nLab
module over a monoidal functor

Contents

Context

Monoidal categories

monoidal categories

With symmetry

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 (i.e.: and action of) that monoid.

Definition

Definition

Let

  1. 𝒞,𝒟\mathcal{C},\mathcal{D} be monoidal categories, hence equipped with tensor product functors 𝒞:𝒞×𝒞𝒞\otimes_{\mathcal{C}}\colon \mathcal{C} \times \mathcal{C}\to \mathcal{C} and 𝒟:𝒟×𝒟𝒟\otimes_{\mathcal{D}}\colon \mathcal{D} \times \mathcal{D}\to \mathcal{D};

  2. 𝒩\mathcal{N} be a left module category over 𝒞\mathcal{C}, hence equipped with a compatible action functor ρ:𝒞×𝒩𝒩\rho \colon \mathcal{C}\times\mathcal{N} \to \mathcal{N};

  3. F:𝒞𝒟F \colon \mathcal{C}\to \mathcal{D} a lax monoidal functor.

Then a left module over FF is

  1. a functor N:𝒩𝒟N \colon \mathcal{N} \longrightarrow \mathcal{D}

  2. a natural transformation

    α:F() 𝒟N()N(ρ(,))\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

References

Last revised on June 14, 2016 at 02:39:53. See the history of this page for a list of all contributions to it.