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I have spelled out a chunk of elementary details at Grothendieck group – For commutative monoids:
wrote out a second version of the definiton, made explicit the proof that it is all well defined and satisfies the universal property of the group completion, added remark on how the definition simplifies in the cancellative case, and wrote out the most basic examples in some detail.
In the course of this I created an entry cancellative monoid with a bare minimum of content.
I also slightly re-structured the remaining bit of the entry. The small section on $\infty$-group completion I simply removed, because that belongs to group completion where in fact the content of the paragraphed that I removed is kept in more polished form.
In view of their easy and elementary nature, might this be an example where the details of proof could be relegated to a separate page? The bare statements, which are short, could certainly remain I think.
okay, I have split off an entry Grothendieck group of a commutative monoid with the elementary construction of the group completion.
This also addresses an old query box discussion that was still sitting at Grothendieck group, which I have removed there and am appending here, just for the record:
[ age old discussion moved from the entry: ]
+– {: .query}
Urs Schreiber: the category of vector bundles is not abelian, but just Quillen exact. I added a clause to that effect above.
Generally, my feeling is that “Grothendieck group” refers to “Grothendieck group completion of a monoid”. The first definition at the Wikipedia entry on Gorthendieck group.
What is described here – the group defined by sequences – is the definition of the K-theory group in algebraic K-theory. This is described at the beginning of K-theory.
I’d think that the logic is rather that in some cases both concepts happen to coincide: the algebraic K-theory of the category of bounded complexes of vector bundles on a space happens to coincide with the Grothendieck group completion of the monoid of isomorphism classes of vector bundles.
My feeling is that the Wikipedia entry is a bit suboptimal in its second part, where it effectively calls algebraic K-theory the “Grothendieck group completion”. I’d think one shouldn’t do that in this generality.
Another thing in this context is the statement about the exact sequences. It is really the homotopy exact sequences namely the fibration sequences that count. Of course some of them can be computed by ordinary exact sequences if these are sufficiently cofibrant. That’s what the structure provided by a Waldhausen category structure provides: a realization of the homotopy exact sequences as pushouts of cofibration morphisms.
So in summary my feeling is: the entry in its current form actually tries to define algebraic K-theory and not the Grothendieck group concept, and that description with a few subtleties taken care of is currently at K-theory and should eventually be copied/moved to algebraic K-theory. The entry titled “Grothendieck group” should discuss the concept where from a monoid $A$ a group structure on $A \times A$ is defined by dividing out the equivalence relation $(a_1,b_1) \sim (a_2,b_2) \Leftrightarrow \exists k : a_1 + b_2 + k = a_2 + b_1 + k$.
But maybe I am mixed up. Let me know what you think.
Toby: Well, I'll say this: I always understood ’Grothendieck group’ in the article text above, and I never understood ’K-theory’. That may have been my fault, of course, and it may be telling that I was John's student.
Urs Schreiber: so Weibel in his book referenced below indeed says “Grothendieck group” synonomously for “algebraic K-theory group” and says just “group completion” (chapter to section 1) for what I suggested should properly be called “Grothendieck group completion”.
I added corresponding remarks to the beginning of the entry. And I created two subsections under “Definintion”. One for plain group completion. The other for algebraic K-theory groups.
Toby: The idea of the free group on an abelian monoid is a very simple algebraic idea that, at least for a cancellative monoid (so that the unit is monic and one can reasonably use the term ‘completion’) certainly predates Grothendieck. That $\mathbb{Z}$ is the group completion of $\mathbb{N}$ goes back at least to Kronecker.
Urs Schreiber: I copied that last paragraph of yours into the section above. I still think that many people will say “Grothendieck group” for this group completion. But maybe I am wrong and we should branch off another page on “free group completion”.
Colin Tan: I suggest branching off another page on “free group completion” and including the noncommutative case in there.
=–
Thanks!
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