An indexed category is a 2-presheaf.
When doing category theory relative to a base topos (or other more general sort of category), the objects of 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 .” An -indexed category is a category together with, for every object , a notion of “-indexed family of objects of .”
Let be a category.
This appears for instance as (Johnstone, def. B1.2.1).
One may also call a prestack in categories over .
Traditionally one writes the image of an object under as and calls it the category of -indexed families of objects of . Similarly, one writes the image of a morphism as .
If has a terminal object we think of as the underlying ordinary category of the -indexed category . Part of the theory of indexed categories is about when and how to extend structures on to all of .
A morphism of -indexed categories is an indexed functor.
If has pullbacks, then its codomain fibration is an -indexed category denoted .
This assigns to an object the corresponding over-category
and to a morphism the functor that sends every to its pullback along .
This indexed category represents itself (or rather its codomain fibration) in the world of -indexed categories.
(change of base)
If is a functor and is a -indexed category, then we have an -indexed category defined by
Indexed category of a functor
Combining these previous examples we get
For a functor and a finitely complete category, there is the -indexed category given by
If the functor preserves pullbacks then this induces a morphism of -indexed categories.
Indexed category of a topos over a base topos
This situation frequently arises when and are toposes and is the inverse image part of a geometric morphism.
In this way, if is a topos, then to be thought of as a base topos, then any topos over (i.e. an object of the slice 2-category Topos) gives rise to a topos relative to , i.e. a “topos object” in the 2-category of -indexed categories, and this operation can be shown to be fully faithful.
See base topos for more on this.
Also, via this indexed category, exhibits as a 2-sheaf (see there) over , with respect to the canonical topology.
Indexed monoidal category
See also indexed monoidal category, indexed closed monoidal category and dependent linear type theory.
Extensions of adjunctions to indexed categories
be a pair of adjoint functors between finitely complete categories. Then extends to an -indexed functor
where is the self-indexing of from example 1 and is the base change indexing of from example 3.
By the general properties of adjunctions on overcategories (see there) we get for each an adjunction
Here is always a -indexed functor , and is if preserves pullbacks (by example 3). If so, we have an -indexed adjunction
This appears as (Johnstone, lemma B1.2.3).
Well-powered indexed categories
This appears as (Johnstone, example. B1.3.14).
Let be a pair of adjoint functors such that preserves pullbacks. Then the -indexed category is well powered if is.
This is (Johnstone, prop. B1.3.17).
Section B1.2 in