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Hatto von Aquitanien
science forum Guru
Joined: 19 Nov 2005
Posts: 410
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 Posted: Mon Jul 03, 2006 11:37 pm Post subject:
Re: What are all the (important) *-morphisms?
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Robert Low wrote:
| Quote: | Lee Rudolph wrote:
I think we're witnessing an instance of polymorphism, but I am not
quite sure who "Hatto"'s previous avatars have been.
If you have a few minutes which you don't value, you might check out
his earlier 'contributions' in other groups, a few of which are:
prussian blue formation in alt.sci.physics
witch trials in history in alt.religion.druid
special relativity as special case of perceptual universe in
sci.physics.particle
He's developed a pretty impressive track record in a rather
short time.
See the sig. |
--
Nil conscire sibi - Arms of the Golden Hind |
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Gene Ward Smith
science forum Guru
Joined: 08 Jul 2005
Posts: 409
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| Posted: Mon Jul 03, 2006 11:15 pm Post subject:
Re: What are all the (important) *-morphisms?
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Arturo Magidin wrote:
| Quote: | Contrary to the categorical pronouncement (no pun intended) of
Mariano, there are many branches of mathematics and many authors for
whom "epimorphism" is used as a synonym for "surjective." This is
inherited from its use as such a synonym by Bourbaki.
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According to this, Bourbaki invented the term:
http://en.wikipedia.org/wiki/Epimorphism#Terminology
But these days it is best to be clear what you mean by it when in a
category where it doesn't equate to surjective. Happily, often it does. |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Mon Jul 03, 2006 11:03 pm Post subject:
Re: What are all the (important) *-morphisms?
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In article <AImdnSvJCLArATTZnZ2dnUVZ_qOdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
| Quote: | mariano.suarezalvarez@gmail.com wrote:
Hatto von Aquitanien wrote:
quote
book="Elements of Abstract Algebra"
author="Allan Clark"
A ring homomorphism ?:R->R' is called:
(1) a monomorphism if ? is one to one;
(2) an epimorphism if ? is onto;
(3) an isomorphism if ? is a one-to-one correspondence;
(4) an endomorphism if R'=R;
(5) an automorphism if R'=R and ? is a one-to-one correspondence;
/quote
I'm not familiar with that book, but item (2) is wrong with respect to
the definitions in use today.
A ring epimorphism (ie, an epimorphism in the category of rings)
phi:R->R' is not necessarily surjective as a map of sets. For example,
the inclusion map of the ring ZZ of the integers into the ring ZZ[1/2]
of rational fractions with denominator a power of two is an
epimorphism yet it is obviously not surjective.
Not surjective with respect to ZZ[1/2]? If being surjective is not the
property that leads to its classification as an epimorphism, then what is
the essential property. I'm not challenging your claim. I'm merely
seeking clarification.
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An epimorphism is a right-cancellable morphism.
In a given category C, a morphism f:A->B is an epimorphism if and only
if for evey object D of C, and every pair of maps g,h:B->C, if the
compositions gf and hf are equal, then it must be the case that g=h.
When the category is set based (the objects are sets, and the
morphisms are maps of the underlying sets), then any surjective map
f:A->B is necessarily an epimorphism from A to B. However, there are
categories where the converse does not hold.
Mariano mentioned one example, from the category of rings (or of
commutative rings). In fact, the most common example is the natural
embedding of Z into Q, or more generally of a domain into its field of
fractions. They are easily shown to be epimorphisms (in the
right-cancellable sense).
Contrary to the categorical pronouncement (no pun intended) of
Mariano, there are many branches of mathematics and many authors for
whom "epimorphism" is used as a synonym for "surjective." This is
inherited from its use as such a synonym by Bourbaki.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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Gene Ward Smith
science forum Guru
Joined: 08 Jul 2005
Posts: 409
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| Posted: Mon Jul 03, 2006 10:44 pm Post subject:
Re: What are all the (important) *-morphisms?
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Hatto von Aquitanien wrote:
| Quote: | Not surjective with respect to ZZ[1/2]? If being surjective is not the
property that leads to its classification as an epimorphism, then what is
the essential property. I'm not challenging your claim. I'm merely
seeking clarification.
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See
http://en.wikipedia.org/wiki/Ring_epimorphism#Types_of_ring_homomorphisms |
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Hatto von Aquitanien
science forum Guru
Joined: 19 Nov 2005
Posts: 410
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| Posted: Mon Jul 03, 2006 10:39 pm Post subject:
Re: What are all the (important) *-morphisms?
