nLab indexed category

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.

Hyperdoctrine

• hyperdoctrine

Indexed monoidal category

See also indexed monoidal category, indexed closed monoidal category and dependent linear type theory.

Indexed $(\infty, 1)$-category

See indexed (infinity, 1)-category.

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).

Proof

(…)

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).

Related concepts

• indexed functor

• indexed topos

• indexed Grothendieck construction

• indexed monoidal category

• hyperdoctrine

References

The notion of indexed categories was introduced in

• Alexander Grothendieck, §A.1 (p. 3) of Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire N. Bourbaki exp. no190 (1960) 299-327 [numdam:SB_1958-1960__5__299_0]

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:

• P. T. Johnstone, R. Paré, R. D. Rosebrugh, D. Schumacher, R. J. Wood, G. C. Wraith, Indexed Categories and Their Applications, Lecture Notes in Mathematics 661 (1978) [doi:10.1007/BFb0061360]

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

• Andrzej Tarlecki, Rod M. Burstall Joseph A. Goguen, Some fundamental algebraic tools for the semantics of computation: Part 3. indexed categories, Theoretical Computer Science 91 2 (1991) 239-264 [doi:10.1016/0304-3975(91)90085-G]

Discussion in a context of topos theory:

• Peter Johnstone, Section B1.2 in: Sketches of an Elephant (2002)