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System (in a category)

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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). M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=System_(in_a_category)&oldid=18745
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098