nLab indexed functor

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 $\mathcal{S}$ be a category. Let $\mathbb{C}$ and $\mathbb{D}$ be $\mathcal{S}$-indexed categories, that is, pseudofunctors $\mathcal{S}^\mathrm{op} \to \mathbf{Cat}$, then an $\mathcal{S}$-indexed functor $F:{\mathbb{C}}\to{\mathbb{D}}$ is a pseudonatural transformation $F \colon \mathbb{C} \Rightarrow \mathbb{D}$: it assigns to each object $A$ of $\mathcal{S}$ a functor $F_A:{\mathbb{C}}(A) \to {\mathbb{D}}(A)$ and to each morphism $f:A\to B$ of $\mathcal{S}$ a natural isomorphism $\mathbb{D}(f) \circ F_B \cong F_A \circ \mathbb{C}(f)$ that is coherent with respect to the structural isomorphisms of $\mathbb{C}$ and $\mathbb{D}$ (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 $\mathbb{C} : \mathcal{S}^{op} \to \mathbf{Cat}$ and $\mathbb{D} : \mathcal{T}^{op} \to \mathbf{Cat}$ be indexed categories. A morphism between them is a pair of a functor $\Phi : \mathcal{S} \to \mathcal{T}$ and a pseudonatural transformation $\Phi^\flat : \Phi^*\mathbb{C} \Rightarrow \mathbb{D}$, where $\Phi^* \mathbb{C} = \mathbb{C} \circ \Phi^{op}$ is reindexing of $\mathbb{C}$ along $\Phi$.

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

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

Related concepts

• indexed adjoint functor theorem

• indexed monoidal category

References

Section B1 of

• Peter Johnstone, Sketches of an Elephant