Английская Википедия:Ind-completion

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Шаблон:Short description 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

Шаблон:See

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 <math>n \le m</math>, is a filtered category.

Direct systems

Шаблон:See also

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

<math>F : I \to C</math>

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

<math>X_0 \to X_1 \to \cdots</math>

of objects in C together with morphisms as displayed.

The ind-completion

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

Two ind-objects

<math> F:I\to C </math>

and

<math display="inline">G:J\to C </math> determine a functor

Iop x J <math>\to</math> Sets,

namely the functor

<math>\operatorname{Hom}_C(F(i),G(j)).</math>

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:

<math>\operatorname{Hom}_{\operatorname{Ind}\text{-}C}(F,G) = \lim_i \operatorname{colim}_j \operatorname{Hom}_C(F(i), G(j)).</math>

More colloquially, this means that a morphism consists of a collection of maps <math>F(i) \to G(j_i)</math> for each i, where <math>j_i</math> 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

<math>\{*\} \to C, * \mapsto X</math>

and therefore to a functor

<math>C \to \operatorname{Ind}(C), X \mapsto (* \mapsto X).</math>

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

<math>\operatorname{Ind}(C) \to C.</math>

However, if C possesses all filtered colimits (also known as direct limits), then sending an ind-object <math>F: I \to C</math> (for some filtered category I) to its colimit

<math>\operatorname {colim}_I F(i)</math>

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

<math>\text{“}\varinjlim_{i \in I} \text{ } F(i). </math>

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 <math>F: C \to D</math> taking values in a category D that has all filtered colimits extends to a functor <math>Ind(C) \to D</math> 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

<math>\operatorname{Hom}_{\operatorname{Ind}(C)}(X, -)</math>

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 <math>\operatorname{Ind}(C_0)</math> for some small category <math>C_0</math>. 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 <math>F: C \to D</math> taking values in a category D that has all filtered colimits, has an extension

<math>\tilde F: \operatorname{Ind}(C) \to D, </math>

that preserves filtered colimits. This extension is unique up to equivalence. First, this functor <math>\tilde F</math> is essentially surjective if any object in D can be expressed as a filtered colimits of objects of the form <math>F(c)</math> for appropriate objects c in C. Second, <math>\tilde F</math> 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

<math>F: \operatorname{FinSet} \subset \operatorname{Set},</math>

the equivalence

<math>\operatorname{Ind}(\operatorname{FinSet}) \cong \operatorname{Set}</math>

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

<math> \operatorname{Pro}(C) := \operatorname{Ind}(C^{op})^{op}.</math>

(The definition of pro-C is due to Шаблон:Harvtxt.[2])

Therefore, the objects of Pro(C) are Шаблон:EmШаблон:Anchor or Шаблон:Em in C. By definition, these are direct system in the opposite category <math>C^{op}</math> or, equivalently, functors

<math>F: I \to C</math>

from a small Шаблон:Em 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.

The appearance of topological notions in these pro-categories can be traced to the equivalence, which is itself a special case of Stone duality,

<math>\operatorname{FinSet}^{op} = \operatorname{FinBool}</math>

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.[4]

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 Шаблон:Harvtxt.

See also

Notes

Шаблон:Reflist

References

Шаблон:Refbegin

Шаблон:Refend

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  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. ↑ Шаблон:Cite book
  3. ↑ Шаблон:Harvtxt
  4. ↑ Шаблон:Harvtxt
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