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 in 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 in need not be inclusion maps. Roughly speaking, the direct union of the sets Sn is obtained by regarding each in as an inclusion, and taking the union.

More formally, let $$\textstyle\coprod S_n$$ denote the disjoint union of the sets Sn:
 * $$\textstyle \coprod S_n \;=\; \displaystyle\bigcup_{n=1}^\infty \,\{n\} \times S_n \;=\; \left\{(n,x) \mid n\in\mathbb{N}\text{ and }x\in S_n\right\}.$$

Let &sim; 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 Sa and injections ia,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 ia,b are the compositions