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Homedirect limit of sets
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direct limit of sets
Let $\mathcal{A}=\{A_{i}\mid i\in I\}$ be a family of sets indexed by a nonempty set $I$. $\mathcal{A}$ is said to be a direct family if
1. $I$ is a directed set,
2. whenever $i\leq j$ in $I$, there is a function $\phi_{{ij}}:A_{i}\to A_{j}$,
3. $\phi_{{ii}}$ is the identity function on $A_{i}$,
4. if $i\leq j\leq k$, 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 $\mathbb{N}=\{1,2,\ldots,n,\ldots\}$ can be regarded as a direct family. Here, for any $i\leq j$, $\phi_{{ij}}:i\to j$ 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 $I$. Take the disjoint union of the members of $\mathcal{A}$ and call it $A$ (this can be achieved even when the members themselves have nonempty intersections, simply form the product $A_{i}\times\{i\}$ first before taking the union). Therefore, $A$ 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 $i\neq j$, then $a\neq b$.
Define a binary relation $\sim$ on $A$ as follows: given that $a\in A_{i}$ and $b\in A_{j}$, $a\sim b$ iff there is $A_{k}$ such that $\phi_{{ik}}(a)=\phi_{{jk}}(b)$.
Proposition 1.
$\sim$ on $A$ 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 $I$ is directed, there is $r\in I$ such that $p,q\leq r$. 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 $a\sim c$. ∎
Definition. Let $\mathcal{A}$ be a direct family of sets indexed by $I$. Let $A$ 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 $A_{i}$ is sometimes written $A_{{\infty}}$, or $\underrightarrow{\lim}A_{i}$. Elements of $A_{{\infty}}$ are sometimes denoted by $[a]_{I}$ or $[a]$ whenever there is no confusion.
Remarks.

This definition is consistent with the formal definition of direct limits in a category. The index $I$, being a directed set, can be viewed as a category whose objects are elements of $I$ and morphisms defined by the partial order on $I$.

The notation $A_{{\infty}}$ comes from the following fact: if $I=n<\infty$, then $\underrightarrow{\lim}A_{i}\cong A_{n}$. Here, $\cong$ stands for bijection.

Let $J$ be a subset of a directed set $I$. Let $\mathcal{A}$ be a direct family indexed by $I$ and $\mathcal{A}^{{\prime}}\subseteq\mathcal{A}$ indexed by $J$. Form the direct limit $A^{{\prime}}_{{\infty}}$ of sets in $\mathcal{A}^{{\prime}}$. Then there is a natural mapping $\phi_{{JI}}:A^{{\prime}}_{{\infty}}\to A_{{\infty}}$ 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 $I$ is a filtered set, and the mappings $\phi_{{ij}}:A_{i}\to A_{j}$ is defined whenever $j\leq i$. An inverse family is also known as an inverse system, or a projective system.
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