"Bill Page" <[EMAIL PROTECTED]> writes: | On March 13, 2006 6:34 AM Ralf Hemmecke asked: | > ... | > But here the question to our category theory experts: | > Since Monoid is something like (*,1) would it make sense | > to speak of a category (in the mathematical sense) of | > | > monoids that have * and 1 as their operations | > (1) | > | > ? Morphisms would respect 1 not just the identity element | > with respect to *. And for each morphism f we would have | > f(a*b) = f(a)*f(b). Of course as operations the two * above | > are different but in that category they have to have the same | > name. (No idea whether this makes sense, but it seems that | > this is the way as "Category" it is implemented in Axiom/Aldor.) | > | > Then, of course, (N, +, 0) is not an object in the category | > given by (1). | > | | I keep trying to answer these questions but I am not sure I | would like to classify myself as an "expert" in category theory. | :) But here goes ... | | In category theory **Mon** (** means written in bold face font) | consists of all monoids (as objects) and all monoid homomorphisms | as morphisms. This does not say anything directly about what | operation are present "inside" the objects of the category.
Agreed. | It is one of the goals of category theory to define what we mean | by 'monoid' entirely in terms of the homomorphisms and perhaps | statements about the existence of "special" objects). | | It turns out that an individual object of **Mon** is also a | category. It consists of a single object. The morphisms are words. | There is an identity morphism that we can denote '1'. Composition | is just the monoid operation. (N, +, 0) is such an object of | **Mon**. yes, but there are not very interesting categories :-) | To make a connection with categories in Axiom, I would want to | say that 'Monoid' represents the category **Mon** while | 'Monoid(*$INT,1)' denotes an object of **Mon**. In fact both | of these are categories in their own right. Indeed. | This brings up a design feature of Axiom and Aldor that I do | not really like. Axiom and Aldor implement a strict two-level | type class hierachy that distinquishes between categories and | domains. Just as in category theory, I think this distinction | is necessary in order to build mathematical constructs. But | categories and domains should not remain strictly separate. | There are many times when we might want to treat a category as | a domain in some other category. We sometimes need to be able | to "flatten" this hierachy and consider really only one kind | of type. I guess it depends on where we want to go. But, you're right that since Aldor already has Type:Type, the distinction between category and domain becomes very artificial. -- Gaby _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
