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New entry spectral cookbook with sketch of some very nice constructions of A. Rosenberg. New stub sheaf on a noncommutative space, pretty contentless so far, and a redirect page noncommutative sheaf, where the latter may have a different meaning (that is why a separate page).
A category C is local if the full subcategory generated by all objects which are not initial, has itself an initial object.
how is this full subcategory generated? Since you are talking what looks like an arbitrary category, I’m not sure what you mean. Is it just the full subcategory on those objects?
Yes, this I said. Full subcategory which has those objects is said to be generated by those objects. This is standard terminology, I think, I come very often across it.
On the other hand, I should put redirect spectral cuisine as alias for spectral cookbook. :)
I’ve never heard it before either.
Although a bit of terminology may be standard in a part of mathematics it may be at odds with terminology in other parts. This is a pain!. ’Generated by’ seems the wrong term. ’Specified by’, ’determined by’ to me seem clearer and more usual.
I also find “specified by” or “determined by” to be more clearer and more usual, but I don’t think “generated by” is incorrect. In general, the X generated by Y, in the context of an ambient Z, is the smallest sub-X of the ambient Z which contains Y. And the smallest full subcategory of a category containing a given collection of objects is, in fact, the full subcategory containing only those objects. If Zoran had written just “the subcategory generated by” then I would say it was wrong, since without the qualifier “full”, the smallest subcategory containing a given collection of objects would be the discrete one. In a similar vein, “the subset of the real numbers generated by the integers” is a roundabout way to say “the set of integers”. (-:
Hmm, yes. After looking at it with the intended meaning it is sort of obvious. But in the context of categories being perhaps abelian, or having other structure, where this will be applied, I immediately leaped to the conclusion that this was some sort of ’generation’ I wasn’t familiar with.
Yes, if the categories have more structure, like being monoidal or additive, then “generated by” would include closure under that structure.
If Zoran had written just “the subcategory generated by” then I would say it was wrong, since without the qualifier “full”
Surely, generation pertains to “full subcategory”. But I feel that your comments that it may confuse a reader are justified (on the other hand specified and determined are not that much better, full phrase is the best if not laisy – the full subcategory whose objects are…).
Well, what is your decision? I am happy in avoiding this phrase in $n$Lab, and saying it with a couple of words more and unambigous. Toby and Mike, it is up to you to advise me, I think you thought much more about basic categorical terminology than I did.
Surely, generation pertains to “full subcategory”.
Only if you say so!
I am happy in avoiding this phrase in nLab, and saying it with a couple of words more and unambigous.
I think unambiguity is good to strive for.
If I might suggest a wording:
A category C is local if the full subcategory whose objects are exactly those objects which are not initial, has itself an initial object.
then in a separate statement:
In this part of the literature, this subcategory is sometimes called the full subcategory generated by the objects which are not initial.
In other words to use unambiguous language in the statement and then use the form that you say is standard in this part of the literature. That way if someone uses that terminology inadvertently later in this, or in a new, entry the meaning is covered.
I agree with Tim, but in general one does not need to emphasise by “exactly” if “the” is present, though in a key definition one is free to emphasise.
A category $C$ is local if the full subcategory whose objects are exactly those objects which are not initial, has itself an initial object.
You are right about the ’exactly’. I do wonder about the use of ’generated’ as if additional structure is present, e.g. monoidal, braided, etc. then ’generated’ can be very ambiguous.
What about just “the full subcategory of non-initial objects”?
Mike #14 - sounds good to me.
14 – this is again somewhat phraseological.
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