Universal Property
In its simplest form, the Grothendieck group of a commutative monoid is the universal way of making that monoid into an abelian group. Let M be a commutative monoid. Its Grothendieck group N should have the following universal property: There exists a monoid homomorphism
- i:M→N
such that for any monoid homomorphism
- f:M→A
from the commutative monoid M to an abelian group A, there is a unique group homomorphism
- g:N→A
such that
- f=gi.
In the language of category theory, the functor that sends a commutative monoid M to its Grothendieck group N is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids.
Read more about this topic: Grothendieck Group
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