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Context

In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there are (at least) 4 (equivalent) definitions for non-abelian gerbe:

  1. 2-Bundles
  2. Bundle-Gerbes
  3. Cocycles
  4. Sheaf of Groupoids

As I am working in the context of number 4, I have been working to find a definition of connection on a sheaf of groupoids which, for certain examples, would line up nicely with the local definitions I have read in the 2-Bundle case.

Question

Following [BM]: Let $S$ be a scheme, $X$ be an $S$-scheme, and $G$ be a sheaf of groups on $X$. They denote by $ \Delta^n_{X/S} $, the $S$- scheme which parametrizes the $(n+1)$-tuples of first order infinitesimally close points of $X$.

After learning Gerbes via Brylinski (and other similar approaches), I want to use the fact that we can cover $X$ with open sets $U_i$ and define $Y = \coprod U_i$, so that we have $$X \xrightarrow{\Delta} Y \times_X Y \xrightarrow{p_1, p_2} X$$ and similar maps for higher fibered products. I am tempted to set $ \Delta^n_{X/S} $ equal to the n-fold fibered product of $Y$ with itself. However, I don't think I am capturing the same information that $ \Delta^n_{X/S} $ is supposed to be capturing. Is there an analogy for this with the $ \Delta^n_{X/S} $?

At the very least, how should I think of "$X$ is an $S$-scheme", and thus, "$X/S$" in the context of an open cover on my space?

Any suggestions are greatly appreciated.

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  • $\begingroup$ That object of infiniteismally close points is a means to talk about differential forms and hence about connection structure on the gerbe. Check out section 4.5.1.3 in this note here ncatlab.org/schreiber/files/cohesivedocumentv032.pdf which might give more intuition as to what is going on. So this is different from the Cech cocycle data that gives the gerbe itself, which is encoded in the cover that you denote by Y. For how to describe that see the introduction in section 1.2.5 of the above note, So I am not really sure what the question regarding Delta^n_X is. Could you maybe expand? $\endgroup$
    – Urs Schreiber
    Oct 4, 2013 at 6:47
  • $\begingroup$ Thank you for the comment and the notes! I suppose I was hoping to arrive at the local data for a non-abelian gerbe with connection (i.e. the cocycles, 2-coycles, connection 1-forms, curving 2-forms) starting with a locally connected, locally non-empty sheaf of groupoids. It sounds like you're saying that in order to proceed I would need to learn this language of synthetic geometry. $\endgroup$
    – cheyne
    Oct 4, 2013 at 14:01
  • $\begingroup$ Hi, so we communicated by email, but let me just re-amplify it here, too: I do not think that one necessarily needs synthertic differential geometry to describe gerbes with connection, no. Just as one does not necessarily need it to discuss bundles with connection. SDG is good for some things, but one may well discuss higher connections without. BM chose to phrase it in SDG style, but there are other choices. In arxiv.org/abs/1011.4735 we discuss general infinity-connections without explicitly using synthetic reasoning. $\endgroup$
    – Urs Schreiber
    Oct 4, 2013 at 16:14
  • $\begingroup$ Thank you for these comments. I suppose that you've answered my question "Is there an analogy for Delta^n_X in the context of intersections of open sets" as a simple "Not really". I don't know how to properly close this question now. $\endgroup$
    – cheyne
    Oct 5, 2013 at 16:31

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