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 $ I $ be a set endowed with a pre-order relation $ \leq $, and suppose that with each element $ i \in I $ a set $ X _ {i} $ is associated and with each pair $ ( i , j ) $, $ i , j \in I $, in which $ i \leq j $, a mapping $ \phi _ {ij} : X _ {i} \rightarrow X _ {j} $ is associated, where the $ \phi _ {ii} $, $ i \in I $, are identity mappings and $ \phi _ {jk} \phi _ {ij} = \phi _ {ik} $ for $ i \leq j \leq k $. A set $ X $ is called a projective limit of the family of sets $ X _ {i} $ and mappings $ \phi _ {ij} $ if the following conditions are satisfied: a) there exists a family of mappings $ \pi _ {i} : X \rightarrow X _ {i} $ such that $ \phi _ {ij} \pi _ {i} = \pi _ {j} $ for any pair $ i \leq j $; b) for any family of mappings $ \alpha _ {i} : Y \rightarrow X _ {i} $, $ i \in I $, from an arbitrary set $ Y $, for which the equalities $ \phi _ {ij} \alpha _ {i} = \alpha _ {j} $ hold for $ i \leq j $, there exists a unique mapping $ \alpha : Y \rightarrow X $ such that $ \alpha _ {i} = \pi _ {i} \alpha $ for every $ i \in I $. The projective limit can be described explicitly as follows. One considers the direct product $ \prod _ {i \in I } X _ {i} $ and selects in it the set of all functions $ f : I \rightarrow \cup _ {i \in I } X _ {i} $ for which the equalities $ \phi _ {ij} ( f ( i) ) = f ( j) $ hold for $ i \leq j $. This subset is the projective limit of the family $ X _ {i} $. If all the $ X _ {i} $ are equipped with an additional structure of the same type and the $ \phi _ {ij} $ 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 $ F : \mathfrak D \rightarrow \mathfrak K $ be a functor from a small category $ \mathfrak D $ to an arbitrary category $ \mathfrak K $. An object $ X \in \mathop{\rm Ob} \mathfrak K $ together with morphisms $ \pi _ {D} : X \rightarrow F ( D) $, $ D \in \mathop{\rm Ob} \mathfrak D $, is called a projective limit (inverse limit, or simply limit) of the functor $ F $ if the following conditions are satisfied: $ \alpha $) $ F ( \phi ) \pi _ {D} = \pi _ {D ^ \prime } $ for any morphism $ \phi : D \rightarrow D ^ \prime $; and $ \beta $) for any family of morphisms $ \alpha _ {D} : Y \rightarrow F ( D) $ satisfying $ F ( \phi ) \alpha _ {D} = \alpha _ {D ^ \prime } $ for all $ \phi : D \rightarrow D ^ \prime $ there exists a unique morphism $ \alpha : Y \rightarrow X $ such that $ \alpha _ {D} = \phi _ {D ^ \prime } \alpha $ for all $ D \in \mathop{\rm Ob} \mathfrak D $. Notation: $ \lim\limits F = ( X , \pi _ {D} ) $.

Examples of projective limits.[edit]

1) Let $ I $ be a discrete category. Then for an arbitrary functor $ F : I \rightarrow \mathfrak K $ the projective limit of the functor $ F $ coincides with the product of the family of objects $ F ( i) $, $ i \in I $( cf. Product of a family of objects in a category).

2) Let $ \mathfrak D $ be a category with two objects $ A , B $ and two non-identity morphisms $ \alpha , \beta : A \rightarrow B $. Then the limit of a functor $ F : \mathfrak D \rightarrow \mathfrak K $ is the equalizer of the pair of morphisms $ F ( \alpha ) , F ( \beta ) $( 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[edit]

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].

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