Inverse system

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. 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 nleq m} , is a filtered category. 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. 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 Grothendieck (1960).