Smoothness in Codifferential Categories - O'Neill, Keith (Université d'Ottawa / University of Ottawa, 2017)
The Hochschild-Kostant-Rosenberg theorem, which relates the Hochschild homology of an algebra to its modules of differential $n$-forms, can be considered a benchmark for smoothness of an algebra. This notion is used here in the search for a conception of smoothness formulated in the context of codifferential categories, both commutative and noncommutative. Since it is desirable to adapt such a conception to the noncommutative context, the theory of this domain is developed considerably; a significant result in this direction establishes a connection between noncommutative codifferential categories and commutative ones. This investigation necessitates, then, both the formulation of a notion of smoothness for $T$-algebras in codifferential categories and an adaptation of the Hochschild-Kostant-Rosenberg theorem to a wide variety of contexts which includes noncommutative ones. The former consideration fosters both the notion of a smooth monad, and a formulation of Andr\'e-Quillen homology in codifferential categories; the latter engenders a highly adaptable version of the Hochschild-Kostant-Rosenberg theorem. Specifically, it is shown that for any algebra modality there exists a corresponding Hochschild-Kostant-Rosenberg theorem. This includes a version of the theorem for the free associative algebra monad, the conclusion of which is satisfied by noncommutative smooth algebras.
Keywords: Smoothness; Category Theory