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 formula_1, is a filtered category.
Direct systems.
A "direct system" or an "ind-object" in a category "C" is defined to be a functor
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
of objects in "C" together with morphisms as displayed.
The ind-completion.
Ind-objects in "C" form a category ind-"C", and pro-objects form a category pro-"C". The definition of pro-"C" is due to .
Two ind-objects
and
formula_5 determine a functor
namely the functor
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:
More colloquially, this means that a morphism consists of a collection of maps formula_9 for each "i", where formula_10 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
and therefore to a functor
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
However, if "C" possesses all filtered colimits (also known as direct limits), then sending an ind-object formula_14 (for some filtered category "I") to its colimit
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
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 formula_17 taking values in a category "D" which has all filtered colimits extends to a functor formula_18 which 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
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 formula_20 for some small category formula_21. 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 formula_17 taking values in a category "D" that has all filtered colimits, has an extension
that preserves filtered colimits. This extension is unique up to equivalence. First, this functor formula_24 is essentially surjective if any object in "D" can be expressed as a filtered colimits of objects of the form formula_25 for appropriate objects "c" in "C". Second, formula_24 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
the equivalence
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
Therefore, the objects of Pro("C") are or in "C". By definition, these are direct system in the opposite category formula_30 or, equivalently, functors
from a 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,
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 object 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 .