#18175: Implement categories for topological and metric spaces and related
categories
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.8
Component: categories | Resolution:
Keywords: geometry, | Merged in:
topology, sd67 | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 95a30aa57fc62f23a884790b57835d107d8bdeef
public/categories/topological_metric_spaces-18175| Stopgaps:
Dependencies: #18174 #17160 |
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Comment (by bpillet):
Hi Eric,
Replying to [comment:28 egourgoulhon]:
> [...] I wonder now if a best strategy would be to leave instead the base
field generic, with the understanding that ''differentiable'' means
'''K'''-''differentiable'', where '''K''' stands for the base field.
As I said, there are case for some fields '''K''' where '''K'''-analytic
is weaker than expected : For example I can make a change of charts x -->
x^p^ over F_p which is actually invertible since x^p^ is the identity on
F_p but when computing its differential it vanishes identically since
px^p-1^ = 0 in caracteristic p.
In my opinion (which is strongly disputable as I don't have a very wide
knowledge of the existing theories) differential geometry is meant for
'''R''' and '''C''' (or maybe in extremal cases over p-adic fields) but
over other field the right way of doing geometry is through algebraic
geometry : Manifolds or varieties are no longer given by charts but by
equations.
However, there is still hope of having '''K'''-manifolds : The essential
piece of information on a scheme (the algebraic geometry object
generalising manifolds) is the ''seaf of functions''. But we do have,
within Sage Manifolds, a similar object : The set of `ScalarFields` over
an open domain. The rest is just algebra over that (sheaf of) ring.
Yet I don't have any idea whether it is actually possible to implement
algebraic geometry, and if there would be any use of such (gigantic) work.
Replying to [comment:27 tscrim]:
> So you're saying there should be a canonical functor from complex
manifolds to almost complex manifolds given by changing the base field
from C to R (as opposed to it being a subcategory)? From what you said,
this seems to be the best course of action.
Yes and actually this is easy :
Take a complex manifold with chart over U given by complex valued
functions z^k^ and let x^k^ + i y^k^ their decompositions into real and
imaginary parts. Then (x^1^,y^1^, ... , x^n^,y^n^) is a chart over U of
the underlying real manifold. On which we get an almost complex structure
for free, given by J \partial_{x^k^} = \partial_{y^k^}
(Structure recalling which coordinate x and y are to be glued together to
make complex coordinates)
This can be seen at the level of the most elementary examples :
The class `Manifold` has a single object subclass : `RealLine` (of
dimension 1) which encodes the manifold '''R'''.
For the complex there should be 2 cases :
* `ComplexLine` (single object subclass of `ComplexManifolds`) of
dimension 1 (over '''C''') with canonical chart ('''C''',z)
* `ComplexPlane`(single object subclass of `AlmostComplexManifolds`) of
dimension 2 (over '''R''') with canonical chart ('''C''',(x,y)) and
complex structure given by J_0 \partial_x = \partial_y
The functor applied to `ComplexLine` would yield the `ComplexPlane`.
--
Ticket URL: <http://trac.sagemath.org/ticket/18175#comment:29>
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