Re: Further to my question on adjoints

Re: Further to my question on adjoints

[Michael Barr]

I have checked this carefully and it works.  To summarize, let F =
Q[2^{1/2}] and E = Q[2^{1/4}].  Then any power of E contains a square
whose square is a square root of 2 and any ring homomorphism between
powers of E preserves it.  (Incidentally, although it may help your
intuition to take the positive fourth of 2, the various fourth roots of 2
are indistinguishable algebraically.)  Thus any ring in EqP(E) contains a
square root of 2 (although not necessarily a fourth root).  Now F is the
equalizer of the two distinct maps E to E, while Q is the equalizer of the
two distinct maps F to F.

This now gives a counter-example for my original question.  Let C be the
category of commutative rings, F = Hom(-,E) : C ---> Set\op and U = E^{-}:
Set\op ---> C are adjoint.  If T is the resultant triple, then F ---> E
===> E is an equalizer between two values of U, while not being the
canonical equalizer.  TF = E x E and T^2F = E x E x E x E.  I haven't done
the computation, but I believe the equalizer of TF ===> T^2 is F x F.

Thanks George,

Michael


On Mon, 12 May 2008, George Janelidze wrote:

> Dear Michael,
>
> Let C be the category of commutative rings (with 1), let t be the unique
> positive real number with tttt = 2 (if I knew TeX better, I would probably
> write t^4 = 2), and E be the smallest subfield in the field of real numbers
> that contains t. Then:
>
> (a) Every power of E has exactly one element x such that xx = 2 and there
> exists y with x = yy. Let us call this x the positive square root of 2.
>
> (b) Every morphism between powers of E preserves the positive square root of
> 2.
>
> (c) Therefore every equalizer of two arrows between powers of E has an
> element x with xx = 2 (note that I am not saying anything about the
> existence of y, since y above is not determined uniquely!).
>
> (d) Therefore the field Q of rational numbers cannot be presented as an
> equalizer of two arrows between powers of E.
>
> (e) On the other hand Q can be presented as an equalizer of two arrows
> between two objects in C that are equalizers of two arrows between powers of
> E. Indeed: the equalizer of the identity morphism of E and the unique
> non-identity morphism of E is the subfield D in E generated by tt (which is
> just the square root of 2); and the equalizer of the identity morphism of D
> and the unique non-identity morphism of D is Q.
>
> (f) This also gives negative answer to the question about "internally
> complete", since no arrow of our subcategory composed with the two morphisms
> D ---> D above will give the same result.
>
> This story is of course based on the fact that there are Galois field
> extensions L/K and M/L, for which M/K is not a Galois extension.
>
> Best regards, George
>
> ----- Original Message -----
> From: "Michael Barr" <barr@math.mcgill.ca>
> To: "Categories list" <categories@mta.ca>
> Sent: Monday, May 12, 2008 2:34 PM
> Subject: categories: Further to my question on adjoints
>
>
>> In March I asked a question on adjoints, to which I have received no
>> correct response.  Rather than ask it again, I will pose what seems to be
>> a simpler and maybe more manageable question.  Suppose C is a complete
>> category and E is an object.  Form the full subcategory of C whose objects
>> are equalizers of two arrows between powers of E.  Is that category closed
>> in C under equalizers?  (Not, to be clear, the somewhat different question
>> whether it is internally complete.)
>>
>> In that form, it seems almost impossible to believe that it is, but it is
>> surprisingly hard to find an example.  When E is injective, the result is
>> relatively easy, but when I look at examples, it has turned out to be true
>> for other reasons.  Probably there is someone out there who already knows
>> an example.
>>
>> Michael
>>
>>
>>
>

re: Further to my question on adjoints

[George Janelidze]

Dear Michael,

Sorry to say, I have no time now - so, just briefly:

I know that "positive" is just an illusion and I only used it to make things
seem more obvious. Moreover, "fields" is also an illusion, since all the
rings involved are (quasi-) separable Q-algebras - and in fact one should
put things inside the dual category of G-sets, where, say, G is a finite
group that has a subgroup whose normalizer is a normal subgroup in G
different from G itself. This would imply that the phenomenon you were
looking for can even be found in a category dual to a Boolean topos.

