Re: right adjoint by forgetting relational symbols

Re: right adjoint by forgetting relational symbols

[Ross Street]

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),

Re: right adjoint by forgetting relational symbols

[Gaucher Philippe]

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.

right adjoint by forgetting relational symbols

[Gaucher Philippe]

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.