In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects.
They are a special case of the concept of limit in category theory.
is called an inverse system of groups and morphisms over
comes equipped with natural projections πi: A → Ai which pick out the ith component of the direct product for each
The inverse limit and the natural projections satisfy a universal property described in the next section.
The inverse limit will also belong to that category.
The inverse limit can be defined abstractly in an arbitrary category by means of a universal property.
be an inverse system of objects and morphisms in a category C (same definition as above).
The inverse limit of this system is an object X in C together with morphisms πi: X → Xi (called projections) satisfying πi =
The pair (X, πi) must be universal in the sense that for any other such pair (Y, ψi) there exists a unique morphism u: Y → X such that the diagram commutes for all i ≤ j.
If it does, however, it is unique in a strong sense: given any two inverse limits X and X' of an inverse system, there exists a unique isomorphism X′ → X commuting with the projection maps.
Inverse systems and inverse limits in a category C admit an alternative description in terms of functors.
Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j if and only if i ≤ j.
An inverse system is then just a contravariant functor I → C. Let
be the category of these functors (with natural transformations as morphisms).
The inverse limit, if it exists, is defined as a right adjoint of this trivial functor.
For an abelian category C, the inverse limit functor is left exact.
If I is ordered (not simply partially ordered) and countable, and C is the category Ab of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms fij that ensures the exactness of
Specifically, Eilenberg constructed a functor (pronounced "lim one") such that if (Ai, fij), (Bi, gij), and (Ci, hij) are three inverse systems of abelian groups, and is a short exact sequence of inverse systems, then is an exact sequence in Ab.
If the ranges of the morphisms of an inverse system of abelian groups (Ai, fij) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j :
The name "Mittag-Leffler" for this condition was given by Bourbaki in their chapter on uniform structures for a similar result about inverse limits of complete Hausdorff uniform spaces.
is non-zero is obtained by taking I to be the non-negative integers, letting Ai = piZ, Bi = Z, and Ci = Bi / Ai = Z/piZ.
More generally, if C is an arbitrary abelian category that has enough injectives, then so does CI, and the right derived functors of the inverse limit functor can thus be defined.
The nth right derived functor is denoted In the case where C satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim1 on AbI to series of functors limn such that It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim.
that lim1 Ai = 0 for (Ai, fij) an inverse system with surjective transition morphisms and I the set of non-negative integers (such inverse systems are often called "Mittag-Leffler sequences").
However, in 2002, Amnon Neeman and Pierre Deligne constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim1 Ai ≠ 0.
Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if C has a set of generators (in addition to satisfying (AB3) and (AB4*)).
Barry Mitchell has shown (in "The cohomological dimension of a directed set") that if I has cardinality
This applies to the I-indexed diagrams in the category of R-modules, with R a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which limn, on diagrams indexed by a countable set, is nonzero for n > 1).
More general concepts are the limits and colimits of category theory.