Smooth and rough modules over self-induced algebras (Q2883473)

An algebra \(A\) in a (closed) monoidal category is self-induced if the multiplication map \(A\otimes A\to A\) induces an isomorphism \(A\otimes_A A\cong A\). Let \(A\) be a self-induced algebra and let \(X\) be a left \(A\)-module. The smoothening and roughening of \(X\) are defined by \(S_A(X):=A\otimes_A X\) and \(R_A(X):=\text{Hom}_A(A,X)\) respectively. These functors generalize some previous author's constructions for group representations on bornological vector spaces. In the paper under review, the author studies the basic properties of these functors.