#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 tscrim):
Replying to [comment:21 egourgoulhon]:
> - `Manifolds`: topological manifolds over a topological field K
> - `Complex`: topological manifolds over K='''C''' with a holomorphic
atlas
> - `Differentiable`: topological manifolds over K='''R''' with a
differentiable atlas
> - `Smooth`: topological manifolds over K='''R''' with a C^oo^ atlas
> - `Analytic`: topological manifolds over K='''R''' with an analytic
atlas
> - `AlmostComplex`: smooth manifolds with an almost complex structure
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. 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.
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Ticket URL: <http://trac.sagemath.org/ticket/18175#comment:22>
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