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User:Jim.belk/Draft:Direct union

From Wikipedia, the free encyclopedia

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 or the infinity sphere .

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

Union of a sequence

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Let

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

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 denote the disjoint union of the sets Sn:

Let ∼ be the equivalence relation on defined by

for each . Then the direct union is the set of equivalence classes

Union of a direct system

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

where is a directed set, together with an injections

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

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