as examples of functors associated to right modules over operads. For the moment, we only give the definition of these structures – enveloping operads in §4.1, representations in §4.2, enveloping algebras in §4.3, Kähler differentials in §4.4 – and we study applications of these operadic constructions for the usual commutative, associative and Lie operads.