Opened 18 months ago
Closed 17 months ago
#31883 closed defect (fixed)
Refine category of ScalarField
Reported by:  Matthias Köppe  Owned by:  

Priority:  major  Milestone:  sage9.4 
Component:  manifolds  Keywords:  
Cc:  Eric Gourgoulhon, Travis Scrimshaw, Michael Jung  Merged in:  
Authors:  Matthias Koeppe  Reviewers:  Travis Scrimshaw 
Report Upstream:  N/A  Work issues:  
Branch:  cd3ca79 (Commits, GitHub, GitLab)  Commit:  cd3ca79ab94f48321cc9aa3f0b34c1b3db4e405e 
Dependencies:  Stopgaps: 
Description
A ScalarField
on a topological manifold is supposed to be a continuous map  but this is not reflected by the category of its parent.
sage: M = Manifold(2, 'R^2', structure='topological') sage: c_cart.<x,y> = M.chart() sage: r_squared = M.scalar_field(x^2+y^2) sage: r_squared.parent().category() Category of commutative algebras over Symbolic Ring sage: r_squared.codomain() AttributeError: 'ScalarFieldAlgebra_with_category.element_class' object has no attribute 'codomain'
Change History (12)
comment:1 Changed 18 months ago by
comment:2 Changed 18 months ago by
Branch:  → u/mkoeppe/refine_category_of_scalarfield 

comment:3 Changed 18 months ago by
Authors:  → Matthias Koeppe 

Commit:  → 1942c714237d5f2996587bfeda80404b57720696 
Status:  new → needs_review 
Anyway, here is a simple solution
New commits:
1942c71  ScalarField.codomain: New, put scalar fields in category of continuous maps

comment:5 Changed 18 months ago by
sage t long randomseed=0 src/sage/graphs/graph.py # 1 doctest failed sage t long randomseed=0 src/sage/manifolds/differentiable/scalarfield_algebra.py # 2 doctests failed sage t long randomseed=0 src/doc/en/thematic_tutorials/vector_calculus/vector_calc_advanced.rst # 1 doctest failed sage t long randomseed=0 src/doc/en/thematic_tutorials/vector_calculus/vector_calc_plane.rst # 1 doctest failed
The one in graph.py
doesn't seem like an error related to this ticket, but the others are trivial. Please make the changes and check that graph.py
is indeed unrelated. Once done, you can set a positive review.
comment:6 Changed 18 months ago by
Commit:  1942c714237d5f2996587bfeda80404b57720696 → cd3ca79ab94f48321cc9aa3f0b34c1b3db4e405e 

Branch pushed to git repo; I updated commit sha1. New commits:
cd3ca79  Update doctests for refined category of ScalarField

comment:7 Changed 18 months ago by
The failing doctest in graph.py
is unrelated and is fixed by #31848
comment:10 Changed 18 months ago by
I agree with the code added in this ticket, but let me seize the opportunity to point out some current inconsistency in the manifold set up, which is due to the lack of a proper real field in Sage (cf. #24456):
 the base field of a real manifold is
RR
 the base ring of the algebra of scalar fields on a manifold is
SR
Of course, from a pure mathematical side, 1 and 2 should be identical.
The basic reason for 1 is RR
being the simplest proxy for the genuine real field, albeit all (symbolic) computations on manifolds have nothing to do with 53 bits of precision...
The basic reason for 2 is to naturally allow for algebra operations like a*f
, where a
is a symbolic variable and f
a scalar field.
If #24456 converges some day, both 1 and 2 should be set to the real field object that will emerge from it. Meanwhile, we have
sage: M = Manifold(2, 'M') sage: CM = M.scalar_field_algebra() sage: M.base_field() Real Field with 53 bits of precision sage: CM.base_ring() Symbolic Ring
With the codomain()
method introduced in this ticket, we have in addition
sage: X.<x,y> = M.chart() sage: f = M.scalar_field(x^2) sage: f.codomain() is f.domain().base_field() True
which is nice, but
sage: f.codomain() is f.parent().base_ring() False
Moreover, for images of scalar fields, we have
sage: p = M((pi, 0)) sage: f(p) pi^2 sage: f(p) in f.codomain() True
but this holds only because pi^2
can be converted to an element of RR
. Otherwise, the test fails:
sage: a = var('a', domain='real') sage: p = M((a, 0)) sage: f(p) in f.codomain() False
f(p)
not lying in f.codomain()
is certainly something we do not want, but this is currently a consequence of the inconsistency 1 <> 2 mentioned above. The latter could be removed by making SR
the base field of manifolds. But then a real manifold and a complex one would appear as having the same base field (according to base_field()
).
Finally, it's worth to stress that in practice, the inconsistency 1 <> 2 is a rather mild one: to my knowledge, it has never triggered any error nor yield any false result in actual computations on manifolds (apart from the trivial CM.base_ring() is M.base_field()
being false).
comment:12 Changed 17 months ago by
Branch:  u/mkoeppe/refine_category_of_scalarfield → cd3ca79ab94f48321cc9aa3f0b34c1b3db4e405e 

Resolution:  → fixed 
Status:  positive_review → closed 
Unfortunately, trying to make
ScalarField
a subclass ofMap
leads to: