On Jun 10, 2004, at 4:23 PM, Ron de Bruijn wrote:
Well, what is it then?For a counter-example, think of the dual category Set^{op}. A morphism f : a -> b in that category means that there is a function f^{op} : b -> a
where a and b are sets, however f probably isn't a function at all.
The short answer is: it's the formal dual of a function, and that's all. You will have to get used to this fact about categorical duality, that it's just a formal construction and has no deep meaning in and of itself.
The short long answer is: it's an antifunction, or a predicate transformer (I think).
What is an antifunction? Well, if you factor a function as a surjection followed by a bijection followed by an injection (as you can always do uniquely in Set using image factorization), then you can understand a function as something which identifies, then renames, then adjoins elements of a set. If you turn this map around, and look at what happens to the elements on their way home, you can see that what happens is some elements get deleted, then renamed and then copied. So a function identifies, renames and adjoins while an antifunction deletes, renames and copies.
To formalize this perspective, you can view a(n antiset) as a boolean algebra using the faithful embedding of Set^{op} in Set via the contravariant powerfunctor. The action on arrows turns an antifunction into what I imagine is called a predicate transformer. This is nicely explained in Vaughan Pratt's paper [2] which, BTW, is about the Chu construction which I mentioned in my last post to Jerzy.
Is the following a good summary? A multiplication is just a name for an operation that is defined or not defined for each mathematical construction in terms of to which laws the operation should comply. The laws are then things like communativity and so on.
Multiplication is just a name often used for any operation which is typically but not always associative and has right and left units, and is perhaps also commutative. Addition is exactly the same. If one has two operations which satisfy these properties, then one often distinguishes them by saying one is "addition" and the other is "multiplication".
It is all informal convention; the important things are the laws.
[1] Andrea Asperti and Giuseppe Longo. Categories, Types and Structures. 1990. Available here: http://www.di.ens.fr/users/longo/download.html
[2] Vaughan Pratt. The Stone Gamut: A Coordinatization of Mathematics. In Proc. LICS '95, 1995.
http://boole.stanford.edu/pub/gamut.ps.gz
Regards, Frank
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