|
|
| View previous topic :: View next topic |
| Author |
Message |
Terry/Padden
Guest
|
 Posted: Thu Jul 03, 2008 6:59 am Post subject: Category Theory - Terminology again ? |
 |
|
Presumably for any given category, C, we can form another category, D, whose
objects are the morphisms of C and whose morphisms are the objects of C.
What is the standard name for D ? |
|
| Back to top |
|
 |
| |
Ads |
Advertising
Sponsor
|
|
Rotwang
Guest
|
| Posted: Thu Jul 03, 2008 6:59 am Post subject: Re: Category Theory - Terminology again ? |
|
|
On 3 Jul, 02:59, Terry/Padden <tpad...@bigpond.net.au> wrote:
| Quote: |
Presumably for any given category, C, we can form another category, D, whose
objects are the morphisms of C and whose morphisms are the objects of C.
What is the standard name for D ?
|
How do you define the various categorical operations in D, i.e.
domains, codomains, composition etc.? |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Rotwang
Guest
|
| Posted: Thu Jul 03, 2008 6:59 am Post subject: Re: Category Theory - Terminology again ? |
|
|
On 3 Jul, 03:05, Rotwang <sg...@hotmail.co.uk> wrote:
| Quote: |
On 3 Jul, 02:59, Terry/Padden <tpad...@bigpond.net.au> wrote:
Presumably for any given category, C, we can form another category, D, whose
objects are the morphisms of C and whose morphisms are the objects of C.
What is the standard name for D ?
How do you define the various categorical operations in D, i.e.
domains, codomains, composition etc.?
|
Come to think of it, if C is a non-discrete category with a finite
number of objects then there are more morphisms than objects, so that
you can't define a category D in this manner (since there must be a
distinct identity morphism in D for each object in D). |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Mariano Suárez-Alvarez
Guest
|
| Posted: Mon Jul 07, 2008 2:30 am Post subject: Re: Category Theory - Terminology again ? |
|
|
On Jul 5, 10:30 pm, Terry/Padden <tpad...@bigpond.net.au> wrote:
| Quote: |
On 3/07/08 12:27 PM, in article
9a13e332-e3e6-4076-8ee0-c15e12f43...@k30g2000hse.googlegroups.com, "Rotwang"
sg...@hotmail.co.uk> wrote:
Thanks for your responses. They are very helpful. As you may surmise I am
not a mathematician and am still stuck on page 1 of the CT books I bought
many years ago.
On 3 Jul, 03:05, Rotwang <sg...@hotmail.co.uk> wrote:
On 3 Jul, 02:59, Terry/Padden <tpad...@bigpond.net.au> wrote:
Presumably for any given category, C, we can form another category, D, whose
objects are the morphisms of C and whose morphisms are the objects of C.
What is the standard name for D ?
How do you define the various categorical operations in D, i.e.
domains, codomains, composition etc.?
Come to think of it, if C is a non-discrete category with a finite
number of objects then there are more morphisms than objects, so that
you can't define a category D in this manner (since there must be a
distinct identity morphism in D for each object in D).
It seems to me there are obvious axiomatic ways around the issues you raise;
but that is not important. For me the (invertible) derivation of D from C
is an intellectual necessity. It seems you are saying that Categories such
as D are mathematically impossible in CT. If this is so ( and it seems from
my skimming of the standard texts to be confirmed by a "Proof by Absence of
Reference" which is why I posted the question), then it is an unacceptable
situation for any reasonable person. I would welcome authoritative
confirmation, or otherwise, that D is not acceptable as a category in CT.
|
Don't you think that getting past (at the very least!)
the first few pages of an introductory textbook on the subject
of categry theory is required of you before anyone even cares to
take whatever it is you see as a problem seriously?
