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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