what happens when a basic sentence is added ?

what happens when a basic sentence is added ?

[Gaucher Philippe]

Dear categorists,

I have a question which is probably obvious except for me... Take a limit
theory T. Add a basic sentence to this theory to obtain a theory T'. So
Mod(T') is accessible, Mod(T) is locally presentable. Is Mod(T') accessibly-
embedded, i.e. does Mod(T')\subset Mod(T) preserve filtered colimits for a big
enough regular cardinal ? Or in other terms, what is going on when a basic
axiom is added. I cannot find any answer in Adamek&Rosicky's book. Of course I
ask you the question because this is my situation. In my situation T and T'
are purely relational and the signature contains the same four relation
symbols. The only difference between T and T' is an additional axiom which is a
basic sentence.

thanks in advance. pg.


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Re: what happens when a basic sentence is added ?

[Jiri Adamek]

Dear Philippe,

Your question about limit theories T:

> Add a basic sentence to this theory to obtain a theory T'. So
> Mod(T') is accessible, Mod(T) is locally presentable. Is Mod(T') accessibly-
> embedded

has an affirmative answer, essentailly due to Coste's 1979 paper.
Take a cardinal k larger than the arities of the symbols of
your signature S, then both Mod(T) and Mod(T') are closed under
k-filtered colimits in Str S. (See the ananlogous argument in part II
of the proof of Theroem 5.9 of Rosicky's and mine book.) Consequently,
the embedding Mod(T') -> Mod(T) preserves k-filtered colimits.

Best regards
Jiri


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