However I still have to study what you say in the second paragraph of your
message...

Thank you for an interesting question-

George

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "George Janelidze" <janelg@telkomsa.net>
Cc: "Categories list" <categories@mta.ca>; "John F. Kennison"
<JKennison@clarku.edu>; "Bob Raphael" <raphael@alcor.concordia.ca>
Sent: Monday, May 12, 2008 9:27 PM
Subject: Re: categories: Further to my question on adjoints


> I have checked this carefully and it works.  To summarize, let F =
> Q[2^{1/2}] and E = Q[2^{1/4}].  Then any power of E contains a square
> whose square is a square root of 2 and any ring homomorphism between
> powers of E preserves it.  (Incidentally, although it may help your
> intuition to take the positive fourth of 2, the various fourth roots of 2
> are indistinguishable algebraically.)  Thus any ring in EqP(E) contains a
> square root of 2 (although not necessarily a fourth root).  Now F is the
> equalizer of the two distinct maps E to E, while Q is the equalizer of the
> two distinct maps F to F.
>
> This now gives a counter-example for my original question.  Let C be the
> category of commutative rings, F = Hom(-,E) : C ---> Set\op and U = E^{-}:
> Set\op ---> C are adjoint.  If T is the resultant triple, then F ---> E
> ===> E is an equalizer between two values of U, while not being the
> canonical equalizer.  TF = E x E and T^2F = E x E x E x E.  I haven't done
> the computation, but I believe the equalizer of TF ===> T^2 is F x F.
>
> Thanks George,
>
> Michael
>
>
> On Mon, 12 May 2008, George Janelidze wrote:
>
> > Dear Michael,
> >
> > Let C be the category of commutative rings (with 1), let t be the unique
> > positive real number with tttt = 2 (if I knew TeX better, I would
probably
> > write t^4 = 2), and E be the smallest subfield in the field of real
numbers
> > that contains t. Then:
> >
> > (a) Every power of E has exactly one element x such that xx = 2 and
there
> > exists y with x = yy. Let us call this x the positive square root of 2.
> >
> > (b) Every morphism between powers of E preserves the positive square
root of
> > 2.
> >
> > (c) Therefore every equalizer of two arrows between powers of E has an
> > element x with xx = 2 (note that I am not saying anything about the
> > existence of y, since y above is not determined uniquely!).
> >
> > (d) Therefore the field Q of rational numbers cannot be presented as an
> > equalizer of two arrows between powers of E.
> >
> > (e) On the other hand Q can be presented as an equalizer of two arrows
> > between two objects in C that are equalizers of two arrows between
powers of
> > E. Indeed: the equalizer of the identity morphism of E and the unique
> > non-identity morphism of E is the subfield D in E generated by tt (which
is
> > just the square root of 2); and the equalizer of the identity morphism
of D
> > and the unique non-identity morphism of D is Q.
> >
> > (f) This also gives negative answer to the question about "internally
> > complete", since no arrow of our subcategory composed with the two
morphisms
> > D ---> D above will give the same result.
> >
> > This story is of course based on the fact that there are Galois field
> > extensions L/K and M/L, for which M/K is not a Galois extension.
> >
> > Best regards, George
> >
> > ----- Original Message -----
> > From: "Michael Barr" <barr@math.mcgill.ca>
> > To: "Categories list" <categories@mta.ca>
> > Sent: Monday, May 12, 2008 2:34 PM
> > Subject: categories: Further to my question on adjoints
> >
> >
> >> In March I asked a question on adjoints, to which I have received no
> >> correct response.  Rather than ask it again, I will pose what seems to
be
> >> a simpler and maybe more manageable question.  Suppose C is a complete
> >> category and E is an object.  Form the full subcategory of C whose
objects
> >> are equalizers of two arrows between powers of E.  Is that category
closed
> >> in C under equalizers?  (Not, to be clear, the somewhat different
question
> >> whether it is internally complete.)
> >>
> >> In that form, it seems almost impossible to believe that it is, but it
is
> >> surprisingly hard to find an example.  When E is injective, the result
is
> >> relatively easy, but when I look at examples, it has turned out to be
true
> >> for other reasons.  Probably there is someone out there who already
knows
> >> an example.
> >>
> >> Michael
> >>
> >>
> >>
> >
>

