Grothendieck Group - Universal Property

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.