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

1. $I$ is a directed set,
2. whenever $i\le j$ 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 $i\le j\le k$ 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 $I$ Take the disjoint union of the members of $\mathcal{A}$ and call it $A$ (this can be achieved even when the members themselves have non-empty intersections, simply form the product $A_i\times \lbrace i\rbrace$ 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 $i\ne j$ then $a\ne b$

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 $p,q\le r$ 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 $a\sim c$

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 $\varinjlim A_i$ 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 $\varinjlim A_i\cong A_n$ 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 $J$ be a subset of a directed set $I$ Let $\mathcal{A}$ be a direct family indexed by $I$ and $\mathcal{A}'\subseteq \mathcal{A}$ indexed by $J$ 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 $I$ 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.