RE: Further to my question on adjoints

[Stephen Lack]

Dear Michael,

I do not remember your original question, but here is an answer to this.
Let C be Cat^op and E be the arrow category 2.

It's easier to work in Cat itself. Then we are interested in the full
subcategory consisting of all categories X which admit a presentation 

I.2 --> J.2 --> X
    -->  

where I and J are sets, and "." is cotensor: e.g. J.2 denotes the 
coproduct of J copies of 2.

But a category admits such a presentation if and only if it is free on 
a graph, and the free categories are of course not closed under
coequalizers.

Steve.

> -----Original Message-----
> From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of 
> Michael Barr
> Sent: Monday, May 12, 2008 10:34 PM
> To: Categories list
> Subject: categories: Further to my question on adjoints
> 
> In March I asked a question on adjoints, to which I have 
> received no correct response.  Rather than ask it again, I 
> will pose what seems to be a simpler and maybe more 
> manageable question.  Suppose C is a complete category and E 
> is an object.  Form the full subcategory of C whose objects 
> are equalizers of two arrows between powers of E.  Is that 
> category closed in C under equalizers?  (Not, to be clear, 
> the somewhat different question whether it is internally complete.)
> 
> In that form, it seems almost impossible to believe that it 
> is, but it is surprisingly hard to find an example.  When E 
> is injective, the result is relatively easy, but when I look 
> at examples, it has turned out to be true for other reasons.  
> Probably there is someone out there who already knows an example.
> 
> Michael
> 
> 
> 

Re: Further to my question on adjoints

[George Janelidze]

Dear Michael,

Let C be the category of commutative rings (with 1), let t be the unique
positive real number with tttt = 2 (if I knew TeX better, I would probably
write t^4 = 2), and E be the smallest subfield in the field of real numbers
that contains t. Then:

(a) Every power of E has exactly one element x such that xx = 2 and there
exists y with x = yy. Let us call this x the positive square root of 2.

(b) Every morphism between powers of E preserves the positive square root of
2.

(c) Therefore every equalizer of two arrows between powers of E has an
element x with xx = 2 (note that I am not saying anything about the
existence of y, since y above is not determined uniquely!).

(d) Therefore the field Q of rational numbers cannot be presented as an
equalizer of two arrows between powers of E.

(e) On the other hand Q can be presented as an equalizer of two arrows
between two objects in C that are equalizers of two arrows between powers of
E. Indeed: the equalizer of the identity morphism of E and the unique
non-identity morphism of E is the subfield D in E generated by tt (which is
just the square root of 2); and the equalizer of the identity morphism of D
and the unique non-identity morphism of D is Q.

(f) This also gives negative answer to the question about "internally
complete", since no arrow of our subcategory composed with the two morphisms
D ---> D above will give the same result.

This story is of course based on the fact that there are Galois field
extensions L/K and M/L, for which M/K is not a Galois extension.

