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1. • CommentRowNumber1.
   • CommentAuthorhilbertthm90
   • CommentTimeJan 7th 2011
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   Author: hilbertthm90
   Format: HtmlHello. The concept of O_X* - gerbes is brought up <a href="http://ncatlab.org/nlab/show/gerbe+(as+a+stack)">here</a> . If you have an abelian (or not) sheaf in general, say A, I was looking for the concept of A-gerbe to be defined somewhere but couldn't find it in any of the gerbe articles. Does it exist somewhere? I thought this would be as good a place as any to write it, and then it would be easier to give the specific examples asked for. I thought I'd ask before writing it just in case it does appear somewhere.

   Hello. The concept of O_X* - gerbes is brought up here . If you have an abelian (or not) sheaf in general, say A, I was looking for the concept of A-gerbe to be defined somewhere but couldn't find it in any of the gerbe articles. Does it exist somewhere? I thought this would be as good a place as any to write it, and then it would be easier to give the specific examples asked for. I thought I'd ask before writing it just in case it does appear somewhere.
2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 7th 2011
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   Author: DavidRoberts
   Format: MarkdownItexApart from the link you gave (gerbe (as a stack)), I don't think gerbes where the structure group is a sheaf of groups is mentioned anywhere else. Go ahead!

   Apart from the link you gave (gerbe (as a stack)), I don’t think gerbes where the structure group is a sheaf of groups is mentioned anywhere else. Go ahead!

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2011
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   Author: Urs
   Format: MarkdownItexI don't think that there is an explicit discussion currently, and it would be good to have one. (It is implicitly subsumed in the discussion at [[principal infinity-bundle]]: for $A$ your sheaf of abelian groups, take $G = \mathbf{B}A$ its delooping as the structure oo-group.)

   I don’t think that there is an explicit discussion currently, and it would be good to have one.

   (It is implicitly subsumed in the discussion at principal infinity-bundle: for $A$ your sheaf of abelian groups, take $G = \mathbf{B}A$ its delooping as the structure oo-group.)

4. • CommentRowNumber4.
   • CommentAuthorDexter Chua
   • CommentTimeDec 25th 2016
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   Author: Dexter Chua
   Format: MarkdownItexIn the definition of a gerbe in [[gerbe (as a stack)]], the locally connected condition says a stack $F$ on $B$ is locally connected if there is an open covering $\{U_i\}$ of B for which every groupoid $F(U_i)$ is connected. However, in both Wikipedia and Moerdijk's notes, the local connectedness condition is seemingly weaker, requiring only that for any $U$ and any particular elements $a, b \in F(U)$, there is an open cover $\{V_i\}$ of $U$ such that $a$ is connected to $b$ when restricted to $V_i$. This also appears to be the condition proved in the example of torsors. It seems unlikely to me that the two conditions are actually equivalent in general, so is this an error in the entry, or perhaps there are competing definitions for gerbes in use?

   In the definition of a gerbe in gerbe (as a stack), the locally connected condition says a stack $F$ on $B$ is locally connected if there is an open covering $\{U_i\}$ of B for which every groupoid $F(U_i)$ is connected. However, in both Wikipedia and Moerdijk’s notes, the local connectedness condition is seemingly weaker, requiring only that for any $U$ and any particular elements $a, b \in F(U)$, there is an open cover $\{V_i\}$ of $U$ such that $a$ is connected to $b$ when restricted to $V_i$. This also appears to be the condition proved in the example of torsors.

   It seems unlikely to me that the two conditions are actually equivalent in general, so is this an error in the entry, or perhaps there are competing definitions for gerbes in use?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 27th 2016
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   Author: Urs
   Format: MarkdownItexYes, the condition is that the sheaf $\pi_0$ is locally constant on the singleton.

   Yes, the condition is that the sheaf $\pi_0$ is locally constant on the singleton.