nLab module over a monoidal functor

Idea

To the extent that a monoidal functor is analogous to a monoid, a module over a monoidal functor is analogous to a module over (hence an action of) that monoid.

Definition

Definition

Let

1. $\mathcal{C},\mathcal{D}$ be monoidal categories, hence equipped with tensor product functors $\otimes_{\mathcal{C}}\colon \mathcal{C} \times \mathcal{C}\to \mathcal{C}$ and $\otimes_{\mathcal{D}}\colon \mathcal{D} \times \mathcal{D}\to \mathcal{D}$;

2. $\mathcal{N}$ be a left module category over $\mathcal{C}$, hence equipped with a compatible action functor $\rho \colon \mathcal{C}\times\mathcal{N} \to \mathcal{N}$;

3. $F \colon \mathcal{C}\to \mathcal{D}$ a lax monoidal functor.

Then a left module over $F$ is

1. a functor $N \colon \mathcal{N} \longrightarrow \mathcal{D}$

2. a natural transformation

   $\alpha \colon F(-) \otimes_{\mathcal{D}} N(-) \longrightarrow N(\rho(-,-))$

satisfying the evident categorification of the action-property. Analogously for right modules and bimodules. (e.g. Yetter 01, def. 39).

Example

• For $(\mathcal{C},\otimes)$ a symmetric monoidal category and the functor category $([\mathcal{C},Set],\otimes_{Day})$ equipped with the induced Day convolution product, then a monoid object with respect to $\otimes_{Day}$ is equivalently a lax monoidal functor from $\mathcal{C}$ to itself, and a module object over that monoid is equivalently a module over that functor in the above sense. See there for more.

References

• David Yetter, Abelian categories of modules over a (Lax) monoidal functor (arXiv:math/0112307)