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I've seen several papers that I would like to read that use the language of gerbes and bands. The wiki page on gerbes is useful, but doesn't even contain the word 'band', so I'm left confused even about what context they come up in.

What's a good (english, preferably short) reference that would introduce these concepts and their function to me?

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    $\begingroup$ I have found the following text by Ieke Moerdijk useful: math.uu.nl/publications/preprints/1264.ps $\endgroup$
    – Dan Petersen
    Commented Jan 30, 2011 at 23:03
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    $\begingroup$ The first section of LECTURES ON SPECIAL LAGRANGIAN SUBMANIFOLDS of Hitchin discusses Gerbes arxiv.org/PS_cache/math/pdf/9907/9907034v1.pdf $\endgroup$
    – Dmitri Panov
    Commented Jan 31, 2011 at 0:59
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    $\begingroup$ You probably don't want to consider bands. They were a misguided attempt by Giraud to characterise the coefficients of second degree non-abelian cohomology, his definition of which turned out to be non-functorial. See the references at ncatlab.org/nlab/show/gerbe+%28general+idea%29, the most useful of which is probably arxiv.org/PS_cache/math/pdf/0611/0611317v2.pdf by Larry Breen. $\endgroup$
    – David Roberts ♦
    Commented Jan 31, 2011 at 1:35
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    $\begingroup$ In what context are you considering them. Chapter V of the book Loop Spaces, Characteristic Classes, and Geometric Quantization by Brylinski is good for some purposes including a general introduction. $\endgroup$
    – Matt
    Commented Jan 31, 2011 at 4:21
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    $\begingroup$ For Giraud a band was (loosely) a sheaf of groups pasted together by a bunch of outer automorphisms. This is just a shadow of the more modern point of view that the 'structure group' of such a gerbe is a 2-group AUT(G), corresponding to the crossed module $G\to Aut(G)$. The assignment $G \mapsto AUT(G)$ is not functorial. So really one should consider 2-groups (or crossed modules, which are sufficient) to be the structure 2-group of a gerbe, or more precisely, consider torsors for 2-groups. Check out arxiv.org/abs/0909.3350 for a modern take on this together with Breen's notes. $\endgroup$
    – David Roberts ♦
    Commented Feb 1, 2011 at 1:24

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