Re: right adjoint by forgetting relational symbols

[Gaucher Philippe <Philippe.Gaucher@pps.jussieu.fr>]

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.