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.

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

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