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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Fred E.J. Linton -
George Janelidze -
Michael Barr -
Prof. Peter Johnstone