#18175: Implement categories for topological and metric spaces and related
categories
-------------------------------------+-------------------------------------
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 egourgoulhon):
Replying to [comment:22 tscrim]:
>
> However, I don't think we should impose the restriction for
differentiable/smooth/analytic as being over '''R'''. See for instance
http://www.iecn.u-nancy.fr/~bertram/simfin.pdf.
If we can reach this level of generality, why not? I guess that when
defining new categories for Sage, it's better to be as general as
possible.
> Specifically, it would allow us to work over '''Q''', where arithmetic
is exact.
>
> > In particular, it seems to me that in the literature, "differentiable
manifold" and "smooth manifold" always mean a real manifold. As mentionned
in comment:18, in the implementation a complex manifold of dimension n
would be canonically associated to an almost complex manifold of dimension
2n.
>
> This says that complex should be a subcategory of almost complex.
However by parameterizing the manifolds category by the base field (like,
e.g., `Algebras`), there will be no danger of ambiguity of considering an
almost complex manifold as being over '''C''' or over '''R'''. This also
means we could do the hierarchy I previously suggested (with a minor
adjustment):
> {{{
> Manifolds
> |
> Differentiable
> |
> Smooth
> / \
> Analytic AlmostComplex
> \ /
> Complex
> }}}
> In particular, this would allow us to consider a manifold over '''R'''
that supports a complex structure (for instance, '''C''' considered as a
2-dim real manifold). Thus natural identifications could be given by
functors.
OK. I realize that my concern was more about the implementation of the
Parent classes than about the categories: a priori, a complex manifold
class cannot inherit from an almost complex class (despite of the name!)
because it probably does not have any vanishing ''complex'' Nijenhuis
tensor (although it has a vanishing ''real'' one), does it?
BTW what is the opinion of John Palmieri about all this?
--
Ticket URL: <http://trac.sagemath.org/ticket/18175#comment:23>
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