nLab indexed functor

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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

Definition

Let 𝒮\mathcal{S} be a category. Let \mathbb{C} and 𝔻\mathbb{D} be 𝒮\mathcal{S}-indexed categories, that is, pseudofunctors 𝒮 opCat\mathcal{S}^\mathrm{op} \to \mathbf{Cat}, then an 𝒮\mathcal{S}-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 𝒮\mathcal{S} a functor F A:(A)𝔻(A)F_A:{\mathbb{C}}(A) \to {\mathbb{D}}(A) and to each morphism f:ABf:A\to B of 𝒮\mathcal{S} 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).

One can also consider indexed functors between categories indexed by different bases, i.e. morphisms of indexed categories.

Definition

Let :𝒮 opCat\mathbb{C} : \mathcal{S}^{op} \to \mathbf{Cat} and 𝔻:𝒯 opCat\mathbb{D} : \mathcal{T}^{op} \to \mathbf{Cat} be indexed categories. A morphism between them is a pair of a functor Φ:𝒮𝒯\Phi : \mathcal{S} \to \mathcal{T} and a pseudonatural transformation Φ :Φ *𝔻\Phi^\flat : \Phi^*\mathbb{C} \Rightarrow \mathbb{D}, where Φ *=Φ op\Phi^* \mathbb{C} = \mathbb{C} \circ \Phi^{op} is reindexing of \mathbb{C} along Φ\Phi.

In other words, it’s a lax commutative triangle over Cat\mathbf{Cat}.

Remark. These morphisms correspond to morphisms of fibrations through the Grothendieck construction.

References

Section B1 of

Last revised on January 9, 2023 at 19:13:39. See the history of this page for a list of all contributions to it.