Difference between revisions of "Projective limit"
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1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526052.png" /> be a [[discrete category]]. Then for an arbitrary functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526053.png" /> the projective limit of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526054.png" /> coincides with the product of the family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526056.png" /> (cf. [[Product of a family of objects in a category|Product of a family of objects in a category]]). | 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526052.png" /> be a [[discrete category]]. Then for an arbitrary functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526053.png" /> the projective limit of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526054.png" /> coincides with the product of the family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526056.png" /> (cf. [[Product of a family of objects in a category|Product of a family of objects in a category]]). | ||
− | 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526057.png" /> be a category with two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526058.png" /> and two non-identity morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526059.png" />. Then the limit of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526060.png" /> is the equalizer of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526061.png" /> (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]). | + | 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526057.png" /> be a category with two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526058.png" /> and two non-identity morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526059.png" />. Then the limit of a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526060.png" /> is the [[equalizer]] of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526061.png" /> (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]). |
If a category has products of arbitrary small families of objects and equalizers of pairs of morphisms, then it has limits for all functors defined on small categories. | If a category has products of arbitrary small families of objects and equalizers of pairs of morphisms, then it has limits for all functors defined on small categories. |
Revision as of 18:30, 22 October 2017
inverse limit
A construction that arose originally in set theory and topology, and then found numerous applications in many areas of mathematics. A common example of a projective limit is that of a family of mathematical structures of the same type indexed by the elements of a pre-ordered set. Let be a set endowed with a pre-order relation
, and suppose that with each element
a set
is associated and with each pair
,
, in which
, a mapping
is associated, where the
,
, are identity mappings and
for
. A set
is called a projective limit of the family of sets
and mappings
if the following conditions are satisfied: a) there exists a family of mappings
such that
for any pair
; b) for any family of mappings
,
, from an arbitrary set
, for which the equalities
hold for
, there exists a unique mapping
such that
for every
. The projective limit can be described explicitly as follows. One considers the direct product
and selects in it the set of all functions
for which the equalities
hold for
. This subset is the projective limit of the family
. If all the
are equipped with an additional structure of the same type and the
preserve it, then the same structure is induced in the projective limit. Therefore it is possible to speak of projective limits of groups, modules, topological spaces, etc.
A natural generalization of this concept of projective limit is that of the projective limit of a functor. Let be a functor from a small category
to an arbitrary category
. An object
together with morphisms
,
, is called a projective limit (inverse limit, or simply limit) of the functor
if the following conditions are satisfied:
)
for any morphism
; and
) for any family of morphisms
satisfying
for all
there exists a unique morphism
such that
for all
. Notation:
.
Examples of projective limits.
1) Let be a discrete category. Then for an arbitrary functor
the projective limit of the functor
coincides with the product of the family of objects
,
(cf. Product of a family of objects in a category).
2) Let be a category with two objects
and two non-identity morphisms
. Then the limit of a functor
is the equalizer of the pair of morphisms
(cf. Kernel of a morphism in a category).
If a category has products of arbitrary small families of objects and equalizers of pairs of morphisms, then it has limits for all functors defined on small categories.
Comments
In most modern work in category theory, the unadorned name "limit" is used for this concept (and the dual concept is called a colimit). The terms "inverse limit" and its dual, direct limit (or inductive limit), are generally restricted to diagrams over directed pre-ordered sets (see Directed order); "projective limit" is best avoided, because of the danger of confusion with the notion of projective object of a category. Inverse and direct limits were first studied as such in the 1930's, in connection with topological concepts such as Čech cohomology; the general concept of limit was introduced in 1958 by D.M. Kan [a1].
References
[a1] | D.M. Kan, "Adjoint functors" Trans. Amer. Math. Soc. , 87 (1958) pp. 294–329 |
[a2] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 |
Projective limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_limit&oldid=42161