User:Jim.belk/Draft:Direct union

In mathematics, the direct union is a method for constructing the nested union of a collection of objects that are not technically contained in one another. The method can be applied to sets, groups, rings, modules, topological spaces, or more generally objects from an category. Direct unions are commonly used to construct various limiting objects, such as the infinite general linear group $\operatorname {GL} (\infty ,\mathbb {F} )$ or the infinity sphere $S^{\infty }\!$.

Direct unions are a special case of the more general direct limit.

Union of a sequence

Let

$S_{1}\;{\overset {i_{1}}{\longrightarrow }}\;S_{2}\;{\overset {i_{2}}{\longrightarrow }}\;S_{3}\;{\overset {i_{3}}{\longrightarrow }}\;\cdots$

be a sequence of sets, where each i_n is an injection. This situation resembles a nested sequence of subsets

$S_{1}\;\subseteq \;S_{2}\;\subseteq \;S_{3}\;\subseteq \;\cdots$

except that the injections i_n need not be inclusion maps. Roughly speaking, the direct union of the sets S_n is obtained by regarding each i_n as an inclusion, and taking the union.

More formally, let $\textstyle \coprod S_{n}$ denote the disjoint union of the sets S_n:

$\textstyle \coprod S_{n}\;=\;\displaystyle \bigcup _{n=1}^{\infty }\,\{n\}\times S_{n}\;=\;\left\{(n,x)\mid n\in \mathbb {N} {\text{ and }}x\in S_{n}\right\}.$

Let ∼ be the equivalence relation on $\textstyle \coprod S_{n}$ defined by

$(n,x)\;\sim \;{\bigl (}n+1,\,i_{n}(x){\bigr )}$

for each $(n,x)\in \textstyle \coprod S_{n}$. Then the direct union is the set of equivalence classes

$\textstyle \left.\coprod S_{n}\;\right/\sim .$

Union of a direct system

More generally, a direct system of sets is a collection of sets

$\{S_{a}\}_{a\in {\mathcal {D}}},$

where ${\mathcal {D}}$ is a directed set, together with an injections

$i_{a,b}\colon S_{a}\to S_{b}\quad {\text{for }}a\leq b.$

Together, the sets S_a and injections i_a,b are required to form a commutative diagram, in the sense that

$i_{b,c}\circ i_{a,b}=i_{a,c}\quad {\text{for }}a\leq b\leq c.$

(For a sequence of sets, the injections i_a,b are the compositions