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=42577