Hi David,
On 8 Dez., 11:49, "David R. Kohel" <ko...@iml.univ-mrs.fr> wrote:
> Is there an article available which sets
> out the definitions, problems or issues, and solution provided
> by the coercion model?
After a talk on coercion that I gave at some Sage Day in Hagenberg, it
was suggested to write such paper. While I think I understand how the
"pushout" construction is implemented, I would definitely not be
competent enough to write a paper on the whole coercion model without
someone else (Robert Bradshaw, e.g.).
> > Question 1:
> > Submodules do not define their embedding into the ambient module as
> > coercion.
> > sage: M = FreeModule(ZZ,3,inner_product_matrix=Matrix(3,3,range(9)))
> > sage: S = M.submodule([M.random_element(),M.random_element()])
> > sage: S.coerce_embedding() is None
> > True
> > Should this be changed?
> Sure.
Yep, seems really natural.
> > Question 2:
> > When should there be a coercion from free (abstract, non-embedded)
> > modules M to N?
> Clearly when:
> M is N
> M is created as a free submodule of N (as above), extended
> to the lattice of submodules of some fixed ambient module.
I think the extension to the lattice of submodules is automatic.
> N is created as a quotient of M (probably not implemented
> in such a way that this can be determined).
No, that's easy. A quotient module V/W knows its quotient map V -> V/
W, hence, it knows V.
But I was talking here about *abstract* (i.e., not embedded and not
quotiented) modules.
> > Certainly we need coercion from M.base_ring() to N.base_ring() and M.
> > Also, we need M.rank()==N.rank() and M.degree()==N.degree().
> This would effectively identify all free modules of the same rank
> and degree. I would have a better understanding of the idea of
> the coercion model (and where it is applied) to know whether
> isomorphic-but-not-canonically-isomorphic objects are supposed
> to have a coercion between them.
Isn't it related with "unique parent structures"? Example:
sage: QQ^2 is QQ^2
True
Certainly there is no canonical isomorphism between two 2-dimensional
vector spaces over QQ. But the vector spaces in Sage all seem to come
with a fixed basis - and then, mapping basis vectors one-to-one seems
rather canonical to me. And in that situation, Sage tends to not only
have a coercion but actually an identification (QQ^2 *is* QQ^2).
> This is analogous to the question
> about distinct but isomorphic number fields with or without the
> same defining polynomials (with the same polynomials they are
> computationally, but not necessarily canonically isomorphic).
Again, if there is no embedding, Sage would actually say that these
objects are not only canonically isomorphic (since there is a fixed
generator, it is canonical to map the fields by sending generator to
generator) but actually identical:
sage: K.<t> = NumberField(x^2+1)
sage: K is NumberField(x^2+1,name='t')
True
> Comment (*):
> This is essentially a question of different categories -- if the
> category of the free module is that of inner product spaces, then
> there are fewer (homo)morphisms. But I don't know if the coercion
> model adopts such a strong categorical hierarchy.
To my understanding, the coercion model says that a coercion should be
a morphism in an *appropriate* category. A priori (hence, on the level
of the model) it is not defined what "appropriate" means -- that's
about the implementation of the model (not about the model itself) and
is the point where the developers need to make decisions, which are
also based on the expectations of the users.
So, the question is whether we want a coercion from an inner product
space to its underlying free module, of course in the category of free
modules. Sub-question: What would the user expect?
Restrictive answer:
If s/he defines an inner product then this structure apparently is
important and should be preserved by coercion.
Permissive answer:
If the user tries to add an element of an inner product space to
an element of its underlying free module, then s/he apparently does
not care much about the inner product, for otherwise s/he would not
use an element of the underlying free module. So, it is OK to let the
sum live in the underlying free module.
> It seems to mix
> canonical homomorphisms in a category with functors between categories.
No, but it relies on *our* decision to what extent we are willing to
consider an object of a category as an object of a super-category.
> This concerns implementation or performance issues of writing out
> sparse matrices as dense ones or expressing a dense matrix in a
> form in which it would be more expensive. But both coercions seem
> important to easily convert between types.
Agreed.
> > Question 3:
> > I guess it is straight forward that pushout(A,B) of two sub-modules A
> > of M and B of N (hence, the structure in which arithmetic between
> > elements of A and B is performed) will be the submodule of
> > pushout(M,N) that is generated by the images of A and B. Agreed?
> My comment (*) makes it clear (to me) that I don't know what the
> coercion model is supposed to model (functors or homomorphisms).
Let me try to rephrase:
Assume that we have two chains of sub-modules, S1 in S2 in M and T1 in
T2 in M.
In question 2 I asked if we agree that the pushout of S2 and T2 should
be the sub-module of M that is spanned by S2 and T2. I think that's
natural.
Now question 3 is: Do we want that the pushout of S1 and T1 is a sub-
module of pushout(S2,T2)? Or a sub-module of M? Or not defined at all
(No!)?
Cheers,
Simon
