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. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory.
By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or inductive limit, and a limit becomes a colimit.
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Transcription
Formal definition
Algebraic objects
We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let be a directed poset (not all authors require I to be directed). Let (A_{i})_{i∈I} be a family of groups and suppose we have a family of homomorphisms for all (note the order) with the following properties:
 is the identity on ,
Then the pair is called an inverse system of groups and morphisms over , and the morphisms are called the transition morphisms of the system.
We define the inverse limit of the inverse system as a particular subgroup of the direct product of the 's:
The inverse limit comes equipped with natural projections π_{i}: A → A_{i} which pick out the ith component of the direct product for each in . The inverse limit and the natural projections satisfy a universal property described in the next section.
This same construction may be carried out if the 's are sets,^{[1]} semigroups,^{[1]} topological spaces,^{[1]} rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding category. The inverse limit will also belong to that category.
General definition
The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let 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 π_{i}: X → X_{i} (called projections) satisfying π_{i} = ∘ π_{j} for all i ≤ j. The pair (X, π_{i}) must be universal in the sense that for any other such pair (Y, ψ_{i}) there exists a unique morphism u: Y → X such that the diagram
commutes for all i ≤ j. The inverse limit is often denoted
with the inverse system being understood.
In some categories, the inverse limit of certain inverse systems does not exist. 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). An object X of C can be considered a trivial inverse system, where all objects are equal to X and all arrow are the identity of X. This defines a "trivial functor" from C to The inverse limit, if it exists, is defined as a right adjoint of this trivial functor.
Examples
 The ring of padic integers is the inverse limit of the rings (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers such that each element of the sequence "projects" down to the previous ones, namely, that whenever The natural topology on the padic integers is the one implied here, namely the product topology with cylinder sets as the open sets.
 The padic solenoid is the inverse limit of the topological groups with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of real numbers such that each element of the sequence "projects" down to the previous ones, namely, that whenever Its elements are exactly of form , where is a padic integer, and is the "remainder".
 The ring of formal power series over a commutative ring R can be thought of as the inverse limit of the rings , indexed by the natural numbers as usually ordered, with the morphisms from to given by the natural projection.
 Profinite groups are defined as inverse limits of (discrete) finite groups.
 Let the index set I of an inverse system (X_{i}, ) have a greatest element m. Then the natural projection π_{m}: X → X_{m} is an isomorphism.
 In the category of sets, every inverse system has an inverse limit, which can be constructed in an elementary manner as a subset of the product of the sets forming the inverse system. The inverse limit of any inverse system of nonempty finite sets is nonempty. This is a generalization of Kőnig's lemma in graph theory and may be proved with Tychonoff's theorem, viewing the finite sets as compact discrete spaces, and then applying the finite intersection property characterization of compactness.
 In the category of topological spaces, every inverse system has an inverse limit. It is constructed by placing the initial topology on the underlying settheoretic inverse limit. This is known as the limit topology.
 The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are discrete, the limit space is totally disconnected. This is one way of realizing the padic numbers and the Cantor set (as infinite strings).
Derived functors of the inverse limit
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 MittagLeffler condition is a condition on the transition morphisms f_{ij} that ensures the exactness of . Specifically, Eilenberg constructed a functor
(pronounced "lim one") such that if (A_{i}, f_{ij}), (B_{i}, g_{ij}), and (C_{i}, h_{ij}) are three inverse systems of abelian groups, and
is a short exact sequence of inverse systems, then
is an exact sequence in Ab.
MittagLeffler condition
If the ranges of the morphisms of an inverse system of abelian groups (A_{i}, f_{ij}) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j : one says that the system satisfies the MittagLeffler condition.
The name "MittagLeffler" 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. MittagLeffler used a similar argument in the proof of MittagLeffler's theorem.
The following situations are examples where the MittagLeffler condition is satisfied:
 a system in which the morphisms f_{ij} are surjective
 a system of finitedimensional vector spaces or finite abelian groups or modules of finite length or Artinian modules.
An example where is nonzero is obtained by taking I to be the nonnegative integers, letting A_{i} = p^{i}Z, B_{i} = Z, and C_{i} = B_{i} / A_{i} = Z/p^{i}Z. Then
where Z_{p} denotes the padic integers.
Further results
More generally, if C is an arbitrary abelian category that has enough injectives, then so does C^{I}, 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*), JanErik Roos generalized the functor lim^{1} on Ab^{I} to series of functors lim^{n} such that
It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications. ) that lim^{1} A_{i} = 0 for (A_{i}, f_{ij}) an inverse system with surjective transition morphisms and I the set of nonnegative integers (such inverse systems are often called "MittagLeffler 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 lim^{1} A_{i} ≠ 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 (the dth infinite cardinal), then R^{n}lim is zero for all n ≥ d + 2. This applies to the Iindexed diagrams in the category of Rmodules, 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 lim^{n}, on diagrams indexed by a countable set, is nonzero for n > 1).
Related concepts and generalizations
The categorical dual of an inverse limit is a direct limit (or inductive limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: inverse limits are a class of limits, while direct limits are a class of colimits.
Notes
 ^ ^{a} ^{b} ^{c} John Rhodes & Benjamin Steinberg. The qtheory of Finite Semigroups. p. 133. ISBN 9780387097800.
References
 Bourbaki, Nicolas (1989), Algebra I, Springer, ISBN 9783540642435, OCLC 40551484
 Bourbaki, Nicolas (1989), General topology: Chapters 14, Springer, ISBN 9783540642411, OCLC 40551485
 Mac Lane, Saunders (September 1998), Categories for the Working Mathematician (2nd ed.), Springer, ISBN 0387984038
 Mitchell, Barry (1972), "Rings with several objects", Advances in Mathematics, 8: 1–161, doi:10.1016/00018708(72)900023, MR 0294454
 Neeman, Amnon (2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne)", Inventiones Mathematicae, 148 (2): 397–420, doi:10.1007/s002220100197, MR 1906154
 Roos, JanErik (1961), "Sur les foncteurs dérivés de lim. Applications", C. R. Acad. Sci. Paris, 252: 3702–3704, MR 0132091
 Roos, JanErik (2006), "Derived functors of inverse limits revisited", J. London Math. Soc., Series 2, 73 (1): 65–83, doi:10.1112/S0024610705022416, MR 2197371
 Section 3.5 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 9780521559874. MR 1269324. OCLC 36131259.