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From: Bill Dubuque on 13 Nov 2009 13:00
A <anonymous.rubbertube(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.

--Bill Dubuque
From: A on 13 Nov 2009 13:49
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.
>
> --Bill Dubuque


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.

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.
From: Bill Dubuque on 13 Nov 2009 14:48
A <anonymous.rubbertube(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.

> 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).

--Bill Dubuque
From: A on 13 Nov 2009 15:15
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.


> > 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."



> --Bill Dubuque


From: Bill Dubuque on 13 Nov 2009 16:04
A <anonymous.rubbertube(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")
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