# 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 dual concept is a direct system.

## The category of inverse systems

Pro-objects in *C* form a category *pro-C*. The general definition was given by Alexander Grothendieck in 1959, in *TDTE*.^{[1]}

Two inverse systems

and

namely the functor

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**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.

## References

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## Notes

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