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Re: [sage-combinat-devel] Re: [Axiom-developer] Re: [sage-devel] Categor

From: Ralf Hemmecke
Subject: Re: [sage-combinat-devel] Re: [Axiom-developer] Re: [sage-devel] Categories for the working programmer
Date: Mon, 10 Nov 2008 01:09:28 +0100
User-agent: Thunderbird (X11/20080914)

What is the relationship between "categories" in Axiom and the
mathematical notion of a category?

None. It is much better to think of categories in Axiom as multisorted algebras. Or to make it simpler, as a first approach you can think of it as universal algebras.

A semigroup in Axiom looks like

SemiGroup(): Category == with
    *: (%, %) -> %

Monoid: Category == SemiGroup with
    1: %


Programmatically, it is nothing else than the "interface" (Java-speak) of a domain, i.e. all the exported function names and their signatures (it's a bit oversimplified).

And I also would not too much draw a distinction between domains and packages. A package is a domain where the special symbol % (which stands for something like ThisDomain, old Axiom use $ instead of %) does not appear.

The type hierarchy in Axiom is actually:


Then there is a hierarchy of domains (only single inheritance is possible) and a hierarchy of categories (multiple inheritance allowed since there is no conflict, because a category (usually) contains no implementation of the signatures).

More details you find in Section 2.5 (p. 28) of


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