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.