Re: The boringness of the dual of exponential

["Reinhard Boerger" <Reinhard.Boerger@FernUni-Hagen.de>]

Hello!

David Leduc is not satisfied:

>> Take any category E where exponentiable is interesting.
>> Then the dual of exponentiable is not boring in E^op.
> 
> Indeed! And this is clearly true of the example given by Thomas, namely
> Set^op.
> 
> However, I am not yet satisfied. Let me precise my thoughts. In the
> textbooks and lecture notes on category category that I have read,
> there are always product and coproduct, pullback and pushout,
> equalizer and coequalizer, monomorphism and epimorphism, and so on.
> However exponential is always left alone. That is why I assumed it is
> boring. If it is not boring, why is it never mentioned in textbooks
> and lecture notes on category theory?

I wonder whether it makes sense to introduce notions, which only in the
duals of familiar categories. Of course, Set^op is equivalent to the
category of complete atomic Boolean algebras, but I do not see that the dual
of exponentiation plays an important role in the theory if these Boolean
algebras.

> Also, in logic, "and" goes in pair with "or", "for all" goes in pair
> with "there exists". But implication is always left alone. Why is it

In classical logic, one can form this "co-implication" but it does not look
very interesting to me. In intuitionistic logic I do not see how to add it
more ore less meaningfully (e.g. in such a way that it is left adjoint to
"or" in the first argument).


Greetings
Reinhard



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