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Changed the page local section to discuss a slightly more general concept than the local sections of a bundle, and over a more general pretopology.
I also edited the link on the page exponential map in the section on logarithms, which pointed to the non-existent partial section, to point to local section.
Changed the page local section to discuss a slightly more general concept than the local sections of a bundle, and over a more general pretopology.
This may not globally be followed, but in principle the notion behind bundle is already the most general: just any map. (Otherwise one should point to fiber bundle.) I have allowed myself to change your edit to read as such:
For a bundle, (often taken to be a fiber bundle or at least typically taken to be a regular epimorphic map)
Okay?
That’s fine. There’s probably an I could add, but will be tomorrow.
Is there any reason why partial section shouldn't just redirect to local section?
Who says “partial section” for “local section”?
Who says “partial section” for “local section”?
Toby, perhaps? :-)
Anyway, it’s fine: a partial section is a kind of partial map. I had begun replacing “partial section” by “local section” in the article logarithm, for instance, but however one wants to handle it is fine by me.
Thanks for that, Todd. I didn’t notice that it was used so many times!
I added the small remark that maps which admit local sections in a finitely complete form a singleton pretopology. I need to dig up the remark in an MO answer that dealt with the relationship between the old coverage and the one of local section and include that
Who says “partial section” for “local section”?
H'm, maybe I invented it.
Wikipedia suggests that ‘partial section’ is used in architecture and structural engineering in the context of blueprints. (And these really are examples of local sections; after all, we more or less took the word ‘section’ from this context.)
The mention of regular epimorphisms strikes me as odd. Even a fibre bundle need not be regular epic, because the fibre might be empty; and while that is just one exception (and one that has no global sections), it makes me think that regular epis are standing in for some other concept here. But I don't know what it should be.
Hmm, you’re right. I think the page was written with the thought that the local section would be over a covering map - originally it was just over an étale map, which of course doesn’t need to onto in any sense. I may have just aimed at a stronger notion because of the result I added.
Even a fibre bundle need not be regular epic, because the fibre might be empty;
In the definition of fiber bundles that I care about this case is explicitly excluded and fiber bundles are indeed required to be effective epis.
@Urs #12: Whaa? Are you sure that’s the right definition of fiber bundle then? Of course a principal bundle can’t have an empty fiber since its fibers are equivalent to the structure group, but shouldn’t you be able to say that the structure group acts on the empty set and therefore produces an associated fiber bundle with empty fibers?
In any case, it does seem odd to mention regular epis at local section, since a bundle certainly doesn’t have to be an epi in order to have local sections.
Indeed covering spaces of nonconnected spaces shouldn’t be assumed to have all fibres inhabited. And it makes perfect sense to ask for local sections of any étale map.
Re “partial section”: it seems to me one should really have this concept around anyway, and it’s more general than “local section”. To me, “local section” implicitly means the domain is an open subset. Not so with “partial section”. For example, in obstruction theory, one tries to see how far one can construct a section of a bundle (say a principal bundle) over a CW-complex , where one might be given a section of the restriction of over the -skeleton (hence a partial section of ), and sees whether there is any obstruction to extending the partial section to , the obstruction being measured by a class in .
Of course, there are numerous other applications of the notion of partial section, and I can’t think of a better name for the notion.
Todd, used in this sense, though, the notion is really different from “local section”. If partial sections in this sense are to be discussed, I would suggest to give them a dedicated entry with that name.
Urs: of course it makes sense to give it a dedicated entry. I would also say that not only are the notions different, but that “local section” is a special case of “partial section” (and therefore it was never wrong to refer to a local section as a partial section, but that it is wrong to have “partial section” to redirect to “local section”).
Yes!
Created a quick stub for partial section.
I agree with what Todd said, FWIW.
That makes sense.
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