An indexed category is a 2-presheaf.
When doing category theory relative to a base topos $S$ (or other more general sort of category), the objects of $S$ are thought of as replacements for sets. Since often in category theory we need to speak of “a set-indexed family of objects” of some category, we need a corresponding notion in “category theory over $S$.” An $S$-indexed category is a category $\mathbb{C}$ together with, for every object $X\in S$, a notion of “$X$-indexed family of objects of $\mathbb{C}$.”
Let $S$ be a category.
An $S$-indexed category $C$ is a pseudofunctor
from the opposite category of $S$ to the 2-category Cat of categories.
Under the Grothendieck construction equivalence this is equivalently a fibered category
over $S$.
Similarly, an $S$-indexed functor $\mathbb{C} \to \mathbb{D}$ is a pseudonatural transformation of pseudofunctors, and an indexed natural transformation is a modification.
This defines the 2-category $S IndCat \coloneqq [S^{op}, Cat]$ of $S$-indexed categories.
This appears for instance as (Johnstone, def. B1.2.1).
One may also call $\mathbb{C}$ a prestack in categories over $S$.
Traditionally one writes the image of an object $X \in S$ under $\mathbb{C}$ as $\mathbb{C}^X$ and calls it the category of $X$-indexed families of objects of $\mathbb{C}$. Similarly, one writes the image of a morphism $u\colon X\to Y$ as $u^*\colon \mathbb{C}^Y\to \mathbb{C}^X$.
If $\mathcal{S}$ has a terminal object $*$ we think of $\mathbb{C}^*$ as the underlying ordinary category of the $\mathcal{S}$-indexed category $\mathbb{C}$. Part of the theory of indexed categories is about when and how to extend structures on $\mathbb{C}^*$ to all of $\mathbb{C}$.
A morphism of $S$-indexed categories is an indexed functor.
(canonical self-indexing)
If $\mathcal{S}$ has pullbacks, then its codomain fibration is an $\mathcal{S}$-indexed category denoted $\mathbb{S}$.
This assigns to an object $I$ the corresponding over-category
and to a morphism $f : I \to J$ the functor $f^*$ that sends every $s \to I$ to its pullback $f^*$ along $f$.
This indexed category represents $\mathcal{S}$ itself (or rather its codomain fibration) in the world of $\mathcal{S}$-indexed categories.
(change of base)
If $F\colon \mathcal{S}\to \mathcal{T}$ is a functor and $\mathbb{C}$ is a $\mathcal{T}$-indexed category, then we have an $\mathcal{S}$-indexed category $F^*\mathbb{C}$ defined by
$(F^*\mathbb{C})^I = \mathbb{C}^{F(I)}$ for every object $I \in \mathcal{S}$;
and $x^* = F(x)^*$ for every morphism $x : I \to J$ in $\mathcal{S}$.
Combining these previous examples we get
For $F : \mathcal{S} \to \mathcal{C}$ a functor and $\mathcal{C}$ a finitely complete category, there is the $\mathcal{S}$-indexed category $F^* \mathbb{C}$ given by
If the functor $F$ preserves pullbacks then this induces a morphism $\mathbb{S} \to F^* \mathbb{C}$ of $\mathcal{S}$-indexed categories.
This situation frequently arises when $\mathcal{S}$ and $\mathcal{C}$ are toposes and $F \coloneqq f^*$ is the inverse image part of a geometric morphism.
In this way, if $\mathcal{S}$ is a topos, then to be thought of as a base topos, then any topos over $\mathcal{S}$ (i.e. an object of the slice 2-category Topos$/S$) gives rise to a topos relative to $\mathcal{S}$, i.e. a “topos object” in the 2-category of $\mathcal{S}$-indexed categories, and this operation can be shown to be fully faithful.
See base topos for more on this.
Also, via this indexed category, $f$ exhibits $\mathcal{C}$ as a 2-sheaf (see there) over $\mathcal{C}$, with respect to the canonical topology.
See also indexed monoidal category, indexed closed monoidal category and dependent linear type theory.
Let
be a pair of adjoint functors between finitely complete categories. Then $R$ extends to an $\mathcal{S}$-indexed functor
where $\mathbb{S}$ is the self-indexing of $\mathcal{S}$ from example 1 and $\mathbb{C}$ is the base change indexing of $\mathcal{C}$ from example 3.
By the general properties of adjunctions on overcategories (see there) we get for each $I \in \mathcal{S}$ an adjunction
Here $\mathbb{R} : I \mapsto R/I$ is always a $\mathcal{S}$-indexed functor $\mathbb{C} \to \mathbb{S}$, and $\mathbb{L} : I \mapsto L/I$ is if $L$ preserves pullbacks (by example 3). If so, we have an $\mathcal{S}$-indexed adjunction
This appears as (Johnstone, lemma B1.2.3).
(…)
An $\mathcal{S}$-indexed category $\mathbb{C}$ is called well-powered if the fibered category $\tilde \mathbb{C} \to \mathcal{S}$ corresponding to it under the Grothendieck construction has the property that the forgetful functor
has a right adjoint, where $Q(2,\tilde \mathbb{C})$ is the full subcategory of $Rect(2, \tilde \mathbb{C})$ on vertical monomorphisms.
This appears as (Johnstone, example. B1.3.14).
Let $(L \dashv R) : \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{S}$ be a pair of adjoint functors such that $L$ preserves pullbacks. Then the $\mathcal{S}$-indexed category $\mathbb{C}$ is well powered if $\mathbb{S}$ is.
hm
This is (Johnstone, prop. B1.3.17).
Section B1.2 in