Contents

category theory

# Contents

## Idea

An indexed category is a 2-presheaf.

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

## Definition

Let $\mathcal{S}$ be a category.

###### Definition

An $\mathcal{S}$-indexed category $\tilde \mathbb{C}$ is a category defined by a pseudofunctor

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

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

Under the Grothendieck construction equivalence this is equivalently a fibered category

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

over $\mathcal{S}$.

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

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

One may also call $\mathbb{C}$ a prestack in categories over $\mathcal{S}$.

Traditionally one writes the image of an object $X \in \mathcal{S}$ 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.

## Examples

### Self indexing

###### Example

(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

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

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

### Base change

###### Example

(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 $\mathcal{S}$.

### Indexed category of a functor

Combining these previous examples we get

###### Example

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

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

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

### Indexed category of a topos over a base topos

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.

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

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.

## Properties

### Extensions of adjunctions to indexed categories

###### Proposition

Let

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

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

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

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

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

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

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 ). If so, we have an $\mathcal{S}$-indexed adjunction

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

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

(…)

### Well-powered indexed categories

###### Definition

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

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

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).

###### Proposition

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.

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

The notion of indexed categories was introduced in

but under the name “fibered category” (catégorie fibrée) which later became the standard term, instead, for the (equivalent) Grothendieck construction on an indexed category, cf.:

• Jean Bénabou, p. 898 (2 of 41) of: Fibrations petites et localement petites, C. R. Acad. Sci. Paris 281 Série A (1975) 897-900 [gallica]

An early monograph:

Early discussion in the context of categorical semantics in computer science:

Discussion in a context of topos theory: