#15801: Categories over a base ring category
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Reporter: nthiery | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.2
Component: categories | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/categories/over-a-base- | 12afbe46ebafe56cf74661bb6d5ea81af583ba83
ring-category-15801 | Stopgaps:
Dependencies: #10963 |
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Comment (by nthiery):
Replying to [comment:19 tscrim]:
> So `Algebras(Rings())` would cover any algebra (well...any assoc &
unital algebra)?
Yup. And at some point we will want weaker things like
{{{Algebras(SemiRings())}}} and so on.
> Also I think we should not loose support for having the category of
algebras over a fixed base ring.
I agree with that, though more if we want to manipulate the category
"mathematically". From a code perspective point of view, since #11900,
the provided abstract classes only depends on the category of the base
ring, and this has not been a limitation so far.
> For example, given two representations `V`, `W` of a group algebra `RG`
over `R`, the morphisms from `V` to `W` as `RG`-modules is (significantly)
smaller than the morphisms as `R`-modules. Currently our set of morphisms
depends on the category, which if we keep the current setup, would give
that `Hom(V, W, Modules(RG)) is Hom(V, W, Modules(R))` (well, maybe only
`==`).
>
> Actually...`RG` would be the category of group algebras whereas `R`
would be in the category of rings, unless `R` was also the group algebra
of some other group (okay, it's very contrived, but possible). My point is
that just passing the category into `Modules` doesn't always tell us the
full structure (of the homset), and is there some way we can work around
this?
Well, as you say, we have two distinct categories:
{{{
sage: Modules(Rings())
sage: Modules(GroupAlgebras(Rings()))
}}}
so we are fine, aren't we?
Cheers,
Nicolas
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Ticket URL: <http://trac.sagemath.org/ticket/15801#comment:20>
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