nLab module category

Idea

By module category may be meant

1. a category equipped with an action of a monoidal category (more commonly called actegory),

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

3. a linear category equipped with an action of a tensor linear category (basically the first bullet point but in the $K\mathbf{Mod}$-enriched setting).

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.

Related concepts

• actegory

• 2-module, n-module

• bimodule category

• module

• module 2-category

• module over a monoidal functor

References

For instance

• G. Janelidze, G.M. Kelly, A note on actions of a monoidal category, Theory and Applications of Categories, Vol. 9, No. 4, 2001, pp. 61–91, pdf.

• Robert Gordon, A.J. Power, Enrichment through variation, J. Pure Appl. Algebra 120 (1997), 167-185.

• Chris Douglas, Chris Schommer-Pries, Noah Snyder, Dualizable tensor categories (arXiv:1312.7188)