# nLab module category

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

By “module category” may be meant

1. a category equipped with an action of a monoidal category.

2. a category of modules of a monoid (e.g. an associative algebra).

Here we consider the first sense. For the second, see at category of modules.

The collection of module categories over a monoidal category forms a 2-category of module categories.

## Definition

Let $\mathcal{M}$ be a monoidal category and $B\mathcal{M}$ its delooping as a bicategory. A (left) module category is then simply a 2-functor $B\mathcal{M} \to Cat$.

Written out, this amounts to:

• A category $\mathcal{C}$
• A monoidal functor $\mathcal{M} \to End(\mathcal{C})$

Further expanding this definition, we have the following data:

• A category $\mathcal{C}$
• A functor $- \triangleright -\colon \mathcal{M} \times \mathcal{C} \to \mathcal{C}$
• A natural isomorphism $\alpha_{A,B,X}\colon A \triangleright (B \triangleright X) \to (A \otimes B) \triangleright X$ satisfying a pentagon axiom involving the associator of $\mathcal{M}$
• A natural isomorphism $\lambda_X\colon I \triangleright X \to X$, where $I$ is the monoidal unit of $\mathcal{M}$, compatible with the left unitor of $\mathcal{M}$.

An obvious example is given by a monoidal category, which has an action on itself by left multiplication.

For instance