View source for Projective limit

''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752601.png" /> be a set endowed with a pre-order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752602.png" />, and suppose that with each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752603.png" /> a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752604.png" /> is associated and with each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752606.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752607.png" />, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752608.png" /> is associated, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p0752609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526010.png" />, are identity mappings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526012.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526013.png" /> is called a projective limit of the family of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526014.png" /> and mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526015.png" /> if the following conditions are satisfied: a) there exists a family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526017.png" /> for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526018.png" />; b) for any family of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526020.png" />, from an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526021.png" />, for which the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526022.png" /> hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526023.png" />, there exists a unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526025.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526026.png" />. The projective limit can be described explicitly as follows. One considers the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526027.png" /> and selects in it the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526028.png" /> for which the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526029.png" /> hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526030.png" />. This subset is the projective limit of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526031.png" />. If all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526032.png" /> are equipped with an additional structure of the same type and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526033.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526034.png" /> be a functor from a [[Small category|small category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526035.png" /> to an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526036.png" />. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526037.png" /> together with morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526039.png" />, is called a projective limit (inverse limit, or simply limit) of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526040.png" /> if the following conditions are satisfied: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526041.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526042.png" /> for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526043.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526044.png" />) for any family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526045.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526046.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526047.png" /> there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526049.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526050.png" />. Notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075260/p07526051.png" />.

===Examples of projective limits.===


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]]).

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|inductive limit]]), are generally restricted to diagrams over directed pre-ordered sets (see [[Directed order|Directed order]]);  "projective limit"  is best avoided, because of the danger of confusion with the notion of [[Projective object of a category|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|Čech cohomology]]; the general concept of limit was introduced in 1958 by D.M. Kan [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.M. Kan,   "Adjoint functors"  ''Trans. Amer. Math. Soc.'' , '''87'''  (1958)  pp. 294–329</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. MacLane,   "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR></table>