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.

Pro-objects in C form a category pro-C. The general definition was given by Alexander Grothendieck in 1959, in TDTE.

Two inverse systems

and

G:J${\displaystyle \to }$ C determine a functor

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${\displaystyle \to }$C. In practice, e.g. if C is a category of algebraic or topological objects, this functor is not an equivalence of categories.

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

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