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mariano.suarezalvarez@gmail.com wrote:
| Quote: | Hatto von Aquitanien wrote:
quote
book="Elements of Abstract Algebra"
author="Allan Clark"
A ring homomorphism ?:R->R' is called:
(1) a monomorphism if ? is one to one;
(2) an epimorphism if ? is onto;
(3) an isomorphism if ? is a one-to-one correspondence;
(4) an endomorphism if R'=R;
(5) an automorphism if R'=R and ? is a one-to-one correspondence;
/quote
I'm not familiar with that book, but item (2) is wrong with respect to
the definitions in use today.
A ring epimorphism (ie, an epimorphism in the category of rings)
phi:R->R' is not necessarily surjective as a map of sets. For example,
the inclusion map of the ring ZZ of the integers into the ring ZZ[1/2]
of rational fractions with denominator a power of two is an
epimorphism yet it is obviously not surjective.
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Not surjective with respect to ZZ[1/2]? If being surjective is not the
property that leads to its classification as an epimorphism, then what is
the essential property. I'm not challenging your claim. I'm merely
seeking clarification.
--
Nil conscire sibi |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Thu Jun 29, 2006 12:58 am Post subject:
Re: What are all the (important) *-morphisms?
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In article <1151537092.645875.213100@b68g2000cwa.googlegroups.com>,
Gene Ward Smith <genewardsmith@gmail.com> wrote:
| Quote: | Hatto von Aquitanien wrote:
I suggest you learn to think abstractly, fool:
The thing is, Hatto, mathematicians have trouble with that. This is
especially true when they start messing around with stuff like
universal algebra or category theory.
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Yes. For example, I certainly have a lot of trouble with the notion
that "thinking abstractly" means "ignore everything said, especially
that which is wrong, and concentrate only on whatever would make it
correct if shuffled, changed, or misinterpreted in some way."
I confess to being utterly crippled that way... We should contact Bill
and Melinda and ask for a grant to help treat us.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Thu Jun 29, 2006 12:38 am Post subject:
Re: What are all the (important) *-morphisms?
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In article <y72dnXD4ks8hmT7ZnZ2dnUVZ_qmdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
| Quote: | Arturo Magidin wrote:
In article <44a2f5be$0$7769$7a628cd7@news.club-internet.fr>,
Denis Feldmann <denis.feldmann.asupprimer@club-internet.fr> wrote:
Arturo Magidin a écrit :
In article <LPudnSB7eIO9dD_ZnZ2dnUVZ_oSdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
[.snip.]
Strange you still dont see that troll for what he is...
I do not tolerate fools gladly, and find willful ignorance, whether
true or faked, particularly irksome.
I suggest you learn to think abstractly, fool:
"We define abstraction as selective ignorance?concentrating on the ideas
that are relevant to the task at hand, and ignoring everything else?and we
think that it is the most important idea in modern programming."
http://www.acceleratedcpp.com/details/preface.html
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Good one!
You've certainly mastered ignorance. Now, if you could only learn how
to be accurately selective about it, instead of the wholesale approach
you seem to favor, you might have an snowball's chance in hell to
stop being an ignoramus sometime before the 21st century ends.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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Gene Ward Smith
science forum Guru
Joined: 08 Jul 2005
Posts: 409
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| Posted: Wed Jun 28, 2006 11:24 pm Post subject:
Re: What are all the (important) *-morphisms?
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Hatto von Aquitanien wrote:
| Quote: | I suggest you learn to think abstractly, fool:
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The thing is, Hatto, mathematicians have trouble with that. This is
especially true when they start messing around with stuff like
universal algebra or category theory. |
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Hatto von Aquitanien
science forum Guru
Joined: 19 Nov 2005
Posts: 410
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| Posted: Wed Jun 28, 2006 10:37 pm Post subject:
Re: What are all the (important) *-morphisms?
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Arturo Magidin wrote:
| Quote: | In article <44a2f5be$0$7769$7a628cd7@news.club-internet.fr>,
Denis Feldmann <denis.feldmann.asupprimer@club-internet.fr> wrote:
Arturo Magidin a écrit :
In article <LPudnSB7eIO9dD_ZnZ2dnUVZ_oSdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
[.snip.]
Strange you still dont see that troll for what he is...
I do not tolerate fools gladly, and find willful ignorance, whether
true or faked, particularly irksome.
|
I suggest you learn to think abstractly, fool:
"We define abstraction as selective ignorance?concentrating on the ideas
that are relevant to the task at hand, and ignoring everything else?and we
think that it is the most important idea in modern programming."
http://www.acceleratedcpp.com/details/preface.html
| Quote: | I'll try; it is immaterial at this point. I am about to start a
vacation in any case...
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You obviously need it.
--
Nil conscire sibi |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Wed Jun 28, 2006 10:25 pm Post subject:
Re: What are all the (important) *-morphisms?
