Inverse system

In mathematics, an inverse system in a category "C" is a functor from a small cofiltered category "I" to "C". An inverse system is sometimes called a "pro-object" in "C".

The category of inverse systems

Pro-objects in "C" form a category "pro-C". Two inverse systems

:"F":I$o$ C"

and

"G":J$o$ C" determine a functor

:"I"^op x "J" $o$ "Sets",

namely the functor

"Hom_C"("F"("i"),"G"("j").

The set of homomorphisms between "F" and "G" in "pro-C" is defined to be the colimit of this functor in the first variable, followed by the limit in the second variable.

If "C" has all inverse limits, then the limit defines a functor "pro-C"$o$"C". In practice, e.g. if "C" is a category of algebraic or topological objects, this functor is not an equivalence of categories.

Direct systems/Ind-objects

An ind-object in "C" is a pro-object in "C"^{op. The category of ind-objects is written "ind-C".}

Examples

* If "C" is the category of finite groups, then "pro-C" is equivalent to the category of profinite groups and continuous homomorphisms between them.

* If "C" is the category of finitely generated groups, then "ind-C" is equivalent to the category of all groups.