#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:20 tscrim]:
>
> However I feel like it would be best for `Manifolds` to be over an
arbitrary topological field (which will require some very mild changes).
So then the heirarchy for manifolds would be:
> {{{
> Manifolds
> |
> Differentiable
> |
> Smooth
> / \
> Analytic AlmostComplex
> |
> Complex
> }}}
> Do you think this what we want?
If manifolds are distinguished by their base field (after all, this is the
base field that defines the dimension), an alternative hierarchy would be
{{{
Manifolds
/ \
Complex Differentiable
|
Smooth
/ \
Analytic AlmostComplex
}}}
with the understanding that
- `Manifolds`: topological manifolds over a topological field K
- `Complex`: topological manifolds over K='''C'''
- `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
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.
>
> Also do we think we should add a stub category for PL and/or (pseudo)
Riemannian manifolds? How about `ManifoldsWithBoundary` as a supercategory
of `Manifolds` (and how many of these extra structures lift to the
boundary)?
Probably at some point, `ManifoldsWithBoundary` will be necessary, but
this could be left for a second stage...
--
Ticket URL: <http://trac.sagemath.org/ticket/18175#comment:21>
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