The Category of Inverse Systems
Pro-objects in C form a category pro-C. Two inverse systems
- F:I C
and
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.
Read more about this topic: Inverse System
Famous quotes containing the words category, inverse and/or systems:
“Despair is typical of those who do not understand the causes of evil, see no way out, and are incapable of struggle. The modern industrial proletariat does not belong to the category of such classes.”
—Vladimir Ilyich Lenin (1870–1924)
“Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
—Ralph Waldo Emerson (1803–1882)
“We have done scant justice to the reasonableness of cannibalism. There are in fact so many and such excellent motives possible to it that mankind has never been able to fit all of them into one universal scheme, and has accordingly contrived various diverse and contradictory systems the better to display its virtues.”
—Ruth Benedict (1887–1948)