nLab
indexed functor

Contents

Idea

An indexed category is a 2-presheaf. An indexed functor is a morphism of 2-presheaves.

The “indexed”-terminology here is traditional in 1-topos theory and hence indexed functors are usually considered only between pseudofunctors (as opposed to more general 2-functors).

Definition

Let SS be a category. Let \mathbb{C} and 𝔻\mathbb{D} be SS-indexed categories, that is, pseudofunctors S opCatS^\mathrm{op} \to Cat.

Then an SS-indexed functor F:𝔻F:{\mathbb{C}}\to{\mathbb{D}} is a pseudonatural transformation F:𝔻F \colon \mathbb{C} \Rightarrow \mathbb{D}: it assigns to each object AA of SS a functor F A: A𝔻 AF^A:{\mathbb{C}}^A\to{\mathbb{D}}^A and to each morphism f:ABf:A\to B of SS a natural isomorphism 𝔻(f)F BF A(f)\mathbb{D}(f) \circ F^B \cong F^A \circ \mathbb{C}(f) that is coherent with respect to the structural isomorphisms of \mathbb{C} and 𝔻\mathbb{D} (see pseudonatural transformation for details).

References

Section B1 of

Revised on January 7, 2014 07:25:55 by Urs Schreiber (82.113.121.219)