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
Famous quotes containing the words universal and/or property:
“Necessity does everything well. In our condition of universal dependence, it seems heroic to let the petitioner be the judge of his necessity, and to give all that is asked, though at great inconvenience.”
—Ralph Waldo Emerson (1803–1882)
“For wisdom is the property of the dead,
A something incompatible with life; and power,
Like everything that has the stain of blood,
A property of the living; but no stain
Can come upon the visage of the moon
When it has looked in glory from a cloud.”
—William Butler Yeats (1865–1939)