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# is the identity on , and
# for all .Then the pair is called a direct system over .

The direct limit of the direct system is denoted by and is defined as follows. Its underlying set is the disjoint union of the 's modulo a certain [[equivalence relation]] $\backslash sim\backslash ,$:

:

Here, if and , then if and only if there is some with and such that .Intuitively, two elements in the disjoint union are equivalent

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