nLab
indexed category

Idea

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 $ℂ$ together with, for every object $X\in S$, a notion of ”$X$-indexed family of objects of $ℂ$.”

Definition

Let $S$ be a category.

Definition

An $S$-indexed category $C$ is a pseudofunctor

$$ℂ:S^{\mathrm{op}}\to \mathrm{Cat}$$\mathbb{C} : S^{op}\to Cat

from the opposite category of $S$ to the 2-category Cat of categories.

Under the Grothendieck construction equivalence this is equivalently a fibered category

$$\begin{array}{c}\widetilde{ℂ}\\ \downarrow \\ S\end{array}$$\array{ \tilde \mathbb{C} \\ \downarrow \\ S }

over $S$.

Similarly, an $S$-indexed functor $ℂ\to 𝔻$ is a pseudonatural transformation of pseudofunctors, and an indexed natural transformation is a modification.

This defines the 2-category $S\mathrm{IndCat}:=[S^{\mathrm{op}},\mathrm{Cat}]$ of $S$-indexed categories.

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

One may also call $ℂ$ a prestack in categories over $S$.

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

If $𝒮$ has a terminal object $*$ we think of ${ℂ}^{*}$ as the underlying ordinary category of the $𝒮$-indexed category $ℂ$. Part of the theory of indexed categories is about when and how to extend structures on ${ℂ}^{*}$ to all of $ℂ$.

A morphism of $S$-indexed categories is an indexed functor.

Examples

Self indexing

Example

(canonical self-indexing)

If $𝒮$ has pullbacks, then its codomain fibration is an $𝒮$-indexed category denoted $𝕊$.

This assigns to an object $I$ the corresponding over-category

$${𝕊}^I:=𝒮/I$$\mathbb{S}^I := \mathcal{S}/I

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 $𝒮$ itself (or rather its codomain fibration) in the world of $𝒮$-indexed categories.

Base change

Example

(change of base)

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

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

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

Indexed category of a functor

Combining these previous examples we get

Example

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

• $(F^{*}ℂ{)}^I=𝒞/F(I)$.

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

Index category of a topos over a base topos

This situation frequently arises when $𝒮$ and $𝒞$ are toposes and $F:=f^{*}$ is the inverse image part of a geometric morphism.

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

In this way, if $𝒮$ is a topos, then to be thought of as a base topos, then any topos over $𝒮$ (i.e. an object of the slice 2-category Topos$/S$) gives rise to a topos relative to $𝒮$, i.e. a “topos object” in the 2-category of $𝒮$-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 $𝒞$ as a 2-sheaf (see there) over $𝒞$, with respect to the canonical topology.

Properties

Extensions of adjunctions to indexed categories

Proposition

Let

$$(L⊣R):𝒞\stackrel{\stackrel{L}{\leftarrow }}{\underset{R}{\to }}𝒮$$(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 $R$ extends to an $𝒮$-indexed functor

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

where $𝕊$ is the self-indexing of $𝒮$ from example 1 and $ℂ$ is the base change indexing of $𝒞$ from example 3.

By the general properties of adjunctions on overcategories (see there) we get for each $I\in 𝒮$ an adjunction

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

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

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

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

Proof

(…)

Well-powered indexed categories

Definition

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

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

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

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

Proposition

Let $(L⊣R):𝒞\stackrel{\stackrel{L}{\leftarrow }}{\underset{R}{\to }}𝒮$ be a pair of adjoint functors such that $L$ preserves pullbacks. Then the $𝒮$-indexed category $ℂ$ is well powered if $𝕊$ is.

hm

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

Related concepts

• indexed functor

• indexed topos

• hyperdoctrine

References

Section B1.2 in

• Peter Johnstone, Sketches of an Elephant