Re: The boringness of the dual of exponential

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

One point that no-one has mentioned yet is that you can't have
exponentiation and its dual in the same category, unless it is a
preorder. If exponentiation exists, then the initial object 0 is
strict, and so 0 x 0 = 0 (read all equality signs as isomorphisms).
But if A + (-) has a left adjoint then it distributes over product,
so
A = A + 0 = A + (0 x 0) = (A + 0) x (A + 0) = A x A
which implies that any two maps into A (with the same domain)
are equal. Of course, bi-Heyting algebras (posets P such that
both P and P^op are cartesian closed) are of some interest, as has
already been mentioned; but if you want to work in non-preordered
categories then you have to choose one or the other.

Peter Johnstone

On Tue, 8 Nov 2011, Paul Taylor wrote:

> When David originally posted his question, I thought it was rather
> a silly one and that it was quite rightly dismissed by various
> people.   On the other hand, he now says
>
>> 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?
>
> In other words, these things are "idioms" or "naturally occurring
> things" in mathematics, but there is a gap in the obvious symmetries.
>
> Looking for gaps in symmetries is a good thing to do.  For example
> Dirac (whose biography by Graham Farmelo I have just started reading)
> predicted the positron this way.
>
> Actually, if we're looking at the categorical structure of the category
> of sets, it isn't very symmetrical at all.  The second edition of
> Paul Cohn's "Universal Algebra" was evidently influenced by Mac Lane's
> famous textbook, but illustrates how categorists had way overemphasised
> duality.
>
> For example the terminal object yields the classical notion of element
> or point, whereas the initial object is strict and boring.
>
> Products and coproducts of sets are very different.
>
> I explored this kind of thing in my book.   For example, the section
> on coproducts shows how different they are in sets/spaces and algebras.
>
> So David's question becomes a good one that deserves an answer if
> we read it as one about the phenomenology of mathematics rather than
> its technicalities.
>
> Paul Taylor
>
> PS There is a boring technical answer that I don't think anyone has
> mentioned, namely copowers, especially of modules.
>


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