The boringness of the dual of exponential

["Paul Taylor" <pt11@PaulTaylor.EU>]

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.



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]