Ind-completion explained

In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category C. The objects in this ind-completed category, denoted Ind(C), are known as direct systems, they are functors from a small filtered category I to C.

The dual concept is the pro-completion, Pro(C).

Definitions

Filtered categories

See also: Filtered category.

Direct systems depend on the notion of filtered categories. For example, the category N, whose objects are natural numbers, and with exactly one morphism from n to m whenever

n \le m , is a filtered category.

Direct systems

A direct system or an ind-object in a category C is defined to be a functor

F : I \to C

from a small filtered category I to C. For example, if I is the category N mentioned above, this datum is equivalent to a sequence

X _{0} \to X _{1} \to …

of objects in C together with morphisms as displayed.

The ind-completion

Ind-objects in C form a category ind-C.

Two ind-objects

F:I\to C and

$G:J\backslash to\; C$ determine a functor

I^op x J

\to Sets,

namely the functor

\operatorname{Hom} _{C(F(i),G(j)).}

The set of morphisms between F and G in Ind(C) is defined to be the colimit of this functor in the second variable, followed by the limit in the first variable:

\operatorname{Hom} _{\operatorname{Ind} - C}(F,G) = \lim _{i} \operatorname{colim} _{j} \operatorname{Hom} _{C(F(i),} G(j)).

More colloquially, this means that a morphism consists of a collection of maps

F(i) \to G(j _{i)} for each i, where j _{i} is (depending on i) large enough.

Relation between C and Ind(C)

The final category I = consisting of a single object * and only its identity morphism is an example of a filtered category. In particular, any object X in C gives rise to a functor

\{*\} \to C, * \mapsto X

and therefore to a functor

C \to \operatorname{Ind}(C), X \mapsto (* \mapsto X). This functor is, as a direct consequence of the definitions, fully faithful. Therefore Ind(C) can be regarded as a larger category than C.

Conversely, there need not in general be a natural functor

\operatorname{Ind}(C) \to C.

However, if C possesses all filtered colimits (also known as direct limits), then sending an ind-object

F: I \to C (for some filtered category I) to its colimit\operatorname {colim} _{I} F(i)

does give such a functor, which however is not in general an equivalence. Thus, even if C already has all filtered colimits, Ind(C) is a strictly larger category than C.

Objects in Ind(C) can be thought of as formal direct limits, so that some authors also denote such objects by

“ \varinjlim _{i} '' F(i). This notation is due to Pierre Deligne.^[1]

Universal property of the ind-completion

The passage from a category C to Ind(C) amounts to freely adding filtered colimits to the category. This is why the construction is also referred to as the ind-completion of C. This is made precise by the following assertion: any functor

F: C \to D taking values in a category D that has all filtered colimits extends to a functor Ind(C) \to D that is uniquely determined by the requirements that its value on C is the original functor F and such that it preserves all filtered colimits.

Basic properties of ind-categories

Compact objects

Essentially by design of the morphisms in Ind(C), any object X of C is compact when regarded as an object of Ind(C), i.e., the corepresentable functor

\operatorname{Hom} _{\operatorname{Ind} (C)}(X, - )

preserves filtered colimits. This holds true no matter what C or the object X is, in contrast to the fact that X need not be compact in C. Conversely, any compact object in Ind(C) arises as the image of an object in X.

A category C is called compactly generated, if it is equivalent to

\operatorname{Ind}(C _{0)} for some small category C _{0} . The ind-completion of the category FinSet of finite sets is the category of all sets. Similarly, if C is the category of finitely generated groups, ind-C is equivalent to the category of all groups.

Recognizing ind-completions

These identifications rely on the following facts: as was mentioned above, any functor

F: C \to D taking values in a category D that has all filtered colimits, has an extension\tilde F: \operatorname{Ind}(C) \to D,

that preserves filtered colimits. This extension is unique up to equivalence. First, this functor

\tilde F is essentially surjective if any object in D can be expressed as a filtered colimits of objects of the form F(c) for appropriate objects c in C. Second, \tilde F is fully faithful if and only if the original functor F is fully faithful and if F sends arbitrary objects in C to compact objects in D.

Applying these facts to, say, the inclusion functor

F: \operatorname{FinSet} \subset \operatorname{Set},

the equivalence

\operatorname{Ind}(\operatorname{FinSet}) \cong \operatorname{Set}

expresses the fact that any set is the filtered colimit of finite sets (for example, any set is the union of its finite subsets, which is a filtered system) and moreover, that any finite set is compact when regarded as an object of Set.

The pro-completion

Like other categorical notions and constructions, the ind-completion admits a dual known as the pro-completion: the category Pro(C) is defined in terms of ind-object as

\operatorname{Pro}(C) : = \operatorname{Ind}(C ^{op} ) ^{op} .

(The definition of pro-C is due to .^[2])

C ^{op} or, equivalently, functorsF: I \to C from a small category I.

Examples of pro-categories

While Pro(C) exists for any category C, several special cases are noteworthy because of connections to other mathematical notions.

• If C is the category of finite groups, then pro-C is equivalent to the category of profinite groups and continuous homomorphisms between them.
• The process of endowing a preordered set with its Alexandrov topology yields an equivalence of the pro-category of the category of finite preordered sets,

\operatorname{Pro}(\operatorname{PoSet} ^{fin)} , with the category of spectral topological spaces and quasi-compact morphisms.

• Stone duality asserts that the pro-category

\operatorname{Pro}(\operatorname{FinSet}) of the category of finite sets is equivalent to the category of Stone spaces.The appearance of topological notions in these pro-categories can be traced to the equivalence, which is itself a special case of Stone duality,\operatorname{FinSet} ^{op} = \operatorname{FinBool}

which sends a finite set to the power set (regarded as a finite Boolean algebra).The duality between pro- and ind-objects and known description of ind-completions also give rise to descriptions of certain opposite categories. For example, such considerations can be used to show that the opposite category of the category of vector spaces (over a fixed field) is equivalent to the category of linearly compact vector spaces and continuous linear maps between them.

Applications

Pro-completions are less prominent than ind-completions, but applications include shape theory. Pro-objects also arise via their connection to pro-representable functors, for example in Grothendieck's Galois theory, and also in Schlessinger's criterion in deformation theory.

Related notions

Tate objects are a mixture of ind- and pro-objects.

Infinity-categorical variants

The ind-completion (and, dually, the pro-completion) has been extended to ∞-categories by .

See also

• • • completions in category theory

References

• • .

Notes and References

1. Illusie, Luc, From Pierre Deligne’s secret garden: looking back at some of his letters, Japanese Journal of Mathematics, vol. 10, pp. 237–248 (2015)
2. Book: C.E. Aull. R. Lowen. Handbook of the History of General Topology. 31 December 2001. Springer Science & Business Media. 978-0-7923-6970-7. 1147.