nLab module category




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 KModK\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.


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

Written out, this amounts to:

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

Further expanding this definition, we have the following data:

  • A category 𝒞\mathcal{C}
  • A functor :×𝒞𝒞- \triangleright -\colon \mathcal{M} \times \mathcal{C} \to \mathcal{C}
  • A natural isomorphism α A,B,X:A(BX)(AB)X\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 λ X:IXX\lambda_X\colon I \triangleright X \to X, where II 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

Last revised on September 27, 2023 at 13:25:53. See the history of this page for a list of all contributions to it.