* Re: right adjoint by forgetting relational symbols
@ 2008-10-16 10:19 Gaucher Philippe
0 siblings, 0 replies; 3+ messages in thread
From: Gaucher Philippe @ 2008-10-16 10:19 UTC (permalink / raw)
To: categories
Le Wednesday 15 October 2008 12:03:23, vous avez écrit :
> Dear all,
>
> I need some references for this problem. Suppose we have a locally
> presentable category C axiomatized by a limits theory T, so C=Mod(T).
> Let us forget some relational symbols in T and all axioms containing these
> relational symbols. One obtains a theory T'. There is a forgetful functor
> Mod(T) --> Mod(T'). Does this functor have always a right adjoint ? If not,
> what conditions must we add ?
>
> Thanks in advance. pg.
Dear all,
Thank you for all answers. But they do not give what I want. I am looking for
a -R-I-G-H-T- adjoint, and by comparing the example I have and the limit
theory axiomatizing the category of small categories (for which the forgetful
functor Mod(T)-->Set does not have any right adjoint since it is not
colimit-preserving), i found (maybe) the following sufficient condition:
If T is a limit theory without equality symbol before the implication signs,
then any forgetful functor Mod(T) --> Mod(T') has a right adjoint.
Indeed, all sentences of T are of the form (Ax)(F(x)=>((E!y)G(x,y)) where F(x)
and G(x,y) are conjunctions of atomic formulas. By hypothesis, F does not
contain the symbol =. So the forgetful functor Mod(T) --> Mod(T') is colimit
preserving. Since Mod(T) is locally presentable, it is cocomplete,
cowellpowered and has a strong generator. So by SAFT, the forgetful functor
Mod(T) --> Mod(T') has a right adjoint.
Does it sound good ?
Thanks in advance. pg.
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: right adjoint by forgetting relational symbols
@ 2008-10-16 22:05 Ross Street
0 siblings, 0 replies; 3+ messages in thread
From: Ross Street @ 2008-10-16 22:05 UTC (permalink / raw)
To: categories
Right adjoints to algebraic functors are very interesting.
Witt vectors and lambda-rings are key words. See
MR0265348 (42 #258) Tall, D. O.; Wraith, G. C.
Representable functors and operations on rings. Proc. London Math.
Soc. (3) 20 1970 619--643.
MR0789309 (86j:13023) Joyal, André(3-PQ)
$\delta$-anneaux et vecteurs de Witt. (French) [$\delta$-rings and
Witt vectors]
C. R. Math. Rep. Acad. Sci. Canada 7 (1985), no. 3, 177--182.
May not be relevant to your examples!
Regards,
Ross
On 16/10/2008, at 9:19 PM, Gaucher Philippe wrote:
> Thank you for all answers. But they do not give what I want. I am
> looking for
> a -R-I-G-H-T- adjoint, and by comparing the example I have and the
> limit
> theory axiomatizing the category of small categories (for which the
> forgetful
> functor Mod(T)-->Set does not have any right adjoint since it is not
> colimit-preserving),
^ permalink raw reply [flat|nested] 3+ messages in thread
* right adjoint by forgetting relational symbols
@ 2008-10-15 10:03 Gaucher Philippe
0 siblings, 0 replies; 3+ messages in thread
From: Gaucher Philippe @ 2008-10-15 10:03 UTC (permalink / raw)
To: categories
Dear all,
I need some references for this problem. Suppose we have a locally
presentable category C axiomatized by a limits theory T, so C=Mod(T).
Let us forget some relational symbols in T and all axioms containing these
relational symbols. One obtains a theory T'. There is a forgetful functor
Mod(T) --> Mod(T'). Does this functor have always a right adjoint ? If not,
what conditions must we add ?
Thanks in advance. pg.
^ permalink raw reply [flat|nested] 3+ messages in thread
end of thread, other threads:[~2008-10-16 22:05 UTC | newest]
Thread overview: 3+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2008-10-16 10:19 right adjoint by forgetting relational symbols Gaucher Philippe
-- strict thread matches above, loose matches on Subject: below --
2008-10-16 22:05 Ross Street
2008-10-15 10:03 Gaucher Philippe
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).