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1. • CommentRowNumber1.
   • CommentAuthorJon Beardsley
   • CommentTimeDec 27th 2013
   • PermaLink
   Author: Jon Beardsley
   Format: MarkdownItexSo, sorry for doing this. I posted this question over on MO, but nobody seemed to know, or be interested in talking about it. Since it references stuff you guys have written on the nlab, I thought I might mention it here and see if anyone knows, or is interested in talking about it: Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that $M^\bullet$ is co-cartesian over $R^\bullet$ if it is true that for every map $[n]\to[m]\in \Delta$, we have that the maps $M^n\to M^m$ and $R^n\to R^m$ induce an isomorphism $M^n\otimes_{R^n} R^m\cong M^m$. On the other hand, suppose I've got a hypercover of simplicial presheaves (on some site, or maybe just simplicial sets, whatever) $Y_\bullet\to X_\bullet$. Then being a hypercover of height zero means that means that the map $Y_n\to (\mathbf{cosk}_{n-1}Y_\bullet)_n\times_{(\mathbf{cosk}_{n-1}X)_n}X_n$ is an isomorphism. So, I might be wrong here, but can this latter statement be made in terms of the former? That is, can we make some kind of statement about $Spec(-)$ of a map of cosimplicial rings such that one is co-cartesian over the other along the given cosimpicial map, giving us a hypercover of height zero in simplicial schemes? I don't really have a good intuition for what I should think of the coskeleton as, which I think is part of why I'm having a hard time making sure of this. Thanks!

   So, sorry for doing this. I posted this question over on MO, but nobody seemed to know, or be interested in talking about it. Since it references stuff you guys have written on the nlab, I thought I might mention it here and see if anyone knows, or is interested in talking about it:

   Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that $M^\bullet$ is co-cartesian over $R^\bullet$ if it is true that for every map $[n]\to[m]\in \Delta$, we have that the maps $M^n\to M^m$ and $R^n\to R^m$ induce an isomorphism $M^n\otimes_{R^n} R^m\cong M^m$.

   On the other hand, suppose I’ve got a hypercover of simplicial presheaves (on some site, or maybe just simplicial sets, whatever) $Y_\bullet\to X_\bullet$. Then being a hypercover of height zero means that means that the map $Y_n\to (\mathbf{cosk}_{n-1}Y_\bullet)_n\times_{(\mathbf{cosk}_{n-1}X)_n}X_n$ is an isomorphism.

   So, I might be wrong here, but can this latter statement be made in terms of the former? That is, can we make some kind of statement about $Spec(-)$ of a map of cosimplicial rings such that one is co-cartesian over the other along the given cosimpicial map, giving us a hypercover of height zero in simplicial schemes? I don’t really have a good intuition for what I should think of the coskeleton as, which I think is part of why I’m having a hard time making sure of this.

   Thanks!