nLab Grothendieck construction for monoidal categories

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

Category theory

Contents

Idea

The Grothendieck construction gives a way of gluing together the constituent categories F(x)F(x) of an indexed category F:X opCatF \colon X^{op} \to \mathsf{Cat} to get a category F\int F which admits an obvious fibration over the base FX\int F \to X. This gives an equivalence between the 2-category of indexed categories and the 2-category of fibrations.

Two ways monoidal structures can join this story is on the total category F\int F, or on the fibres F(x)F(x). The 2-equivalence then lifts to two monoidal variants, one where the fibres are equipped with a monoidal structure, and one where the total category is equipped with a monoidal structure. Under certain conditions on the base category, these two settings are equivalent to each other as well. That is to say, under the right conditions, one can glue together the monoidal structures on the fibres to get a monoidal structure on the total category.

References

Last revised on May 29, 2022 at 14:33:59. See the history of this page for a list of all contributions to it.