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somebody asked the category theory mailing list what the word “encodes” is supposed to mean that appears a few times in the entry on sieves.
I guess I had written that a long time ago, and I think I had meant to amplify that while equivalent, the notion of sieve and the notion of subfunctor of a representable are different notions, even though one does determine or “encode” the other.
I should probably go through that whole entry and try to improve the exposition. But right now I don’t have the time.
I think the exposition at sieve is excellent, and doesn’t need significant improvement. So I’ve replaced the three uses of ’encode’ with ’present’, ’talk about’ and ’encapsulate’ respectively. I hope makes things clearer, or at least doesn’t obscure them.
Also, I added the abstract definition of a sieve as a fully faithful discrete fibration, which I think is cute.
I think the exposition at sieve is excellent,
That’s in fact a relief. Believe it or not, I did not even dare to look at the entry again after I saw the message on the category theory mailing list. Because it was so long ago that I had written (parts of) it, I assumed it must have been all awkward and everyone would jump on me for how badly written it is, but I just did not (and do not) have the time and energy to look after it.
So many thanks indeed for taking care of it!
I wish there were some way that we could more widely spread the message that any questions and/or comments about $n$Lab entries should be posted to the $n$Forum.
I agree, this is one of the best $n$Lab entries.
I have edited the sub-section layout a bit, trying to make it be better structured
okay, good. I’ll post a brief reply to the cattheory mailing list then
Urs said:
I wish there were some way that we could more widely spread the message that any questions and/or comments about nLab entries should be posted to the nForum.
I know what you mean -- I get the impression that there are lots of people who use the Lab (witness e.g. how often it's mentioned on MathOverflow) but don't contribute to discussion about it here. Maybe they don't know that the Forum is the place to discuss the Lab, or they do but are reluctant to contribute for some reason. There must be some way to make newcomers and potential contributors feel more welcome, but I don't know what it is.
On the other hand, I like the idea of having a certain amount of cross-over between the categories list and the nLab/nForum. There was some discussion about this a while ago, if I remember rightly, though I don't remember what conclusions were reached, if any.
I have edited the very beginning of the Idea-section at sieve a little, in order to clarify the notion(s) of “sieve in a category” (as opposed to on an object).
Came here just to praise the amazingly clear and helpful overview section. Wish the whole website was written this way :)
Looking over the entry again, I noticed that the paragraph here was really explaining “sieve” as a concept with an attitude, and so I added the hyperlink.
Often there are indeed two different notions which in certain MODEL and/or generality become mathematically identical. To say that there are the same notion with an attitude does not reflect different origin, generality and motivation and even possible difference in future, more complete, picture. An example is nonempty versus inhabited set which are the same in standard ZF (nonconstructively). Still it is not only “attitude” but it is an artifact of the formalization in classical logic and ZF that the two are the same. Who tells you also for example that in future viewpoint of fundamental physics, quantum fields would stay within the framework of fibre bundles and not some modified, more adequate notion ? (you may disagree with example, but I am caricaturing to get the point). Dynamical system one identifies with some process which has time parameter of some sort, but operator algebraic dynamical systems seem to model more general notion (still involving semigroups though) which does not fit with the parameter description.
Physical notion of a field in field theory is certainly not introduced as sections of fibre bundles but via theory of measurement, where the study of local measurement is carefully examined and results in a notion (as it is introduced in physics books). Then abstraction to do some mathematical models, in certain generality, leads to fibre bundles. Not the original notion.
Finally, one can take an axiomatic theory, where one has primitive notions, and then one can introduce abbreviations for various special classes of objects with characterizations. Now, one may say that two characterizations are equivalent and define the same class. Though the definitions taken are different. The equivalence may disappear once we change the axiomatic framework a bit. And this IS the practice of mathematics, not to be bound by formal system but to create new systems as the need to describe various true content arises. Thus nonempty versus inhabited set again.
There are also the sieves from sieve theory and number theory:
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