Limes (category theory)

In algebra or, more generally, in category theory , the projective limes (or inverse limes or simply limes ) is a construction that can be used to connect different structures that belong together in a certain way. The result of this connection process is mainly determined by images between these structures.

Projective limits for sets and simple algebraic structures

The following construction defines the limit for sets or any algebraic structures that are defined with the help of limits ( products , end objects , difference kernels ). Groups are treated as an example .

Given a semi-ordered amount , for each group and for each two indices with a group homomorphism ${\displaystyle (I,>)}$ ${\displaystyle i\in I}$ ${\displaystyle X_{i}}$ ${\displaystyle i,j\in I}$ ${\displaystyle i>j}$

${\displaystyle f_{ij}\colon X_{i}\to X_{j}.}$

These homomorphisms are also compatible in the sense that : ${\displaystyle i>j>k}$

${\displaystyle f_{ik}=f_{jk}\circ f_{ij}}$

("To get from to , you can also take a detour "). ${\displaystyle i}$ ${\displaystyle k}$ ${\displaystyle j}$

The projective limit is the set of all families with with the property ${\displaystyle \varprojlim _{i\in I}X_{i}}$ ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle x_{i}\in X_{i}}$

${\displaystyle f_{ij}(x_{i})\,=\,x_{j}}$for . ${\displaystyle i>j}$

Through the component-wise definition of its link via the links in the components , it becomes a group. ${\displaystyle X_{i}}$ ${\displaystyle \varprojlim _{i\in I}X_{i}}$

The universal quality

The projective limit together with the homomorphisms ${\displaystyle \varprojlim _{i\in I}X_{i}}$

${\displaystyle \mathrm {pr} _{i}\colon \varprojlim _{i\in I}X_{i}\to X_{i},\quad (x_{j})_{j\in I}\mapsto x_{i},}$

the canonical projections , has the following universal property :

For each group and homomorphisms , for which applies to all , there is a clearly determined homomorphism , so that applies. ${\displaystyle T}$ ${\displaystyle t_{i}\colon T\to X_{i}}$ ${\displaystyle t_{j}=f_{ij}\circ t_{i}}$ ${\displaystyle i>j}$ ${\displaystyle c\colon T\to \varprojlim _{i\in I}X_{i}}$ ${\displaystyle t_{i}=\mathrm {pr} _{i}\circ c}$

Commutative diagram for the definition of the Limes in category theory

Projective limits in any category

With the help of the concept of the projective limit for sets, one can define projective limits in any category: If objects of a category and transition morphisms are given, the limit of this projective system is characterized by a natural equivalence ${\displaystyle X_{i}}$ ${\displaystyle C}$ ${\displaystyle f_{i,j}}$

${\displaystyle \operatorname {Hom} _{C}(T,\lim X_{i})=\lim \operatorname {Hom} _{C}(T,X_{i})}$

of functors in ; the Limes on the right is the already defined Limes term for quantities. The limit defined in this way fulfills the analogous universal property. ${\displaystyle T}$

For "simple" algebraic structures such as vector spaces , groups or rings , this Limes term agrees with the set-based one defined above.

However, there are categories in which projective limits do not exist, for example the category of finite Abelian groups: Let it be the projective system ${\displaystyle (X_{i},f_{i,j})}$

${\displaystyle \ldots \to (\mathbb {Z} /2\mathbb {Z} )^{3}\to (\mathbb {Z} /2\mathbb {Z} )^{2}\to \mathbb {Z} /2\mathbb {Z} }$

with the projection onto the first factors as transition images. For is ${\displaystyle T:=\mathbb {Z} /2\mathbb {Z} }$

${\displaystyle \lim \operatorname {Hom} (T,X_{i})}$

infinite, so not right away

${\displaystyle \operatorname {Hom} (T,L)}$

for some finite Abelian group . ${\displaystyle L}$

Examples

• There are limits in the category of topological spaces : The set-based limit was constructed as a subset of the Cartesian product. If one provides the product with the product topology and the quantity limits with the subspace topology, one obtains the categorical limit. If all are compact and Hausdorff-like , then the projective Limes is also compact and Hausdorff-like. ${\displaystyle A_{i}}$ ${\displaystyle A}$
• Every compact topological group is a projective limit of compact Lie groups.
• For the ring of p -adic integers is the projective limit of the remainder class rings , where the semi-ordered index set is provided with the natural order and the morphisms are the remainder class maps. The natural topology on is that of the discrete topology on the induced product topology, and is dense in . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle X_{i}\;:=\;\mathbb {Z} /p^{i}}$ ${\displaystyle I\;:=\;\mathbb {N} }$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle \mathbb {Z} /p^{i}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} _{p}}$
• → Main article : Profinite number

