Inverse system

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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



G:J C determine a functor

Iop x J Sets,

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-CC. 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 Cop. The category of ind-objects is written ind-C.


  • 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.


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