completion viewed functor math g mathbf abmon mathbf abgrp math property left adjoint corresponding forgetful functor math u mathbf abgrp mathbf abmon math means given morphism math phi u b math abelian monoid math math underlying abelian monoid abelian group math b math exists unique abelian group morphism math g b math example natural numbers illustrative example look grothendieck completion math n math see math g n z math pair math ab math find minimal representative math a'b' math using invariance scaling
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