#10668: Refactor category support for morphisms (Hom is not a functorial
construction!)
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Reporter: nthiery | Owner: nthiery
Type: defect | Status: new
Priority: major | Milestone:
Component: categories | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by SimonKing):
I just uploaded a preliminary patch, implementing my approach. The patch
isn't finished (lacking doc tests), but is ready for discussion. It
depends on #10667 (though it might apply with some noise without #10667 -
test, if you like).
Features:
'''__Structure of hom sets__'''
As I announced, I introduced an attribute for categories `C` that
determines a category which `C.hom_category()` is sub-category of. I call
this attribute `C.hom_structure`:
{{{
# This was already fixed by #10667
sage: LeftModules(ZZ).hom_category()
Category of hom sets in Category of left modules over Integer Ring
sage: type(LeftModules(ZZ).hom_category())
<class 'sage.categories.sets_cat.Sets.HomCategory'>
# This is new:
sage: LeftModules(ZZ).hom_category().is_subcategory(RightModules(ZZ))
True
sage: issubclass(LeftModules(ZZ).hom_category().parent_class,
RightModules(ZZ).parent_class)
True
# Reason for it working:
sage: LeftModules(ZZ).hom_structure
Category of right modules over Integer Ring
}}}
The attribute `C.hom_structure` is used in
`sage.categories.HomCategory.extra_super_categories()`.
'''Hierarchy of hom-categories'''
Recall that currently, the hierarchy of hom-categories goes parallel with
the hierarchy of their base categories, which is wrong.
However, if `C1` is a ''full'' sub-category of `C2` then (and ''only''
then) we should indeed have
`C1.hom_category().is_subcategory(C2.hom_category())`. Similar to the
method `C.super_categories()`, I introduce an attribute
`C.full_subcategory_of`, that provides a list of immediate super
categories in which `C` is full.
With that, I have:
{{{
sage: IntegralDomains().full_subcategory_of
[Category of commutative rings, Category of domains]
sage: Domains().full_subcategory_of
[Category of rings]
sage:
IntegralDomains().hom_category().is_subcategory(CommutativeRings().hom_category())
True
sage:
IntegralDomains().hom_category().is_subcategory(Rings().hom_category())
True
}}}
and, in particular
{{{
sage: issubclass(IntegralDomains().hom_category().parent_class,
Rings().hom_category().parent_class)
True
}}}
'''Self-criticism'''
It is not very pythonic to do those things with an attribute - usually,
methods are better. However, an attribute is easier to add, and it is
faster to access.
I am not sure whether I got the "full subcategory" business right in all
cases.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10668#comment:15>
Sage <http://www.sagemath.org>
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