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.
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