Difference between revisions of "Inductive limit"
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Every pre-ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077069.png" /> can be regarded as a category whose objects are the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077070.png" /> and whose morphisms are all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077071.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077073.png" />, with the obvious law of composition. In an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077074.png" />, a family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077076.png" />, and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077078.png" />, can be regarded as the image of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077079.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077083.png" /> is the category of sets (groups, topological spaces, etc.), then the inductive limit of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077084.png" /> coincides with the construction of the inductive limit defined above. | Every pre-ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077069.png" /> can be regarded as a category whose objects are the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077070.png" /> and whose morphisms are all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077071.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077073.png" />, with the obvious law of composition. In an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077074.png" />, a family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077076.png" />, and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077078.png" />, can be regarded as the image of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077079.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077083.png" /> is the category of sets (groups, topological spaces, etc.), then the inductive limit of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077084.png" /> coincides with the construction of the inductive limit defined above. | ||
− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077085.png" /> is a small discrete category, then an inductive limit of any functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077086.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077087.png" /> into an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077088.png" /> is a coproduct of the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077090.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077091.png" /> is empty, then an inductive limit is a left null or an initial object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077092.png" />. Cokernels of pairs of morphisms of any category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077093.png" /> are inductive limits of functors defined on the category with two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077095.png" /> and four morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077097.png" />. | + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077085.png" /> is a small [[discrete category]], then an inductive limit of any functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077086.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077087.png" /> into an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077088.png" /> is a coproduct of the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077090.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077091.png" /> is empty, then an inductive limit is a left null or an initial object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077092.png" />. Cokernels of pairs of morphisms of any category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077093.png" /> are inductive limits of functors defined on the category with two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077095.png" /> and four morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077097.png" />. |
Every covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077098.png" /> from an arbitrary small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077099.png" /> into a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770100.png" /> has an inductive limit if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770101.png" /> has coproducts and cokernels of pairs of morphisms. | Every covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077098.png" /> from an arbitrary small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i05077099.png" /> into a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770100.png" /> has an inductive limit if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050770/i050770101.png" /> has coproducts and cokernels of pairs of morphisms. |
Revision as of 18:39, 27 May 2016
A construction that first appeared in set theory, and then became widely used in algebra, topology and other areas of mathematics. An important special case of an inductive limit is the inductive limit of a directed family of mathematical structures of the same type. Let be a directed pre-ordered set, that is, a reflexive transitive relation
is defined on
and for any two elements
there exists an element
such that
and
. Suppose further that a structure
is associated with each
(for definiteness, suppose that the
are groups) and that for each
homomorphisms
are given satisfying the two conditions:
for any
and
for any
in
. An equivalence relation
is introduced on the set
: The element
is equivalent to
if
for some
. The quotient set
can then be endowed with a group structure: If
,
and
,
, then the product of the equivalence classes represented by
and
is defined to be the equivalence class with representative
. The resulting group
is called the inductive limit of the family of groups
. There exists for each
a natural homomorphism
associating to an element
its equivalence class. The group
together with its homomorphisms
has the following property: For any system of homomorphisms
,
, for which
for
, there exists a unique homomorphism
such that
for any
.
A generalization of the above construction of an inductive limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object of a category
is called an inductive limit of the covariant functor
if:
1) there exist morphisms , where
, such that
for any morphism
in
; and
2) for any family of morphisms , where
, such that
for any
in
, there exists a unique morphism
such that
,
.
An inductive limit is denoted by or
or
. An inductive limit of a contravariant functor
is defined as an inductive limit of the covariant functor
from the dual category
into the category
.
Every pre-ordered set can be regarded as a category whose objects are the elements of
and whose morphisms are all pairs
for which
and
, with the obvious law of composition. In an arbitrary category
, a family of objects
,
, and morphisms
, where
, can be regarded as the image of a functor
if
and
for
. If
is the category of sets (groups, topological spaces, etc.), then the inductive limit of the functor
coincides with the construction of the inductive limit defined above.
If is a small discrete category, then an inductive limit of any functor
from
into an arbitrary category
is a coproduct of the objects
,
. In particular, if
is empty, then an inductive limit is a left null or an initial object of
. Cokernels of pairs of morphisms of any category
are inductive limits of functors defined on the category with two objects
and
and four morphisms
and
.
Every covariant functor from an arbitrary small category
into a category
has an inductive limit if and only if
has coproducts and cokernels of pairs of morphisms.
References
[1] | P.S. Aleksandrov, "Topological duality theorems" Trudy Mat. Inst. Steklov. , 48 (1955) pp. Chapt. 1 (In Russian) |
[2] | I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) |
[3] | M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian) |
Comments
In the main article above, stands for
: the value of the homomorphism
on the element
.
Similarly, composition of homomorphisms is written backwards: means first apply
and then
.
In English, the term "inductive limit" is usually restricted to limits over directed pre-ordered sets, the more general categorical concept being called a colimit. "Cokernels of pairs of morphisms" are commonly called coequalizers.
Dual to the notion of an inductive limit is that of a projective limit, also called an inverse limit.
References
[a1] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 |
Inductive limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inductive_limit&oldid=38853