indexed category



An indexed category is a 2-presheaf.

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


Let 𝒮\mathcal{S} be a category.


An 𝒮\mathcal{S}-indexed category CC is a pseudofunctor

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

from the opposite category of 𝒮\mathcal{S} to the 2-category Cat of categories.

Under the Grothendieck construction equivalence this is equivalently a fibered category

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

over 𝒮\mathcal{S}.

Similarly, an 𝒮\mathcal{S}-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]\mathcal{S} IndCat \coloneqq [\mathcal{S}^{op}, Cat] of 𝒮\mathcal{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\mathcal{S}.

Traditionally one writes the image of an object X𝒮X \in \mathcal{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 September 23, 2016 09:50:41 by Bram Geron (