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.

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

  • there exists an element h of H such that a = inn_h o b,
  • there exist elements g of G and h of H such that a = inn_h o b o inn_g.

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 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, sometimes called the category of outer groups. 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).

Let C be a site. Consider the fibred category PreBands 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 to get the stack of bands Bands over C. A band is then an object of the fibre category of Bands over a final object of C (and slightly more complicated if C does not have a final object).

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, φ_{ij}) where {X_i —> X} is a covering, each G_i sheaf of groups on X_i, each φ_{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, φ_{ii’}) —> ({Y_j —> X}, H_j, ψ_{jj’}) consider the following two kinds of morphisms of systems

  1. one given by a refinement of coverings {Y_j —> X} —> {X_i —> X} (note reversal arrow) where the H_j are the pullbacks of the G_i, and
  2. one of the form ({X_i —> X}, G_i, φ_{ii’}) —> ({X_i —> X}, H_i, ψ_{ii’}) given by outer homomorphisms of sheaves of groups G_i —> H_i compatible with φ_{ii’} and ψ_{ii’}.

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.

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), φ_{ii’}), where the outer isomorphisms φ_{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.

Then, given a fixed band B, we say that a gerb \cX is banded by 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!

[Edit: I stole this description of bands from the paper by Max Lieblich and Brian Osserman, see arXiv:0807.4562. 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).]

3 thoughts on “Bands

    • Not sure exactly what you mean… For example what is G? Maybe G is a sheaf of groups on X? I think then perhaps H^1(X, Out(G)) classifies a certain type of band, namely one which locally looks like G (don’t want to be too precise here because I haven’t given it enough thought). But I think if you have a “general” band then there may not be any sheaf of groups G on X so that it locally looks like G. OK?

      PS: Part of the reason for specifying what a band is using the procedure above, is that all the prerequisites for using the definition above are already in the stacks project.

      PPS: Of course there is a huge simplification in all of this if you only look at commutative sheaves of groups.

      • By the way, your comment about the relationship with H^1(X, Out(G)) is very helpful in clarifying part of the picture. I certainly should have mentioned this! Somehow it completely eluded me to think about it.

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