The pro-finite completion of the ring of integers is the projective limit of the remainder class rings , whereby the index set is provided with the partial order of the divisibility and the morphisms are the remainder class maps. More precisely: If with , then the remainder class mappings are a compatible system of homomorphisms as above. turns out to be the direct product (addition and multiplication work component-wise - the latter with zero divisors). The natural topology on is that of the discrete topology on the induced product topology, and is dense in . ${\displaystyle {\hat {\mathbb {Z} }}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle X_{m}\;:=\;\mathbb {Z} /m}$ ${\displaystyle I\;:=\;\mathbb {N} }$ ${\displaystyle m,n\in \mathbb {N} }$ ${\displaystyle m\mid n}$ ${\displaystyle f_{nm}\colon \mathbb {Z} /n\to \mathbb {Z} /m}$ ${\displaystyle {\hat {\mathbb {Z} }}}$ ${\displaystyle \prod _{p\in \mathbb {P} }\mathbb {Z} _{p}}$
${\displaystyle {\hat {\mathbb {Z} }}}$ ${\displaystyle \mathbb {Z} /m}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle {\hat {\mathbb {Z} }}}$

Proof of the tightness of in ${\displaystyle \mathbb {Z} }$ ${\displaystyle {\hat {\mathbb {Z} }}}$

The prime numbers are numbered for the purposes of the proof: . The embedding throws an integer in each factor space in the place : with Be an element . For each is an -adic integer. The approximating sequence is with . A sequence term approximates with the approximation quality if the following congruences hold for simultaneous. This is feasible because the modules are coprime pairs. For each and there is a goodness of approximation such that . The component can therefore be approximated arbitrarily, namely to an exact degree. Hence the sequence converges for against . ■ ${\displaystyle \{p_{i}\mid i\in \mathbb {N} \}:=\mathbb {P} }$ ${\displaystyle \iota \colon \mathbb {Z} \to {\hat {\mathbb {Z} }}}$ ${\displaystyle m}$ ${\displaystyle \mathbb {Z} _{p_{i}}}$ ${\displaystyle m}$       ${\displaystyle \iota (m)=(x_{i})_{i\in \mathbb {N} }}$ ${\displaystyle x_{i}:=m\;\forall i\in \mathbb {N} .}$ ${\displaystyle x=(x_{i})_{i\in \mathbb {N} }}$ ${\displaystyle {\hat {\mathbb {Z} }}}$ ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle x_{i}=:\sum _{\nu =0}^{\infty }{x_{i,\nu }\,p_{i}^{\nu }}\;\in \;\mathbb {Z} _{p_{i}}}$ ${\displaystyle p_{i}}$ ${\displaystyle (y_{n})_{n\geq 1}}$ ${\displaystyle y_{n}\in \mathbb {Z} }$ ${\displaystyle y_{n}}$ ${\displaystyle x}$ ${\displaystyle n}$ ${\displaystyle 1,\ldots ,i,\ldots ,n}$       ${\displaystyle {\begin{array}{llll}y_{n}&\equiv \sum _{\nu =0}^{n-1}&{x_{1,\nu }\,p_{1}^{\nu }}&\operatorname {mod} p_{1}^{n}\\y_{n}&\equiv \sum _{\nu =0}^{n-i}&{x_{i,\nu }\,p_{i}^{\nu }}&\operatorname {mod} p_{i}^{n-i+1}\\y_{n}&\equiv &x_{n,0}&\operatorname {mod} p_{n}\end{array}}}$ ${\displaystyle p_{i}^{n-i+1}}$ ${\displaystyle i}$ ${\displaystyle m}$ ${\displaystyle n\geq i+m}$ ${\displaystyle y_{n}\;\equiv \;\sum _{\nu =0}^{m}{x_{i,\nu }\,p_{i}^{\nu }}\;\operatorname {mod} p_{i}^{m+1}}$ ${\displaystyle x_{i}}$ ${\displaystyle \operatorname {mod} p_{i}^{m+1}}$ ${\displaystyle (y_{n})_{n\geq 1}}$ ${\displaystyle n\to \infty }$ ${\displaystyle \lim _{n\to \infty }y_{n}=x}$

