Re: [Axiom-developer] "has" and "with" (was curious algebra failure)

[Gabriel Dos Reis]

William Sit <address@hidden> writes:

| I think I understand the objection raised by Gaby. In matching Monad to
| SetCategory with ..., Axiom is applying a forgetful functor that, given a
| monad, forgets all of its operations except those featured in SetCategory with
| ....   In this sense, applying a forgetful functor is a form of automatic
| coercion on the level of categories.  Gaby is correct in saying that strictly
| speaking, the forgetful functor should also not be automatic, just like the
| usual coercion between domains. In fact, how one forgets the structure of a
| ring to the structure of a Monad would be ambiguous, since there are two
| possible ways. For more complicated algebraic structures (like Hopf algebra,
| differential algebra) if it is possible to specify precisely the forgetful
| functor, it would allow us to have only one Monoid structure instead of an
| AbelianMonoid and a Monoid (and many others distinguished by different names
| for the monoid operation).
| 
| Gaby, is that a fair statement?

Yes, that is close.

| You may be interested in this old 1986 article by Kamal Abdali et al., on An
| Object Oriented Approach to Algebra System Design:
| 
| http://portal.acm.org/ft_gateway.cfm?id=32444&type=pdf&coll=GUIDE&dl=ACM
| 
| One of the stated features of Views is:
| 
| "Views do not depend on operation names. That is, locally (i.e. in their
| domains) two different Rings may have different names for their 'additon
| operation', but, when viewed as Rings, their addition operation is accessed by
| the same identifier."

Thanks; I had a copy, but I did not get back to it.

-- Gaby