Holmstrom Algebra over a monoidal category

Let $C$ be a Monoidal category. A $C$-algebra structure on a category $D$ is a monoidal structure on $D$ together with a monoidal functor $i: C \to D$. A $C$-algebra functor is a monoidal functor “commuting with $i$ up to natural isomorphism”. The $C$-algebras form a 2-category, and there is a forgetful 2-functor to $C$-modules.

Example: Let $C$ be $R$-mod ($R$ a commutative ring), and consider a map of $R$-algs, $S \to T$. Get a $C$-algebra functor from $S-mod$ to $T-mod$ by tensoring with $T$ over the base ring $S$.

If $C$ is symmetric monoidal, then a symmetric $C$-algebra structure on a category $D$ is a symmetric monoidal structure on $D$ together with a symmetric monoidal functor $C \to D$. A symmetric $C$-algebra functor is a symmetric monoidal functor that is also a $C$-algebra functor. We can also define a central $C$-algebra structure.

Example: The category of modules over a commutative Hopf algebra (over a field) is a central algebra over the category of vector spaces. It is symmetric iff the Hopf algebra is cocommutative.

nLab page on Algebra over a monoidal category