Best regards, George

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "Categories list" <categories@mta.ca>
Sent: Monday, May 12, 2008 2:34 PM
Subject: categories: Further to my question on adjoints


> In March I asked a question on adjoints, to which I have received no
> correct response.  Rather than ask it again, I will pose what seems to be
> a simpler and maybe more manageable question.  Suppose C is a complete
> category and E is an object.  Form the full subcategory of C whose objects
> are equalizers of two arrows between powers of E.  Is that category closed
> in C under equalizers?  (Not, to be clear, the somewhat different question
> whether it is internally complete.)
>
> In that form, it seems almost impossible to believe that it is, but it is
> surprisingly hard to find an example.  When E is injective, the result is
> relatively easy, but when I look at examples, it has turned out to be true
> for other reasons.  Probably there is someone out there who already knows
> an example.
>
> Michael
>
>
>

Re: Further to my question on adjoints

[Michael Barr]

No.  For example, in the category of topological abelian groups, Z is far
from injective.  Nonetheless, if you say that a group is Z-compact when it
is an equalizer of two maps between powers of Z, then an equalizer of two
maps between Z-compact abelian groups is again Z-compact.  The proof is
not direct.  As it happens, I am talking on this in our seminar tomorrow.

Even though the reals are not injective in hausdorff spaces, a space is
realcompact iff it is a closed subspace of a power of R, which turns out
to be equivalent to being an equalizer of two maps between powers of R
(that is a cokernel pair of such a closed inclusion has enough real-valued
functions to separate points) and it is clear that a closed subspace of a
realcompact space is again realcompact.  Same thing for N-compact.  In
fact, for every example I have looked at sufficiently closely.

Michael

On Mon, 12 May 2008, Eduardo Dubuc wrote:

> Consider the dual finitary question: In universal algebra in order to show
> that finitely presented objects  are closed under coequalizers it is
> essential  that a amorphism of finitely presented objects lift to a
> morphism  between the free. Is this the only way to prove it ? :
>
> " but when I look at examples, it has turned out to be true
> for other reasons."
>
> greetings  e.d.
>
>
>>
>> In March I asked a question on adjoints, to which I have received no
>> correct response.  Rather than ask it again, I will pose what seems to be
>> a simpler and maybe more manageable question.  Suppose C is a complete
>> category and E is an object.  Form the full subcategory of C whose objects
>> are equalizers of two arrows between powers of E.  Is that category closed
>> in C under equalizers?  (Not, to be clear, the somewhat different question
>> whether it is internally complete.)
>>
>> In that form, it seems almost impossible to believe that it is, but it is
>> surprisingly hard to find an example.  When E is injective, the result is
>> relatively easy, but when I look at examples, it has turned out to be true
>> for other reasons.  Probably there is someone out there who already knows
>> an example.
>>
>> Michael
>>
>>
>

Re: Further to my question on adjoints

[Eduardo Dubuc]

Consider the dual finitary question: In universal algebra in order to show
that finitely presented objects  are closed under coequalizers it is
essential  that a amorphism of finitely presented objects lift to a
morphism  between the free. Is this the only way to prove it ? :

" but when I look at examples, it has turned out to be true
 for other reasons."

greetings  e.d.


>
> In March I asked a question on adjoints, to which I have received no
> correct response.  Rather than ask it again, I will pose what seems to be
> a simpler and maybe more manageable question.  Suppose C is a complete
> category and E is an object.  Form the full subcategory of C whose objects
> are equalizers of two arrows between powers of E.  Is that category closed
> in C under equalizers?  (Not, to be clear, the somewhat different question
> whether it is internally complete.)
>
> In that form, it seems almost impossible to believe that it is, but it is
> surprisingly hard to find an example.  When E is injective, the result is
> relatively easy, but when I look at examples, it has turned out to be true
> for other reasons.  Probably there is someone out there who already knows
> an example.
>
> Michael
>
>

Further to my question on adjoints

[Michael Barr]

In March I asked a question on adjoints, to which I have received no
correct response.  Rather than ask it again, I will pose what seems to be
a simpler and maybe more manageable question.  Suppose C is a complete
category and E is an object.  Form the full subcategory of C whose objects
are equalizers of two arrows between powers of E.  Is that category closed
in C under equalizers?  (Not, to be clear, the somewhat different question
whether it is internally complete.)

In that form, it seems almost impossible to believe that it is, but it is
surprisingly hard to find an example.  When E is injective, the result is
relatively easy, but when I look at examples, it has turned out to be true
for other reasons.  Probably there is someone out there who already knows
an example.

Michael