Andrea Marino
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Is the geometric realization of simplicial functors interesting?
I am looking for simplicial functors at the intersection of these two categories. Nothing fancy, just examples to play around with my lemma and see if it is significant or trivial in meaningful contexts. As an example of what I mean, the classifying space of a group is the nerve of a particular category, but its cohomology has interesting implications in the study of G-bundles. If this distinction is apparent, and the 'algebraic simplicial sets' always have interesting cohomology/homotopy groups, I'd be more than happy to know! Hope I made myself clearer now! Thanks :)
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Is the geometric realization of simplicial functors interesting?
Maybe I am wrong, but I think of simplicial sets being used for two different purposes: algebraic and geometrical. Algebraically they can be used e.g. as a non-commutative version of chain complexes, or to provide resolutions in higher context. The nerve of a category provides a nice algebraic example. Geometrical simplicial sets, instead, are mainly employed to give a combinatorial basis to the (co)homology of their geometric realizations. [cont'd]
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Is the geometric realization of simplicial functors interesting?
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Is the geometric realization of simplicial functors interesting?
On a side note, I see a possible source of misunderstanding in my question, as there are two "higher" involved. First is replacing objects with simplicial resolutions that want to capture higher data. Second is replacing diagrams with homotopy coherent diagrams. I am interested in examples of strict diagrams of simplicial objects, so there is only the first "higher" involved. These are the 'input' of the lemma I came up with. The output of has also the second higher involved, but it's not quite relevant. Sorry if that was misleading.
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Is the geometric realization of simplicial functors interesting?
Thanks. Two examples that Breen's notes made me think about are the classifying space and the nerve of a category. These are maybe different from the initial spirit I had in mind, since the domain is large category and not a geometric object (e.g. poset of open sets), but it can still be a nice source of examples. In the first case the geometric realization is relevant as e.g. the cohomology is interesting. As far as I understand, you seem to confirm that the combinatorics of these 'simplicial resolutions' is more interesting than its geometry, and so the realization has limited interest.
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Kakuro puzzles and sheaf cohomology
Regarding the geometric-combinatorics criticism, what I hope is that some combinatorics from the correspondence gives some insight on the original combinatorics. I agree that it is definitely not obvious that something interesting can pop up, however I have the vague sensation that the information is packed in a weird way in the kakuro puzzle, and is not a "free" description that could appear in a decompositon-like theorem for these kind of sheaves.
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Kakuro puzzles and sheaf cohomology
First of all, thanks for the thorough comments. I haven't checked if it does yield a sheaf, but the value on a segment should precisely encode the compatibility relations: the possible pairs on $cc'$ are exactly the restrictions of sequences coming from $\mathcal{X}_a$. Not sure if it works. Alternatively, we could use higher dimensional simplices to encode longer strips of sequences. In the $10-4$ example, I would have $\mathcal{F}(\Delta^3) =L(10,4)$ and restrictions to faces as we expect. The difference is now that the stalk at a general point is $L(10,4)$.
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One-step problems in geometry
Thanks for pointing that out. I have edited accordingly
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One-step problems in geometry
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Which revolutions in topology and geometry can we expect in the next 20 years?
@YemonChoi: I don't think changes in perspective and language are narrative-based, but I agree that the conditions allowed the change to happen. I made that point in my question, and I was exactly trying to ask for "which soil is preparing a shift?". As a metaphor, there are people studying the same in geopolitics; it is very hard to predict the future, but the discussion is nevertheless worthwhile. People writing the past (and the narrative) work in a different field.
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Which revolutions in topology and geometry can we expect in the next 20 years?
I agree that the question is research level, so I guess it should be closed for other reasons (e.g. subjectivity). Nevertheless, I take the closing act as an indirect answer to the question "Can we predict ..?" with a bold "We can't". I theoretically disagree, but there are many experienced researchers that expressed this opinion, so I guess in practice it is very hard (probably, it is only possible in hindsight). Also, I am glad not to open MO to handle controversy, so I agree with its closure.
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A starting point for research in Graph Theory as a high schooler
I suggest you get in touch with professors in graph theory from your area, go to seminars of the group (if any), and propose yourself to make some humble calculations. I think free work is always welcomed, and if you catch a Professor who encourages your ambition, it could also mentor you through your early career.
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A starting point for research in Graph Theory as a high schooler
In the same vein of Tian Vlasic, a nice starting point (if you are tailored at publishing) could be working out details of so-called "folklore results". That is, take a result that is almost known to experts, but nobody has ever really worked out the details. Study the involved definitions, make the needed calculations, and submit to a minor journal. While this is not satisfactory from a problem-solving point of view, it would allow you to become at ease with research topics and practices while publishing (it is also a service to the community). Knowing of such results is not easy, but [...]