Inverse system
Encyclopedia
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.
and
G:J
C
determine a functor
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 limit
s, then the limit defines a functor pro-C
C. In practice, e.g. if C is a category of algebraic or topological objects, this functor is not an equivalence of categories.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an inverse system in a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
C is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from a small cofiltered category
Filtered category
In category theory, filtered categories generalize the notion of directed set understood as a category .A category J is filtered when...
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. Two inverse systems- F:I
C
and
G:J
Cdetermine 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 limit
Inverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
s, then the limit defines a functor pro-C
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.