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.

Two inverse systems

F:I$\to$ C

and

G:J$\to$ C determine a functor

I^op x J $\to$ Sets,

namely the functor

$\mathrm {Hom} _{C}(F(i),G(j))$.

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$\to$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

• Bourbaki, Nicolas (1968), Elements of mathematics. Theory of sets, Translated from the French, Paris: Hermann, MR 0237342.
• "System (in a category)", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Segal, Jack; Mardešić, Sibe (1982), Shape theory, North-Holland Mathematical Library, vol. 26, Amsterdam: North-Holland, ISBN 978-0-444-86286-0