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In article <44a2f5be$0$7769$7a628cd7@news.club-internet.fr>,
Denis Feldmann <denis.feldmann.asupprimer@club-internet.fr> wrote:
| Quote: | Arturo Magidin a écrit :
In article <LPudnSB7eIO9dD_ZnZ2dnUVZ_oSdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
|
[.snip.]
| Quote: | Strange you still dont see that troll for what he is...
|
I do not tolerate fools gladly, and find willful ignorance, whether
true or faked, particularly irksome.
| Quote: | For instance, compare my analysis of the article in "an isomorphism..."
and his own 6 hours later.
|
I'll try; it is immaterial at this point. I am about to start a
vacation in any case...
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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Denis Feldmann2
science forum addict
Joined: 23 Apr 2006
Posts: 87
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| Posted: Wed Jun 28, 2006 9:34 pm Post subject:
Re: What are all the (important) *-morphisms?
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Arturo Magidin a écrit :
| Quote: | In article <LPudnSB7eIO9dD_ZnZ2dnUVZ_oSdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
Arturo Magidin wrote:
In article <hcOdnVSZCvvWQD_ZnZ2dnUVZ_oudnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
[.snip.]
I was right. You were wrong! Nya nya, nya nya!
Sigh...
No, Hatto. Rather: you have proven yourself to be a woefully
inadequate judge of what you understand and what you do not
understand in mathematics. You are attempting to read texts well
beyond your understanding and ability. While doing so, you have
committed grievous error after grievous error. You have also wasted
the time of many a person who in good faith
I hope you do not count yourself among such people.
I care nothing, at this point, about what you may hope.
If your goal were to
help me understand my errors, then you would not be so insulting and
condescending.
My goal at first was both in good faith to answer questions you posed,
and to correct the mistakes you made. Your replies quickly eroded all
of that. You reaped what you sowed with your contempt.
Strange you still dont see that troll for what he is... |
For instance, compare my analysis of the article in "an isomorphism..."
and his own 6 hours later. |
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Gene Ward Smith
science forum Guru
Joined: 08 Jul 2005
Posts: 409
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| Posted: Wed Jun 28, 2006 9:33 pm Post subject:
Re: What are all the (important) *-morphisms?
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Hatto von Aquitanien wrote:
| Quote: | What I should have written - which I have already clarified - is
that "Isomorphism is an equivalence relation."
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You shouldn't have written that either, as it's wrong.
| Quote: | "Group isomorphisms define an equivalence relation on groups"
I prefer "isomorphism satisfies the requirements of an equivalence
relation".
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Mine is better because it is correct. Yours is worse because it is
intolerably vague, and in the most obvious reading wrong. |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Wed Jun 28, 2006 9:23 pm Post subject:
Re: What are all the (important) *-morphisms?
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In article <E6adnVBJ66gpbD_ZnZ2dnUVZ_r2dnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
| Quote: | Arturo Magidin wrote:
My goal at first was both in good faith to answer questions you posed,
and to correct the mistakes you made. Your replies quickly eroded all
of that. You reaped what you sowed with your contempt.
It is not only the seed which determines the fruit. Field in which it is
sown is of significant consequence.
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And no matter how good or bad the field, erosion will quickly take
care of that.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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Hatto von Aquitanien
science forum Guru
Joined: 19 Nov 2005
Posts: 410
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| Posted: Wed Jun 28, 2006 9:16 pm Post subject:
Re: What are all the (important) *-morphisms?
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Arturo Magidin wrote:
| Quote: | My goal at first was both in good faith to answer questions you posed,
and to correct the mistakes you made. Your replies quickly eroded all
of that. You reaped what you sowed with your contempt.
|
It is not only the seed which determines the fruit. Field in which it is
sown is of significant consequence.
--
Nil conscire sibi |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Wed Jun 28, 2006 9:03 pm Post subject:
Re: What are all the (important) *-morphisms?
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In article <LPudnSB7eIO9dD_ZnZ2dnUVZ_oSdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
| Quote: | Arturo Magidin wrote:
In article <hcOdnVSZCvvWQD_ZnZ2dnUVZ_oudnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
[.snip.]
I was right. You were wrong! Nya nya, nya nya!
Sigh...
No, Hatto. Rather: you have proven yourself to be a woefully
inadequate judge of what you understand and what you do not
understand in mathematics. You are attempting to read texts well
beyond your understanding and ability. While doing so, you have
committed grievous error after grievous error. You have also wasted
the time of many a person who in good faith
I hope you do not count yourself among such people.
|
I care nothing, at this point, about what you may hope.
| Quote: | If your goal were to
help me understand my errors, then you would not be so insulting and
condescending.
|
My goal at first was both in good faith to answer questions you posed,
and to correct the mistakes you made. Your replies quickly eroded all
of that. You reaped what you sowed with your contempt.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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