Brian Conrad just emailed me to ask about gerbes in the stacks project. Unfortunately this is not yet in the stacks project (feel free to send your own write-up to stacks.project@gmail.com if you have one).

This is how I would define them:

(1) A gerbe over a site C is a stack in groupoids p : S —> C with the following properties:

• for every object U of C there exists a covering {U_i —> U} such that each fibre category S_{U_i} is nonempty, and
• for every object U of C and objects x, y in S_U there exists a covering {U_i —> U} such that for every i in I we have that x|_{U_i} is isomorphic to y|_{U_i} in the fibre category S_{U_i}.

Once this has been defined there should be a brief discussion of the “band” of a gerbe. In the case where the band is commutative it should be explained carefully that you get a sheaf of groups over the site. Actually, another important case is the case where you are given a sheaf of groups G on C and you consider gerbes whose band “is” G (this should be precisely defined). Also, it should be defined what is a “trivial” gerbe. I suggest we try to avoid cocycles as much as possible. For an informal discission for gerbes over topological spaces, see Lawrence Breen’s notes or Ieke Moerdijk’s notes. Another, less informal, reference would be the book by Giraud entitled “Cohomologie non abelienne”.

(2) Let \cX —> X be a morphism from an algebraic stack \cX to an algebraic space X. Then we say that \cX is a gerbe over X if and only \cX viewed as a stack in groupoids on (Sch/X)_{fppf} is a gerbe as defined above. Moreover, all the notions defined in the abstract setting can be used in this setting also.

This may not always correspond to the geometric picture of a gerbe, especially if the band (i.e., the automorphism group of an object) isn’t flat! But is it really always the case that gerbes in algebraic geometry have flat automorphism groups?

As usual comments are welcome.

[Edit: Brian adds that we could for instance prove (via erasable delta-functors) that gerbes describe H^2 with commutative coefficients (including its group structure *and* functoriality in both group and base space). And similarly give a torsor/gerbe description of a 7-term exact sequence of pointed sets associated to a central extension of group sheaves. Most of this can be done without using cocycles. On pp. 144-145 of Milne’s book on etale cohomology he gives a nice little summary of the highlights on this aspect of Giraud’s book.

Another great suggestion is: Explain as an example why Artin’s work shows that for an fppf group scheme (or algebraic space group) A, the A-gerbes are Artin stacks.]