PlanetMath: direct limit of sets

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direct limit of sets

(Example)

Let $\mathcal{A}=\lbrace A_i\mid i\in I\rbrace$ be a family of sets indexed by a non-empty set . $\mathcal{A}$ is said to be a direct family if

is a directed set,
whenever $i\le j$ in , there is a function $\phi_{ij}:A_i\to A_j$ ,
$\phi_{ii}$ is the identity function on ,
if $i\le j\le k$ , then $\phi_{jk}\circ \phi_{ij}=\phi_{ik}$ .

In the last condition, if we write $a \phi_{ij}:=\phi_{ij}(a)$ for $a\in A_i$ , then the equation can be rewritten as $\phi_{ij}\phi_{jk}=\phi_{ik}$ .

For example, the natural numbers $\mathbb{N}=\lbrace 1,2,\ldots, n,\ldots \rbrace$ can be regarded as a direct family. Here, for any $i\le j$ , $\phi_{ij}:i\to j$ is given by the natural injection $\phi_{ij}(\ell):=\ell$ for any $\ell \in i$ .

Let $\mathcal{A}$ be a direct family of sets, indexed by . Take the disjoint union of the members of $\mathcal{A}$ and call it (this can be achieved even when the members themselves have non-empty intersections, simply form the product $A_i\times \lbrace i\rbrace$ first before taking the union). Therefore, has the properties that

for any $a\in A$ , $a\in A_i$ for some $i\in I$ , and
if $a\in A_i$ and $b\in A_j$ and $i\ne j$ , then $a\ne b$ .

Define a binary relation $\sim$ on

as follows: given that $a\in A_i$ and $b\in A_j$ , $a\sim b$ iff there is

such that $\phi_{ik}(a)=\phi_{jk}(b)$ .

Proposition 1 $\sim$ on is an equivalence relation.

Proof. Clearly, $\sim$ is symmetric. By condition 2 of a direct family, $\sim$ is also reflexive. Now, suppose $a\sim b$ and $b\sim c$ with $a\in A_i$ , $b\in A_j$ and $c\in A_k$ . So there are $p,q\in I$ such that $\phi_{ip}(a)=\phi_{jp}(b)$ and $\phi_{jq}(b)=\phi_{kq}(c)$ . Since

is directed, there is $r\in I$ such that $p,q\le r$ . From this, we have $\phi_{ir}(a)=\phi_{pr}(\phi_{ip}(a))=\phi_{pr}(\phi_{jp}(b))=\phi_{jr}(b)$ . Similarly, $\phi_{kr}(c)=\phi_{qr}(\phi_{kq}(c))=\phi_{qr}(\phi_{jq}(b))$ . Hence $a\sim c$ . $\qedsymbol$

Definition. Let $\mathcal{A}$ be a direct family of sets indexed by . Let and $\sim$ be defined as above. Then the quotient $A/\sim$ is called the direct limit of the sets in $\mathcal{A}$ . The direct limit of sets is sometimes written $A_{\infty}$ , or $\varinjlim A_i$ . Elements of $A_{\infty}$ are sometimes denoted by or whenever there is no confusion.

Remarks.

This definition is consistent with the formal definition of direct limits in a category. The index , being a directed set, can be viewed as a category whose objects are elements of and morphisms defined by the partial order on .
The notation $A_{\infty}$ comes from the following fact: if $\vert I\vert=n<\infty$ , then $\varinjlim A_i\cong A_n$ . Here, $\cong$ stands for bijection.
For every $i\in I$ , there is a natural mapping $A_i\to A_{\infty}$ , given by $a\mapsto [a]_I$ . This map may be variously denoted by $\phi_{i}$ , $\phi_{i\infty}$ , or $\phi_{iI}$ .
Let be a subset of a directed set . Let $\mathcal{A}$ be a direct family indexed by and $\mathcal{A}'\subseteq \mathcal{A}$ indexed by . Form the direct limit $A'_{\infty}$ of sets in $\mathcal{A}'$ . Then there is a natural mapping $\phi_{JI}:A'_{\infty}\to A_{\infty}$ such that for any $j\in J$ , $\phi_{JI}\circ \phi_{jJ}=\phi_{jI}$ .

The dual notion of a direct limit of sets is that of an inverse limit. Instead of starting from a direct family of sets, we start with an inverse family of sets, which is defined similarly to that to of a direct family, except is a filtered set, and the mappings $\phi_{ij}:A_i\to A_j$ is defined whenever $j\le i$ . An inverse family is also known as an inverse system, or a projective system.