#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: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.
This seems also the best to me.
>
> For the example with finite fields, do they have a reasonable topology
and could that define a transition map? If so, then I think we should
enforce differentiable as being over '''R'''.
If we enforce this, then we are back to the diagram of comment:21. But 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. Then
the diagram becomes
{{{
Manifolds
|
Differentiable
|
Smooth
/ \
Analytic AlmostComplex
|
Complex
}}}
with
- `Manifolds`: topological manifolds over a topological field '''K'''
- `Differentiable`: topological manifolds over '''K''' with a
'''K'''-differentiable atlas
- `Smooth`: topological manifolds over '''K''' with a '''K'''-infinitely
differentiable atlas
- `Analytic`: topological manifolds over '''K''' with a '''K'''-analytic
atlas
- `AlmostComplex`: smooth manifolds over '''K'''='''R''' with an almost
complex structure
- `Complex`: '''C'''-analytic manifolds over '''K'''='''C'''
--
Ticket URL: <http://trac.sagemath.org/ticket/18175#comment:28>
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