# System (in a category)

*direct and inverse system in a category *

A direct system in consists of a collection of objects , indexed by a directed set , and a collection of morphisms in , for in , such that

a) for ;

b) for in .

There exists a category, , whose objects are indexed collections of morphisms such that if in and whose morphisms with domain and range are morphisms such that for . An initial object of is called a direct limit of the direct system . The direct limits of sets, topological spaces, groups, and -modules are examples of direct limits in their respective categories.

Dually, an inverse system in consists of a collection of objects , indexed by a directed set , and a collection of morphisms in , for in , such that

a) for ;

b) for in .

There exists a category, , whose objects are indexed collections of morphisms such that if in and whose morphisms with domain and range are morphisms of such that for . A terminal object of is called an inverse limit of the inverse system . The inverse limits of sets, topological spaces, groups, and -modules are examples of inverse limits in their respective categories.

The concept of an inverse limit is a categorical generalization of the topological concept of a projective limit.

#### References

[1] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |

#### Comments

There is a competing terminology, with "direct limit" replaced by "colimit" , and "inverse limit" by "limit" .

#### References

[1a] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |

**How to Cite This Entry:**

System (in a category).

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=System_(in_a_category)&oldid=42577