Inverse system

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an inverse system in a category

Category (mathematics)

In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...

 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 in the category of small categories....

 from a small cofiltered category

Filtered category

In category theory, filtered categories generalize the notion of directed set.A category is filtered when* it is not empty,* for every two objects and in there exists an object and two arrows and in ,...

 I to C. An inverse system is sometimes called a pro-object in C.

Pro-objects in C form a category pro-C. Two inverse systems

F:I C

and

G:J C
determine a functor

I^op 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 mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an inverse system in a category

Category (mathematics)

In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...

 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 in the category of small categories....

 from a small cofiltered category

Filtered category

In category theory, filtered categories generalize the notion of directed set.A category is filtered when* it is not empty,* for every two objects and in there exists an object and two arrows and in ,...

 I to C. An inverse system is sometimes called a pro-object in C.

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

I^op 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.