# nLab module over a monoidal functor

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

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

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

Last revised on May 29, 2022 at 18:56:17. See the history of this page for a list of all contributions to it.