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From: A on 13 Nov 2009 21:17
On Nov 13, 4:04 pm, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> A <anonymous.rubbert...(a)yahoo.com> wrote:
> > On Nov 13, 2:48 pm, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> >> A <anonymous.rubbert...(a)yahoo.com> wrote:
> >>> On Nov 13, 1:00 pm, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote:
> >>>> A <anonymous.rubbert...(a)yahoo.com> wrote:
>
> >>>>> [...] Finite sets are defined as those which do not admit a proper
> >>>>> subset together with a bijection between the proper subset and the
> >>>>> whole set. Union of sets and Cartesian product of finite sets is
> >>>>> defined, and on passing to the isomorphism classes of finite sets,
> >>>>> one gets a set of isomorphism classes together with two operations,
> >>>>> addition and multiplication, coming from the union and Cartesian
> >>>>> product of sets. This set of isomorphism classes, together with
> >>>>> addition and multiplication operations, is what we usually identify as
> >>>>> the nonnegative integers (we identify the integer n with the
> >>>>> isomorphism class of finite sets with n elements). This definition is
> >>>>> precise, rigorous, effective, and standard. To define the rest of the
> >>>>> ring of integers (rather than just the nonnegative integers) one can
> >>>>> simply apply the Grothendieck group completion (which turns
> >>>>> commutative monoids to commutative groups, and commutative semirings
> >>>>> to commutative rings) to the semiring of nonnegative integers.
>
> >>>> Correction: a commutative monoid may be embedded in a group iff it is
> >>>> _cancellative_. It is grossly historically inaccurate to attribute
> >>>> the corresponding formal difference construction to Grothendieck.
> >>>> Indeed, Ore already published noncommutative generalizations in 1931
> >>>> when G was only a few years old, so it certainly predates G.
>
> >>> I don't think I said anything about anything being an embedding; the
> >>> forgetful functor from groups to monoids has a left adjoint, usually
> >>> called the Grothendieck group completion (it also has a right adjoint:
> >>> taking the group of units of the monoid). You are right that the unit
> >>> map of this adjunction--which takes a monoid M to the underlying
> >>> monoid of its Grothendieck group completion--is injective iff M is
> >>> cancellative.
>
> >> But if you apply the more general construction then you need to
> >> explicitly invoke the lemma that  cancellative => injective  since
> >> it is crucial in the above application (i.e. constructing Z from N).
> >> The point of my post was to highlight this requirement.
>
> > Sure, that's how you know that the natural numbers are actually a
> > subset of the integers.
>
> And, just like in the multiplicative analog of fraction fields of domains
> (vs. total fraction rings of commutative rings, or general localizations)..
> one often teaches the simpler injective case first.
>
>
>
> >>> The idea behind this formal difference construction is no doubt much
> >>> older than Grothendieck, but somehow the version of it defined on the
> >>> entire category of monoids has come to be named after him, perhaps
> >>> because he was the first or one of the first to use it in the study of
> >>> vector bundles and K-theory.
>
> >> It's still historically incorrect even in that more general form
> >> (as of course are many "named" theorems / constructions).
>
> > This seems very believable to me, but unfortunately, the only
> > alternative I know of is to call it this operation or this functor the
> > "group completion," as some people do; but this seems even worse to
> > me, because there is already a completely different "group
> > completion," the completion (in the sense of completing a metric
> > space) of a group equipped with some metric on it, e.g. to produce the
> > p-adic integers from integers with the p-adic metric. So even while
> > the phrase "Grothendieck group completion" probably gives Grothendieck
> > more credit that he's strictly due, it's at least unambiguous, so I
> > still prefer it to simply saying "group completion." I am afraid that
> > a totally different phrase like "formal difference construction" won't
> > be recognized by others unless I actually give the definition, at
> > which point the reader/listener will say "Aha, he means the
> > Grothendieck group completion."
>
> What's wrong with "difference group"? (analogous to "fraction field")


Again, if someone said "difference group" during a talk, I wouldn't be
sure what they were talking about unless they either defined it or
said "you know, the group completion" or something like this. Another
option would be, given a monoid M, "the free group on the monoid M,"
since functors left adjoint to a forgetful functor are usually called
free functors; but again this is open to misinterpretation ("the free
group on the monoid M" is not necessarily itself actually a free group
in the usual sense, i.e., a group in the image of the functor from
sets to groups which is left adjoint to the forgetful functor). I wish
I knew of a really good name for this construction but unfortunately I
don't think there is one.

In any case, originally I said "Grothendieck group completion" in this
thread to explain something to a poster who apparently took no
interest in the explanation, anyway; so for the purposes of this
thread I probably could have said "magically transform the monoid into
a group using the spell you learned at Hogwarts" and it would have
come to the same thing.
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