-- m |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Guest
|
| Posted: Mon Jul 07, 2008 6:15 am Post subject: Re: Category Theory - Terminology again ? |
|
|
On 6 Jul, 02:30, Terry/Padden <tpad...@bigpond.net.au> wrote:
| Quote: |
Presumably for any given category, C, we can form another category, D, whose
objects are the morphisms of C and whose morphisms are the objects of C.
What is the standard name for D ?
For me the (invertible) derivation of D from C
is an intellectual necessity.
|
It may be an intellectual necessity for you, but the fact is
that while there is a category with one object and two morphisms
there is no category with one morphism and two objects.
| Quote: |
I would welcome authoritative
confirmation, or otherwise, that D is not acceptable as a category in CT.
|
I can confirm authoritatively that D is not a category.
Victor Meldrew
"I don't believe it!" |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Terry/Padden
Guest
|
| Posted: Mon Jul 07, 2008 10:02 am Post subject: Re: Category Theory - Terminology again ? |
|
|
On 7/07/08 12:30 PM, in article
7012a2ad-c35f-4f14-870a-8079fba1f49a@t54g2000hsg.googlegroups.com, "Mariano
Suárez-Alvarez" <mariano.suarezalvarez@gmail.com> wrote:
| Quote: |
Don't you think that getting past (at the very least!)
the first few pages of an introductory textbook on the subject
of categry theory is required of you before anyone even cares to
take whatever it is you see as a problem seriously?
-- m
|
No. |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
Terry/Padden
Guest
|
| Posted: Sun Jul 13, 2008 6:57 am Post subject: Re: Category Theory - Terminology again ? |
|
|
On 7/07/08 8:39 AM, in article
6293ed58-d0af-4bbd-8f17-6113a996e2b1@27g2000hsf.googlegroups.com,
"galathaea" <galathaea@gmail.com> wrote:
| Quote: |
On Jul 5, 11:50 pm, Rotwang <sg...@hotmail.co.uk> wrote:
On 6 Jul, 04:58, galathaea <galath...@gmail.com> wrote:
is the arrow category an acceptable alternative?
the arrow category C-> of a category C
has for objects all arrows f of C
and has morphisms m: f -> f' such that
there exist morphisms gd: dom(f) -> dom(f')
gc: cod(f) -> cod(f')
with
gd
dom(f)---->dom(f')
| |
f| |f'
V V
cod(f)---->cod(f')
gc
commuting
it doesn't meet all your requirements
but it is the natural category-theoretic language
in which to study arrows as objects
My understanding is that the "arrow category" is just a special case
of the comma category in which both of the functors are the identity,
i.e. it has as objects the arrows of C and as morphisms f -> f' those
pairs of arrows gd and gc which make the diagram you give commute. Is
this right? If so your statement that it "has morphisms m: f -> f'
such that there exist..." seems slightly awkward, since it could be
taken to mean that there is a single morphism for all such pairs (gd,
gc), rather than a distinct morphism for each pair. Sorry if this
seems like a nitpick.
you are absolutely right
i should have: "such that" --> "if"
these logical screw ups are very meaningful in math
and is something i really need to work harder on
there is nothing nitpicking about it
My news-server only shows the last post of these 3. So it was a bit |
confusing. The Arrow category suitably qualified may do the job (by
axiomatising the "if" ?) - but I think this approach based on functors etc.
is a bit too complicated - and more importantly not sufficiently
generalised.
Thanks for your help. I have developed my own simplistic answer that
suffices for my purposes.
As D is the most natural of Natural Transformations, I suggest real CT
mathematicians should devote more time to this PAGE 1 issue. This excludes
pompous idiots who think reading more pages is more important than
understanding the first page of any text; or as Halmos sort of said - the
purpose of a text book is to enable the reader to understand the
pre-requisites ! |
|
| Back to top |
|
|
| |
Ads |
Advertising
Sponsor
|
|
|
|
You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum
|
|
FAQ
Search
Memberlist
Usergroups
Register
Profile
Log in to check your private messages
Log in
