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.
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
(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.
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
taking values in a category D that has all filtered colimits extends to a functor
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 preserves filtered colimits.
A category C is called compactly generated, if it is equivalent to
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.
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 (The definition of pro-C is due to Grothendieck (1960).
or, equivalently, functors 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.
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).
[4] Pro-completions are less prominent than ind-completions, but applications include shape theory.