indexed category



An indexed category is a 2-presheaf.

When doing category theory relative to a base topos SS (or other more general sort of category), the objects of SS 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 SS.” An SS-indexed category is a category \mathbb{C} together with, for every object XSX\in S, a notion of “XX-indexed family of objects of \mathbb{C}.”


Let SS be a category.


An SS-indexed category CC is a pseudofunctor

:S opCat \mathbb{C} : S^{op}\to Cat

from the opposite category of SS to the 2-category Cat of categories.

Under the Grothendieck construction equivalence this is equivalently a fibered category

˜ S \array{ \tilde \mathbb{C} \\ \downarrow \\ S }

over SS.

Similarly, an SS-indexed functor 𝔻\mathbb{C} \to \mathbb{D} is a pseudonatural transformation of pseudofunctors, and an indexed natural transformation is a modification.

This defines the 2-category SIndCat[S op,Cat]S IndCat \coloneqq [S^{op}, Cat] of SS-indexed categories.

This appears for instance as (Johnstone, def. B1.2.1).

One may also call \mathbb{C} a prestack in categories over SS.

Traditionally one writes the image of an object XSX \in S under \mathbb{C} as X\mathbb{C}^X and calls it the category of XX-indexed families of objects of \mathbb{C}. Similarly, one writes the image of a morphism u:XYu\colon X\to Y as u *: Y Xu^*\colon \mathbb{C}^Y\to \mathbb{C}^X.

If 𝒮\mathcal{S} has a terminal object ** we think of *\mathbb{C}^* as the underlying ordinary category of the 𝒮\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 SS-indexed categories is an indexed functor.


Self indexing


(canonical self-indexing)

If 𝒮\mathcal{S} has pullbacks, then its codomain fibration is an 𝒮\mathcal{S}-indexed category denoted 𝕊\mathbb{S}.

This assigns to an object II the corresponding over-category

𝕊 I𝒮/I \mathbb{S}^I \coloneqq \mathcal{S}/I

and to a morphism f:IJf : I \to J the functor f *f^* that sends every sIs \to I to its pullback f *f^* along ff.

This indexed category represents 𝒮\mathcal{S} itself (or rather its codomain fibration) in the world of 𝒮\mathcal{S}-indexed categories.

Base change


(change of base)

If F:𝒮𝒯F\colon \mathcal{S}\to \mathcal{T} is a functor and \mathbb{C} is a 𝒯\mathcal{T}-indexed category, then we have an 𝒮\mathcal{S}-indexed category F *F^*\mathbb{C} defined by

  • (F *) I= F(I)(F^*\mathbb{C})^I = \mathbb{C}^{F(I)} for every object I𝒮I \in \mathcal{S};

  • and x *=F(x) *x^* = F(x)^* for every morphism x:IJx : I \to J in 𝒮\mathcal{S}.

Indexed category of a functor

Combining these previous examples we get


For F:𝒮𝒞F : \mathcal{S} \to \mathcal{C} a functor and 𝒞\mathcal{C} a finitely complete category, there is the 𝒮\mathcal{S}-indexed category F *F^* \mathbb{C} given by

  • (F *) I=𝒞/F(I)(F^* \mathbb{C})^I = \mathcal{C}/F(I).

If the functor FF preserves pullbacks then this induces a morphism 𝕊F *\mathbb{S} \to F^* \mathbb{C} of 𝒮\mathcal{S}-indexed categories.

Indexed category of a topos over a base topos

This situation frequently arises when 𝒮\mathcal{S} and 𝒞\mathcal{C} are toposes and Ff *F \coloneqq f^* is the inverse image part of a geometric morphism.

f:𝒞f *f *𝒮. f : \mathcal{C} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{S} \,.

In this way, if 𝒮\mathcal{S} is a topos, then to be thought of as a base topos, then any topos over 𝒮\mathcal{S} (i.e. an object of the slice 2-category Topos/S/S) gives rise to a topos relative to 𝒮\mathcal{S}, i.e. a “topos object” in the 2-category of 𝒮\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, ff exhibits 𝒞\mathcal{C} as a 2-sheaf (see there) over 𝒞\mathcal{C}, 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



(LR):𝒞RL𝒮 (L \dashv R) : \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{S}

be a pair of adjoint functors between finitely complete categories. Then RR extends to an 𝒮\mathcal{S}-indexed functor

:𝕊 \mathbb{R} : \mathbb{C} \to \mathbb{S}

where 𝕊\mathbb{S} is the self-indexing of 𝒮\mathcal{S} from example 1 and \mathbb{C} is the base change indexing of 𝒞\mathcal{C} from example 3.

By the general properties of adjunctions on overcategories (see there) we get for each I𝒮I \in \mathcal{S} an adjunction

(L/IR/I): I=𝒞/R(I)𝒮/I=𝕊 I. (L/I \dashv R/I) : \mathbb{C}^I = \mathcal{C}/R(I) \to \mathcal{S}/I = \mathbb{S}^I \,.

Here :IR/I\mathbb{R} : I \mapsto R/I is always a 𝒮\mathcal{S}-indexed functor 𝕊\mathbb{C} \to \mathbb{S}, and 𝕃:IL/I\mathbb{L} : I \mapsto L/I is if LL preserves pullbacks (by example 3). If so, we have an 𝒮\mathcal{S}-indexed adjunction

(𝕃):𝕊 (\mathbb{L} \dashv \mathbb{R}) : \mathbb{C} \to \mathbb{S}

This appears as (Johnstone, lemma B1.2.3).



Well-powered indexed categories


An 𝒮\mathcal{S}-indexed category \mathbb{C} is called well-powered if the fibered category ˜𝒮\tilde \mathbb{C} \to \mathcal{S} corresponding to it under the Grothendieck construction has the property that the forgetful functor

U:Q(2,˜)Rect(*,˜) U : Q(2, \tilde \mathbb{C}) \to Rect(*,\tilde \mathbb{C})

has a right adjoint, where Q(2,˜)Q(2,\tilde \mathbb{C}) is the full subcategory of Rect(2,˜)Rect(2, \tilde \mathbb{C}) on vertical monomorphisms.

This appears as (Johnstone, example. B1.3.14).


Let (LR):𝒞RL𝒮(L \dashv R) : \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{S} be a pair of adjoint functors such that LL preserves pullbacks. Then the 𝒮\mathcal{S}-indexed category \mathbb{C} is well powered if 𝕊\mathbb{S} is.


This is (Johnstone, prop. B1.3.17).


Section B1.2 in

Revised on April 9, 2016 12:02:22 by Ingo Blechschmidt (