• For an arbitrary Galois field extension that is Galois isomorphic to the projective limit the Galois , wherein all finite and Galois intermediate extensions of passes through the semi-ordered index set, the amount of this intermediate body with the inclusion order, and the morphism for is given by (the limitation of an automorphism on the smaller body). If one considers all of them as discrete topological groups , then a product topology is induced which is called the Krull topology . Since all finite extensions of a finite field are cyclic , the Galois group of the algebraic closure of a finite field is isomorphic to (as an additive group). ${\displaystyle E/K}$ ${\displaystyle G(E/K)}$ ${\displaystyle G(L/K)}$ ${\displaystyle L}$ ${\displaystyle E/K}$ ${\displaystyle M/L}$ ${\displaystyle f_{M,L}:G(M/K)\to G(L/K),s\to s\mid _{L}}$ ${\displaystyle G(L/K)}$ ${\displaystyle G(E/K)}$
  ${\displaystyle {\hat {\mathbb {Z} }}}$
• Kolmogorov's extension theorem : Given a non-empty index set and Borel spaces for . Let be the set of all non-empty, finite subsets of . If a projective family of probability measures is given, then there is a uniquely determined probability measure on the measurement space for which applies to each . The projection on the components of the index set denotes . One then writes and calls the probability measure the projective limit. ${\displaystyle I}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle i\in I}$ ${\displaystyle {\mathcal {E}}(I)}$ ${\displaystyle I}$ ${\displaystyle (P_{J})_{J\in {\mathcal {E}}(I)}}$ ${\displaystyle P}$ ${\displaystyle (\Omega ,{\mathcal {A}}):=\left(\prod _{i\in I}\Omega _{i},\bigotimes _{i\in I}{\mathcal {A}}_{i}\right),}$ ${\displaystyle P_{J}=P\circ (\pi _{J}^{I})^{-1}}$ ${\displaystyle J\in {\mathcal {E}}(I)}$ ${\displaystyle \pi _{J}^{I}}$ ${\displaystyle J}$ ${\displaystyle \varprojlim _{J\uparrow I}P_{J}=:P}$ ${\displaystyle P}$

Limits with index categories

In generalizing the limit for subordinate index sets, one can consider limits for any index categories:

Let it be a small category , any category and a functor. Then a limit of is a representing object for the functor ${\displaystyle I}$ ${\displaystyle C}$ ${\displaystyle X\colon I\to C}$ ${\displaystyle X}$

${\displaystyle C^{\mathrm {op} }\to \mathrm {(Mengen)} ,\quad T\mapsto \mathrm {Mor} _{\mathbf {Mor} (I,C)}(\mathrm {const} _{T},X);}$

denote the constant functor with value . The Limes is therefore an object together with a natural equivalence ${\displaystyle \mathrm {const} _{T}}$ ${\displaystyle I\to C}$ ${\displaystyle T}$ ${\displaystyle L}$

${\displaystyle \mathrm {Mor} _{C}(T,L)=\mathrm {Mor} _{\mathbf {Mor} (I,C)}(\mathrm {const} _{T},X)}$

of functors in . ${\displaystyle T}$

From this natural equivalence, one also obtains the canonical projections (as the equivalent of on the left). ${\displaystyle T=L}$ ${\displaystyle L\to X(i)}$ ${\displaystyle id_{L}}$

The natural equivalence is essentially only a compact notation of the universal property: Morphisms in a Limes object correspond to compatible systems of morphisms in the individual objects, just like in the special case of partially ordered index sets.

This Limes term includes some other universal constructions as special cases:

${\displaystyle I}$	universal construction
Any number of objects, just identities	product
${\displaystyle \varnothing }$	End object
	Difference core
	Fiber product

If the index category has an initial object , the limit is the same . ${\displaystyle A}$ ${\displaystyle X(A)}$

References and comments

1. ↑ Some authors define the projective Limes only in the case where it is directed . For the basic properties of Limes in abstract categories presented in this article, this requirement is unnecessary. However, it may be necessary for topological questions. Jon Brugger: Pro-finite groups Remark 3.5 ${\displaystyle (I,>)}$

See also

• Kolimes