id summary reporter owner description type status priority milestone component resolution keywords cc merged author reviewer upstream work_issues branch commit dependencies stopgaps
10668 Refactor category support for morphisms (Hom is not a functorial construction!) nthiery nthiery "Before this ticket, Hom was implemented as a covariant functorial construction:
{{{
sage: Rings().hom_category()
Category of hom sets in Category of rings
sage: Rings().hom_category().super_categories()
[Category of hom sets in Category of sets]
}}}
The intention was to model the fact that a morphism in a category is
also a morphism in any super category, via the forgetful functor. With
the example above, if A and B are rings, then a ring morphism phi:
A->B is also a set morphism. However, at the level of parents, and as
noted in the documentation of sage.category.HomCategory, this is
mathematically plain wrong: if A and B are rings, then Hom(A,B) in the
category of rings does not coincide with Hom(A,B) in the category of
sets.
I, Nicolas, take full blame for this misfeature; it was just the
shortest route to implement some urgently needed features about
morphisms, and still get that huge chunk of category code done.
Status of the current reimplementation:
- Add support for a MorphismsMethods subclass, similar to
ElementMethods and ParentMethods. If Cat is a category, then
Cat.MorphismMethods will provide generic methods for morphisms in
Cat and in any subcategory. This can be achieved by:
- Building of a hierarchy of abstract classes Cat.morphism_class,
similar to Cat.element_class and Cat.parent_class (10 lines of
code; see Category.element_class).
- Having morphisms in Cat inherit from `Cat.morphism_class`; this
is implemented by overriding `Parent.morphism_class`. An
alternative would have been to override `Category.element_class`
in `HomsetsCategory.element_class`.
- Rename both HomCategory and hom_category to Homsets, for consistency
with the other constructions like CartesianProducts, ...
- Fix HomCategory: for Cat a category, the purpose of
Cat.hom_category() shall be to provide mathematical information
about its homsets (e.g. that a homset in the category of vector
spaces is also a vector space). It is only a subcategory of the
homsets of the full super categories of Cat (a homset in the
category of finite groups is also a homset in the category of
groups).
- Remove HomCategory from the global namespace.
Potential further desirable features for a later ticket:
- If CatA is a subcategory of CatB, add automatic coercion (or just
conversion?) from Hom(A,B, CatA) to Hom(A, B, CatB), modeling the
appropriate forgetful functor. There are too many such coercions
for them to be registered explicitly. So this probably needs to be
implemented through a specific CatB._has_coerce_map_from(A) for
homsets.
- Extend/use sage.categories.pushout to handle mixed morphism
arithmetic (e.g. having the sum of an algebra morphism and a
coalgebra morphism return a vector space morphism).
" defect closed major sage-6.4 categories fixed sage-combinat SimonKing Nicolas M. Thiéry Simon King N/A a9379383081f2ae7d42c0e17601c6e549269899c #16340