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 ≤ m {\displaystyle 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 → 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.
The ind-completion 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
Iop 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.
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
{ ∗ } → 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.
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 → 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.
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
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