Ind-completion

{{Short description|In mathematics, process for extending a category}}

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=

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

: $X_0 \to X_1 \to \cdots$

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

$G:J\to C$ 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:

: $\operatorname{Hom}_{\operatorname{Ind}\text{-}C}(F,G) = \lim_i \operatorname{colim}_j \operatorname{Hom}_C(F(i), G(j)).$

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

: $\text{“}\varinjlim_{i \in I} \text{'' } F(i).$

This notation is due to Pierre Deligne.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)

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 pro-completion Pro(C) can defined in terms of the ind-completion as

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

(The original definition of pro-C is due to {{harvtxt|Grothendieck|1960}}.{{cite book|author1=C.E. Aull|author2=R. Lowen|title=Handbook of the History of General Topology|url=https://books.google.com/books?id=dV6WtepcZLkC&pg=PA1147|date=31 December 2001|publisher=Springer Science & Business Media|isbn=978-0-7923-6970-7|page=1147}})

Therefore, the objects of Pro(C) are {{em|inverse systems}}{{anchor|Inverse system}} or {{em|pro-objects}} in C. By definition, these are direct system in the opposite category $C^{op}$ or, equivalently, functors

: $F: I \to C$

from a small {{em|cofiltered}} 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, $\operatorname{Pro}(\operatorname{FinPreord})$ , 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.{{harvtxt|Johnstone|1982|loc=§VI.2}}

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

: $\operatorname{FinSet}^{\text{op}} = \operatorname{FinBool}$

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.{{harvtxt|Bergman|Hausknecht|1996|loc=Prop. 24.8}}

=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 {{harvtxt|Lurie|2009}}.

Notes

References

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Category:Functors

Category:Limits (category theory)