Inmathematics, aninverse limit (also called aprojective limit) is a construction that allows one to "glue together" several relatedobjects, the precise gluing process being specified bymorphisms between the objects. Inverse limits can be defined in anycategory, although their existence depends on the category that is considered. They are a special case of the concept of alimit in category theory.

By working in thedual category—that is, by reversing the arrows—an inverse limit becomes adirect limit orinductive limit, and alimit becomes acolimit.

We start with the definition of aninverse system (or projective system) ofgroups andhomomorphisms. Let${\displaystyle (I,\le )}$ be adirectedposet (not all authors requireI to be directed). Let (A_i)_i∈I be afamily of groups and suppose we have a family of homomorphisms${\displaystyle f_{ij}:A_j\to A_i}$ for all${\displaystyle i\le j}$ (note the order) with the following properties:

1. ${\displaystyle f_{ii}}$ is the identity on${\displaystyle A_i}$,
2. ${\displaystyle f_{ik}=f_{ij}\circ f_{jk}\text{for all }i\le j\le k.}$

Then the pair${\displaystyle ((A_i{)}_{i\in I},(f_{ij}{)}_{i\le j\in I})}$ is called aninverse system of groups and morphisms over${\displaystyle I}$, and the morphisms${\displaystyle f_{ij}}$ are called the transition morphisms of the system.

Theinverse limit of the inverse system${\displaystyle ((A_i{)}_{i\in I},(f_{ij}{)}_{i\le j\in I})}$ is thesubgroup of thedirect product of the⁠${\displaystyle A_i}$⁠'s defined as

${\displaystyle A=\underset{i\in I}{\underleftarrow{\lim }}{A}_i=\left\{\overrightarrow{a}\in \prod _{i\in I}{A}_i|a_i=f_{ij}(a_j)\text{ for all }i\le j\text{ in }I\right\}.}$

The definition above of an inverse system implies, that${\displaystyle A}$ is closed under pointwise multiplication, and therefore a group, since

${\displaystyle f_{i,j}(a_j\cdot b_j)=f_{i,j}(a_j)\cdot f_{i,j}(b_j)=a_i\cdot b_i}$

for all⁠⁠ and every⁠${\displaystyle \overrightarrow{a},\overrightarrow{b}\in A}$⁠

The inverse limit${\displaystyle A}$ comes equipped withnatural projectionsπ_i:A →A_i which pick out theith component of the direct product for each${\displaystyle i}$ in${\displaystyle I}$. The inverse limit and the natural projections satisfy auniversal property described in the next section.

This same construction may be carried out if the${\displaystyle A_i}$'s aresets,semigroups,topological spaces,rings,modules (over a fixed ring),algebras (over a fixed ring), etc., and thehomomorphisms are morphisms in the correspondingcategory. The inverse limit will also belong to that category.^[1] More generally, this construction applies when the⁠${\displaystyle A_i}$⁠ belong to avariety in the sense ofuniversal algebra, that is, a type of algebraic structures, whose axioms are unconditional (fields do not form an algebra, since zero does not have amultiplicative inverse).

The inverse limit can be defined abstractly in an arbitrarycategory by means of auniversal property. Let$(X_i,f_{ij})$ be an inverse system of objects andmorphisms in a categoryC (same definition as above). Theinverse limit of this system is an objectX inC together with morphismsπ_i:X →X_i (calledprojections) satisfyingπ_i =${\displaystyle f_{ij}}$ ∘π_j for alli ≤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 morphismu:Y →X such that the diagram

commutes for alli ≤j. The inverse limit is often denoted

${\displaystyle X=\underleftarrow{\lim }{X}_i}$

with the inverse system$(X_i,f_{ij})$ and the canonical projections${\displaystyle {\pi }_i}$ 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 limitsX andX' of an inverse system, there exists auniqueisomorphismX′ →X commuting with the projection maps.

Inverse systems and inverse limits in a categoryC admit an alternative description in terms offunctors. Any partially ordered setI can be considered as asmall category where the morphisms consist of arrowsi →jif and only ifi ≤j. An inverse system is then just acontravariant functorI →C. Let${\displaystyle C^{I^{\mathrm{o}\mathrm{p}}}}$ be the category of these functors (withnatural transformations as morphisms). An objectX ofC can be considered a trivial inverse system, where all objects are equal toX and all arrow are the identity ofX. This defines a "trivial functor" fromC to${\displaystyle C^{I^{\mathrm{o}\mathrm{p}}}.}$ The inverse limit, if it exists, is defined as aright adjoint of this trivial functor.

