Ind-completion

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 $n\leq m$ , 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

${\textstyle G:J\to C}$ 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.^[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 Grothendieck (1960).^[2])

Therefore, the objects of Pro(C) are inverse systems or 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 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, $\operatorname {Pro} (\operatorname {PoSet} ^{\text{fin}})$ , 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.^[3]

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.^[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 Lurie (2009).

Notes

^ 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)
^ C.E. Aull; R. Lowen (31 December 2001). Handbook of the History of General Topology. Springer Science & Business Media. p. 1147. ISBN 978-0-7923-6970-7.
^ Johnstone (1982, §VI.2)
^ Bergman & Hausknecht (1996, Prop. 24.8)

References

Bergman; Hausknecht (1996), Cogroups and Co-rings in Categories of Associative Rings, Mathematical Surveys and Monographs, vol. 45, doi:10.1090/surv/045, ISBN 9780821804957
Bourbaki, Nicolas (1968), Elements of mathematics. Theory of sets, Translated from the French, Paris: Hermann, MR 0237342.
Blom, Thomas; Moerdijk, Ieke (2023), "Simplicial model structures on pro-categories", Algebraic & Geometric Topology, 23 (8): 3849–3908, arXiv:2009.07539, doi:10.2140/agt.2023.23.3849
Grothendieck, Alexander (1960), "Technique de descente et théoèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules", Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204 (in French), Sociétée mathématique de France, pp. 369–390, MR 1603480, Zbl 0234.14007
"System (in a category)", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Johnstone, Peter T. (1982), Stone Spaces, ISBN 0521337798
Lurie, Jacob (2009), Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659
Segal, Jack; Mardešić, Sibe (1982), Shape theory, North-Holland Mathematical Library, vol. 26, Amsterdam: North-Holland, ISBN 978-0-444-86286-0

[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] C.E. Aull; R. Lowen (31 December 2001). Handbook of the History of General Topology. Springer Science & Business Media. p. 1147. ISBN 978-0-7923-6970-7.

[3] Johnstone (1982, §VI.2)

[4] Bergman & Hausknecht (1996, Prop. 24.8)

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