I think there is some confusion in the literature about what a band is, although likely this is just me. I just googled a bit and found, I think, at least two inequivalent definitions. I have also had at least one very confusing conversation with somebody (can’t remember whom), which I now think is due to us having different definitions. For the stacks project, I would prefer the band of a gerbe to be the finest possible invariant of the gerbe. I think this basically tells us what to do. Please don’t read the rest of this post if you already know how to do this.<\/p>\n
First, let us make the following basic observation (and you are going to laugh at me for even pointing this out). For an element g of a group G let inn_g : G —> G be the map x |—> gxg^{-1}. Suppose that G, H are groups and that a, b : G —> H are homomorphisms of groups. Then the following are equivalent<\/p>\n
The reason is that b o inn_g = inn_{b(g)} o b and that inn_{hh’} = inn_h o inn_{h’}. If the equivalent conditions hold we say that a, b define the same outer homomorphism<\/em> from G to H. You can compose outer homomorphisms because if you have a : G —> H and b : H —> F and g, h, h’, f in G, H, H, F, then we have (inn_f o b o inn_{h’}) o (inn_h o a o inn_g) = inn_{fb(h’)b(h)b(a(g))} o b o a. OK, so this gives us the category of exterior groups<\/em>, sometimes called the category of outer groups<\/em>. An automorphism in the category of exterior groups is often called an outer automorphism. It is clear how to generalize this to sheaves of groups over a site (you have to localize to get the correct notion of an outer homomorphism of sheaves of groups).<\/p>\n Let C be a site. Consider the fibred category PreBands<\/em> over C whose category of sections over an object U is the category of exterior sheaves of groups over U, so objects are sheaves of groups on U and morphisms are outer homomorphisms. Stackify PreBands<\/em> to get the stack of bands Bands<\/em> over C. A band<\/em> is then an object of the fibre category of Bands<\/em> over a final object of C (and slightly more complicated if C does not have a final object).<\/p>\n What does such a band B look like? Let X be a final object of C. Then B is given by a system ({X_i —> X}, G_i, \u03c6_{ij}) where {X_i —> X} is a covering, each G_i sheaf of groups on X_i, each \u03c6_{ij} is an outer isomorphism of G_i|_{X_i \\times_X X_j} —> G_j|_{X_i \\times_X X_j} satisfying a cocycle condition. To get morphisms of bands ({X_i —> X}, G_i, \u03c6_{ii’}) —> ({Y_j —> X}, H_j, \u03c8_{jj’}) consider the following two kinds of morphisms of systems<\/p>\n Then (roughly) you have to invert the ones of the first kind and the ones of the second kind where all the G_i —> H_i are outer isomorphisms to get morphisms of bands.<\/p>\n Finally, given a gerbe \\cX over C we get a band B(\\cX) by choosing a covering {X_i —> X} and objects x_i over the members of the covering in \\cX. The associated band is ({X_i —> X}, Aut(x_i), \u03c6_{ii’}), where the outer isomorphisms \u03c6_{ij} come from the existence of local isomorphisms between the pullbacks of x_i and x_j over X_i \\times_X X_j. This band is well defined up to unique isomorphism of bands.<\/p>\n Then, given a fixed band B, we say that a gerb \\cX is banded by<\/em> B if there exists a isomorphism θ: B —> B(\\cX) to the band associated to the gerbe \\cX. But be careful: If we try to classify all gerbes banded by B we could mean either of the following two things: Classify pairs (\\cX, θ) or classify \\cX’s such that a θ exists!<\/p>\n [Edit: I stole this description of bands from the paper by Max Lieblich and Brian Osserman, see arXiv:0807.4562<\/a>. Unfortunately, there is a typo in their description of outer morphisms, they divide out by Aut(G) and Aut(H) but the should have used Inn(G) and Inn(H).]<\/p>\n","protected":false},"excerpt":{"rendered":" I think there is some confusion in the literature about what a band is, although likely this is just me. I just googled a bit and found, I think, at least two inequivalent definitions. I have also had at least … Continue reading \n