Terminological question, and more

Terminological question, and more

[Fred E.J. Linton]

I've been asked for "... the name given in an arbitrary category 
to an object A for which every mono B----->A is an isomorphism."

I'm stumped. Any ideas?

I've also been asked to comment on whether "the inclusion Q >-------> R of 
the ring of rational numbers into that of real ones is a bimorphism, in the 
category Rng of rings with units and units preserving ring homomorphisms."

Reflexively I think: monic, yes; epic, no, as permuting any two independent
transcendentals should extend to a non-identity automorphism of R over Q.  

Am I missing something here?

TIA; and cheers, -- Fred



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Re: Terminological question, and more

[Prof. Peter Johnstone]

On Wed, 3 Nov 2010, Fred E.J. Linton wrote:

> I've been asked for "... the name given in an arbitrary category
> to an object A for which every mono B----->A is an isomorphism."
>
> I'm stumped. Any ideas?
>
The obvious thing is to borrow a word from ordered set theory and
call it a  minimal object. (All morphisms in a poset are monic.)

The term "strict initial object" is well-established for an initial
object 0 such that *every* morphism A --> 0 is an isomorphism. I've
sometimes been tempted to use "strict object" for this property
without the assumption of initiality; but the trouble is that you
then have to say "costrict object" for the dual property, which
doesn't seem right.

Peter Johnstone


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Re: Terminological question, and more

[George Janelidze]

Dear Fred,

(my comment is about "more" - not about the terminological question)

The group Aut(R) of ring automorphisms of R is well known to be trivial.
Proof:

(a) Every automorphism takes squares to squares.

(b) A real number is a square if and only if it is non-negative.

(c) As follows from (a) and (b), every ring automorphism of R preserves
order.

(d) Aut(Q) is trivial

(e) As follows from (c) and (d), Aut(R) is trivial.

However, you are right that Q >---> R is not an epimorphism of course. Just
use the morphisms into the field C of complex numbers in the same way as you
used automorphisms of R.

Similarly, using the fact that every algebraic extension of fields of
characteristic 0 is separable, it is easy to show that a field extension of
characteristic 0 is an epimorphism in the category of commutative rings if
and only if it is an isomorphism.

Greetings - George

----- Original Message -----
From: "Fred E.J. Linton" <fejlinton@usa.net>
To: "categories" <categories@mta.ca>
Sent: Wednesday, November 03, 2010 11:40 PM
Subject: categories: Terminological question, and more


I've been asked for "... the name given in an arbitrary category
to an object A for which every mono B----->A is an isomorphism."

I'm stumped. Any ideas?

I've also been asked to comment on whether "the inclusion Q >-------> R of
the ring of rational numbers into that of real ones is a bimorphism, in the
category Rng of rings with units and units preserving ring homomorphisms."

Reflexively I think: monic, yes; epic, no, as permuting any two independent
transcendentals should extend to a non-identity automorphism of R over Q.

Am I missing something here?

TIA; and cheers, -- Fred



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Re: Terminological question, and more

[Michael Barr]

I don't know any name for the first question.  As for the second, I think
the conclusion is correct, but the reasoning is not.  The argument that
any automorphism of R is the identity does not depend on multiplicative
inverses.  But any automorphism of a transcendence basis will extend to a
homomorphism of R into C.  This takes place even in the category of
fields.

As a side comment (and the context in which I learned this), the German
translation of Pontrjagin's Topological Groups contains a chapter on
topological fields that was omitted in the English translation (unless it
has been added in the last 50 years).  In that chapter is a flawed
argument for the theorem that the only division algebras containing R are
R, C, and H.  The proof is flawed because the hypothesis did not assume,
but the proof used, that R is in the center of H.  C.T. Yang eventually
came up with the above argument and showed that whichever copy of R in C
you used, you got non-isomorphic quaternions!  Of course, all versions of
C are isomorphic since it is always the algebraic closure of R.  I might
add that, in the European style, "field" did not included commutativity.
So H was called a field.

Michael

On Wed, 3 Nov 2010, Fred E.J. Linton wrote:

> I've been asked for "... the name given in an arbitrary category
> to an object A for which every mono B----->A is an isomorphism."
>
> I'm stumped. Any ideas?
>
> I've also been asked to comment on whether "the inclusion Q >-------> R of
> the ring of rational numbers into that of real ones is a bimorphism, in the
> category Rng of rings with units and units preserving ring homomorphisms."
>
> Reflexively I think: monic, yes; epic, no, as permuting any two independent
> transcendentals should extend to a non-identity automorphism of R over Q.
>
> Am I missing something here?
>
> TIA; and cheers, -- Fred
>
>


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Re: Terminological question, and more

[Fred E.J. Linton]

Thanks again to all who responded to my prior terminological query.

The same interlocutor now inquires reqarding ...

> ... the appropriate name given in an arbitrary X to an object A  
> for which every X-morphism B --> A is an epimorphism. 

Once again, I come up dry, but I'll gratefully transmit 
any suggestions ... :-) .

Cheers, -- Fred



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Re: Terminological question, and more

[Fred E.J. Linton]

Thanks to all who've responded.

> I've been asked for "... the name given in an arbitrary category
> to an object A for which every mono B----->A is an isomorphism."

Peter Johnstone has suggested that such an object (whose poset of
subobjects reduces to just the object itself) is simply "minimal".

And I'm embarrassed at having lost so much of my former grasp of Galois
theory:

> ... whether "the inclusion Q >-------> R of
> the ring of rational numbers into that of real ones is a bimorphism, in  the
> category Rng of rings with units and units preserving ring homomorphisms."
>
> Reflexively I think: monic, yes; epic, no, as permuting any two independent
> transcendentals should extend to a non-identity automorphism of R over Q.

No! There is no "non-identity automorphism of R over Q."  Fortunately,
as Mike Barr and George Janelidze have pointed out, each permutation of a  
transcendence basis of R over Q extends to an injection, over Q, of R 
into its algebraic closure C, which is good enough.

Cheers, and thanks again, -- Fred



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