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:
“The truth is, no matter how trying they become, babies two and under dont have the ability to make moral choices, so they cant be bad. That category only exists in the adult mind.”
—Anne Cassidy (20th century)
“Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
—Ralph Waldo Emerson (18031882)
“The only people who treasure systems are those whom the whole truth evades, who want to catch it by the tail. A system is just like truths tail, but the truth is like a lizard. It will leave the tail in your hand and escape; it knows that it will soon grow another tail.”
—Ivan Sergeevich Turgenev (18181883)