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) |
System (in a category). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=System_(in_a_category)&oldid=48941