4

On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid.

What actually does this mean formally?

Mikhail Katz
  • 47,573
Alec
  • 41
  • 4
    It means it satisfies a universal property. In this case, the Grothendieck group of $M$ is the unique (up to isomorphism) abelian group $G$ and a homomorphism $M \to G$ such that every monoid homomorphism $M \to H$ with $H$ a group factors uniquely through $M \to G$. – Zhen Lin Jun 25 '12 at 06:48
  • Paolo Aluffi has a nice (elementary) discussion of the Grothendieck group in Algebra Chapter 0. I recommend that you check it out. The exercises deals with some elementary instances of it too. – Dedalus Jun 25 '12 at 07:00
  • @Dedalus Can you please share the link of the note of Paolo Aluffi? – Babai Mar 06 '16 at 20:16

1 Answers1

8

Statements like this usually refer to universality. Note that given a monoid $M$ the Grothendieck group $G$ comes with a map of underlying monoids $s:M\to G$.

Now given $H$ an abelian group and $f:M\rightarrow H$ a map of underlying monoids. Then there exists a unique group homomorphism $g:G\to H$ making the diagram commutative, i.e $g\circ s=f$. This also makes $G$ unique up to isomorphism.

You will find universal properties of that kind at many places in mathematics.

Addendum: The following more or less reflects my personal taste and therefore might or might not be helpful. Whenever I see a universal property I don't fully understand, I convince myself that it makes sense in a simpler setting I can grasp better. So if you want to replace the words monoid by subset of $\mathbb R^n$, homomorphism by inclusion and group by closed set. What is the closed set which approximates a given arbitrary set in the best possibly way? Does this satisfy a similar universal property?

Rudy the Reindeer
  • 48,301
Simon Markett
  • 10,916
  • 1
  • 24
  • 32