Talk:Hypercovering

Coskeleton and limit

In the entry Hypercovering, section "Formal definition", second paragraph, $\operatorname {cosk} _{n}\circ \operatorname {sk} _{n}$ isn't necessary (and depending on the definition, is probably wrong, since non-bold coskeleton is a functor from the n-truncated simplicial sets). I think the original author wanted to write ${\operatorname {\mathbf {cosk} } }_{n}:=\operatorname {cosk} _{n}\circ \operatorname {tr} _{n}$ but was confused with the notations (?). Also in the third paragraph of the same section, the diagram on which one takes the limit has to include the augmentation map $U_{0}\to X$ too. Anyone concurs? --Fourier-Deligne Transgirl (talk) 20:20, 5 November 2022 (UTC)

Edited. Anyone care to check out? --Fourier-Deligne Transgirl (talk) 20:33, 5 November 2022 (UTC)

Okay, sorry for wikiblaming, but @157.27.211.149:, in your edit, you said that it was a typo to not precomposing cosk with sk; why? Below is a relevant screenshot of the SGA.

File:Sga 4 expose 5 7.3.1.1.png

Fourier-Deligne Transgirl (talk) 01:12, 6 November 2022 (UTC)

Ohhh.. Did I just violate Wikipedia licensing rule by uploading an image of SGA? I'm not sure though as Grothendieck is already dead... Fourier-Deligne Transgirl (talk) 02:06, 6 November 2022 (UTC)

Somebody just told me that I did violate that, and that I should delete it and upload to Wikipedia instead of Commons. Does anyone here know how to do that? Or if you can do that directly, you have my permission to do so immediately. Sorry and thank you. Fourier-Deligne Transgirl (talk) 11:19, 6 November 2022 (UTC)

Okay, found it. Somebody email de Jong lol... Fourier-Deligne Transgirl (talk) 01:12, 6 November 2022 (UTC)

Actually, why does a hypercovers have to be a simplicial object? The original definition is that they're semisimplicial. I don't see how in general one can define the degeneracy maps canonically (for all hypercovers that can be used to compute derived functor cohomology at least). @Ax-ax: care to elaborate? Fourier-Deligne Transgirl (talk) 04:12, 6 November 2022 (UTC)

It seems that all this discussion would better belong in the Talk page of the article itself (with a link to it from here), to keep things more focused for future editors of that page. PatrickR2 (talk) 09:02, 6 November 2022 (UTC)

@PatrickR2: I mean other people above my thread is also posting similar discussion so… —Fourier-Deligne Transgirl (talk) 12:33, 7 November 2022 (UTC)

@PatrickR2: moved, per your advice. --Fourier-Deligne Transgirl (talk) 18:35, 7 November 2022 (UTC)

I believe the definition I wrote is from Artin-Mazur or from Friedlander, but it's five years ago so who knows. Of course there is always some more general definition available, but "more general" is not always better. It's common to not bother with semi-simplicial objects in contexts when you are only interested in specific geometric constructions anyway, e.g. the nlab page says "A morphism in the category of simplicial objects [..] is called a hypercover if [..]" and if you look in resarch you'll find the same thing, e.g. https://arxiv.org/abs/math/0205027. But if you think your definitions are better I'm not going to stop you because I don't really care. Ax-ax (talk) 15:05, 8 November 2022 (UTC)

@Ax-ax: Even in the arxiv document you linked, Definition 4.2, they still didn't specify what the degeneracy maps are and why they're relevant/important. I assume they meant to take the left adjoint of simplicial $\to$ semisimplicial? Also still you need to include the augmentation map in the diagram in the third paragraph. And if you want to see it in writing, check out SGA 4 Expose 5 7.3.1.1. --Fourier-Deligne Transgirl (talk) 23:25, 8 November 2022 (UTC)

Maybe if you post your question on math overflow or similar, someone who cares will elaborate. Ax-ax (talk) 11:06, 9 November 2022 (UTC)