what happens when a basic sentence is added ?
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Gaucher Philippe -
Jiri Adamek