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
- F:I[math]\displaystyle{ \to }[/math] C
and
G:J[math]\displaystyle{ \to }[/math] C determine a functor
- Iop x J [math]\displaystyle{ \to }[/math] Sets,
namely the functor
- [math]\displaystyle{ \mathrm{Hom}_C(F(i),G(j)) }[/math].
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[math]\displaystyle{ \to }[/math]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 Cop. 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.
Notes
- ↑ C.E. Aull; R. Lowen (31 December 2001). Handbook of the History of General Topology. Springer Science & Business Media. p. 1147. ISBN 978-0-7923-6970-7. https://books.google.com/books?id=dV6WtepcZLkC&pg=PA1147.
References
- Bourbaki, Nicolas (1968), Elements of mathematics. Theory of sets, Translated from the French, Paris: Hermann.
- Hazewinkel, Michiel, ed. (2001), "System (in a category)", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=S/s091930
- Segal, Jack; Mardešić, Sibe (1982), Shape theory, North-Holland Mathematical Library, 26, Amsterdam: North-Holland, ISBN 978-0-444-86286-0
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