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.