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).

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


 * Iop 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.

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.

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 . )

Therefore, the objects of Pro(C) are or  in C. By definition, these are direct system in the opposite category $$C^{op}$$ or, equivalently, functors


 * $$F: I \to C$$

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.

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


 * $$\operatorname{FinSet}^{op} = \operatorname{FinBool}$$

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.