nLab
module category

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

Written out, this amounts to:

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

Further expanding this definition, we have the following data:

  • A category 𝒞\mathcal{C}
  • A functor :𝒞×- \triangleright -\colon \mathcal{C} \times \mathcal{M} \to \mathcal{M}
  • 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.

References

For instance

Revised on August 29, 2017 09:57:38 by Manuel Baerenz (131.130.16.226)