• The ring ofp-adic integers is the inverse limit of the rings${\displaystyle \mathbb{Z}/p^n\mathbb{Z}}$ (seemodular arithmetic) with the index set being thenatural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers${\displaystyle (n_1,n_2,\dots )}$ such that each element of the sequence "projects" down to the previous ones, namely, that${\displaystyle n_i\equiv n_j\text{ mod }{p}^i}$ whenever The natural topology on thep-adic integers is the one implied here, namely theproduct topology withcylinder sets as the open sets.
• Thep-adic solenoid is the inverse limit of the topological groups${\displaystyle \mathbb{R}/p^n\mathbb{Z}}$ 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${\displaystyle (x_1,x_2,\dots )}$ such that each element of the sequence "projects" down to the previous ones, namely, that${\displaystyle x_i\equiv x_j\text{ mod }{p}^i}$ whenever Its elements are exactly of form${\displaystyle n+r}$, where${\displaystyle n}$ is ap-adic integer, and${\displaystyle r\in [0,1)}$ is the "remainder".
• The ring${\displaystyle R[[t]]}$ offormal power series over a commutative ringR can be thought of as the inverse limit of the rings${\displaystyle R[t]/t^{n}R[t]}$, indexed by the natural numbers as usually ordered, with the morphisms from${\displaystyle R[t]/t^{n+j}R[t]}$ to${\displaystyle R[t]/t^{n}R[t]}$ given by the natural projection.
• Pro-finite groups are defined as inverse limits of (discrete) finite groups.
• Let the index setI of an inverse system (X_i,${\displaystyle f_{ij}}$) have agreatest elementm. Then the natural projectionπ_m:X →X_m is an isomorphism.
• In thecategory 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 non-empty finite sets is non-empty. This is a generalization ofKőnig's lemma in graph theory and may be proved withTychonoff's theorem, viewing the finite sets as compact discrete spaces, and then applying thefinite intersection property characterization of compactness.
• In thecategory of topological spaces, every inverse system has an inverse limit. It is constructed by placing theinitial topology (with respect to the projection maps into the constituent spaces of the inverse system) on the underlying set-theoretic inverse limit. This is known as thelimit topology.
  • The set of infinitestrings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces arediscrete, the limit space istotally disconnected. This is one way of realizing thep-adic numbers and theCantor set (as infinite strings).

For anabelian categoryC, the inverse limit functor

${\displaystyle \underleftarrow{\lim }:C^I\to C}$

isleft exact. IfI is ordered (not simply partially ordered) andcountable, andC is the categoryAb of abelian groups, the Mittag-Leffler condition is a condition on the transition morphismsf_ij that ensures the exactness of${\displaystyle \underleftarrow{\lim }}$. Specifically,Eilenberg constructed a functor

${\displaystyle \underleftarrow{\lim }{}^1:{\mathrm{Ab}}^I\to \mathrm{Ab}}$

(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

${\displaystyle 0\to A_i\to B_i\to C_i\to 0}$

is ashort exact sequence of inverse systems, then

${\displaystyle 0\to \underleftarrow{\lim }{A}_i\to \underleftarrow{\lim }{B}_i\to \underleftarrow{\lim }{C}_i\to \underleftarrow{\lim }{}^1{A}_i}$

is an exact sequence inAb.

If the ranges of the morphisms of an inverse system of abelian groups (A_i,f_ij) arestationary, that is, for everyk there existsj ≥k such that for alli ≥j :${\displaystyle f_{kj}(A_j)=f_{ki}(A_i)}$ one says that the system satisfies theMittag-Leffler condition.

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. Mittag-Leffler used a similar argument in the proof ofMittag-Leffler's theorem.

The following situations are examples where the Mittag-Leffler condition is satisfied:

• a system in which the morphismsf_ij are surjective
• a system of finite-dimensionalvector spaces or finite abelian groups or modules of finitelength orArtinian modules.

An example where${\displaystyle \underleftarrow{\lim }{}^1}$ is non-zero is obtained by takingI to be the non-negativeintegers, lettingA_i =pⁱZ,B_i =Z, andC_i =B_i /A_i =Z/pⁱZ. Then

${\displaystyle \underleftarrow{\lim }{}^1{A}_i={\mathbf{Z}}_p/\mathbf{Z}}$

whereZ_p denotes thep-adic integers.

More generally, ifC is an arbitrary abelian category that hasenough injectives, then so doesC^I, and the rightderived functors of the inverse limit functor can thus be defined. Thenth right derived functor is denoted

${\displaystyle R^n\underleftarrow{\lim }:C^I\to C.}$

In the case whereC satisfiesGrothendieck's axiom(AB4*),Jan-Erik Roos generalized the functor lim¹ onAb^I to series of functors limⁿ such that

${\displaystyle \underleftarrow{\lim }{}^n\cong R^n\underleftarrow{\lim }.}$

It was thought for almost 40 years that Roos had proved (inSur les foncteurs dérivés de lim. Applications.) that lim¹A_i = 0 for (A_i,f_ij) an inverse system with surjective transition morphisms andI the set of non-negative integers (such inverse systems are often called "Mittag-Leffler sequences"). However, in 2002,Amnon Neeman andPierre Deligne constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim¹A_i ≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct ifC 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 ifI hascardinality${\displaystyle {\mathrm{\aleph }}_d}$ (thedthinfinite cardinal), thenRⁿlim is zero for alln ≥d + 2. This applies to theI-indexed diagrams in the category ofR-modules, withR 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).

Thecategorical dual of an inverse limit is adirect limit (or inductive limit). More general concepts are thelimits 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.

• Bourbaki, Nicolas (1989),Algebra I, Springer,ISBN 978-3-540-64243-5,OCLC 40551484
• Bourbaki, Nicolas (1989),General topology: Chapters 1-4, Springer,ISBN 978-3-540-64241-1,OCLC 40551485
• Mac Lane, Saunders (September 1998),Categories for the Working Mathematician (2nd ed.), Springer,ISBN 0-387-98403-8
• Mitchell, Barry (1972), "Rings with several objects",Advances in Mathematics,8:1–161,doi:10.1016/0001-8708(72)90002-3,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, Jan-Erik (1961), "Sur les foncteurs dérivés de lim. Applications",C. R. Acad. Sci. Paris,252:3702–3704,MR 0132091
• Roos, Jan-Erik (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 ofWeibel, Charles A. (1994),An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press,ISBN 978-0-521-55987-4,MR 1269324,OCLC 36131259

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