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).

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 ≤ m {\displaystyle n\leq m} , is a filtered category.

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

F : I → C {\displaystyle 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 → X 1 → ⋯ {\displaystyle X_{0}\to X_{1}\to \cdots }

of objects in C together with morphisms as displayed.

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

Two ind-objects

F : I → C {\displaystyle F:I\to C}

and

G : J → C {\textstyle G:J\to C} determine a functor

I^op x J → {\displaystyle \to } Sets,

namely the functor

Hom C ⁡ ( F ( i ) , G ( j ) ) . {\displaystyle \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:

Hom Ind ⁡ - C ⁡ ( F , G ) = lim i colim j ⁡ Hom C ⁡ ( F ( i ) , G ( j ) ) . {\displaystyle \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 ) → G ( j i ) {\displaystyle F(i)\to G(j_{i})} for each i, where j i {\displaystyle j_{i}} is (depending on i) large enough.

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

{ ∗ } → C , ∗ ↦ X {\displaystyle \{*\}\to C,*\mapsto X}

and therefore to a functor

C → Ind ⁡ ( C ) , X ↦ ( ∗ ↦ X ) . {\displaystyle 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

Ind ⁡ ( C ) → C . {\displaystyle \operatorname {Ind} (C)\to C.}

However, if C possesses all filtered colimits (also known as direct limits), then sending an ind-object F : I → C {\displaystyle F:I\to C} (for some filtered category I) to its colimit

colim I ⁡ F ( i ) {\displaystyle \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

“ lim → i ∈ I ⁡ ''  F ( i ) . {\displaystyle {\text{“}}\varinjlim _{i\in I}{\text{'' }}F(i).}

This notation is due to Pierre Deligne.^[1]

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 → D {\displaystyle F:C\to D} taking values in a category D that has all filtered colimits extends to a functor I n d ( C ) → D {\displaystyle 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.

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

Hom Ind ⁡ ( C ) ⁡ ( X , − ) {\displaystyle \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 Ind ⁡ ( C 0 ) {\displaystyle \operatorname {Ind} (C_{0})} for some small category C 0 {\displaystyle 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.

These identifications rely on the following facts: as was mentioned above, any functor F : C → D {\displaystyle F:C\to D} taking values in a category D that has all filtered colimits, has an extension

F ~ : Ind ⁡ ( C ) → D , {\displaystyle {\tilde {F}}:\operatorname {Ind} (C)\to D,}

that preserves filtered colimits. This extension is unique up to equivalence. First, this functor F ~ {\displaystyle {\tilde {F}}} is essentially surjective if any object in D can be expressed as a filtered colimits of objects of the form F ( c ) {\displaystyle F(c)} for appropriate objects c in C. Second, F ~ {\displaystyle {\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 : FinSet ⊂ Set , {\displaystyle F:\operatorname {FinSet} \subset \operatorname {Set} ,}

the equivalence

Ind ⁡ ( FinSet ) ≅ Set {\displaystyle \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.

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

Pro ⁡ ( C ) := Ind ⁡ ( C o p ) o p . {\displaystyle \operatorname {Pro} (C):=\operatorname {Ind} (C^{op})^{op}.}

(The original 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 o p {\displaystyle C^{op}} or, equivalently, functors

F : I → C {\displaystyle F:I\to C}

from a small cofiltered category I.

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, Pro ⁡ ( FinPreord ) {\displaystyle \operatorname {Pro} (\operatorname {FinPreord} )} , with the category of spectral topological spaces and quasi-compact morphisms.
• Stone duality asserts that the pro-category Pro ⁡ ( FinSet ) {\displaystyle \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,

FinSet op = FinBool {\displaystyle \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.^[4]

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.

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

The ind-completion (and, dually, the pro-completion) has been extended to ∞-categories by Lurie (2009).

• Direct limit – Special case of colimit in category theory
• Inverse limit – Construction in category theory
• completions in category theory