How should one think about the band of a gerbe?

Question

Let $X$ be a topological space. Let $\mathcal{F}$ be a fibered category over $X$; seen as an assignment of a category $\mathcal{F}(U)$ for each open $U\subseteq X$.

A fibered catgeory $\mathcal{F}$ over $X$ satisfying some properties, is called a stack over $X$. Further, if $\mathcal{F}(U)$ is a groupoid for each open $U\subseteq X$, we call $\mathcal{F}$ to be a stack of groupoids over the topological space $X$.

A stack $\mathcal{F}$ over a topological space $X$ is said to be a gerbe over $X$, if the following conditions are satisfied:

1. there exists an open cover $\{U_\alpha\}$ of $X$ such that $Obj(\mathcal{F}(U_\alpha))\neq \emptyset$ for every $\alpha$.
2. Fix an open set $U\subseteq X$ and objects $a,b$ of $\mathcal{F}(U)$. Then there exists an open cover $\{V_\alpha\}$ of $U$ such that $\text{Hom}_{\mathcal{F}(V_\alpha)}\left(a|_{V_\alpha},b|_{V_\alpha}\right)\neq \emptyset$ for every $\alpha$.

Given a gerbe $\mathcal{G}$ over $X$, one can choose an open cover $\{U_\alpha\}$ of $X$ and objects $a_\alpha$ of $\mathcal{G}(U_\alpha)$. This gives sheaves of groups $\underline{\text{Aut}}(a_\alpha)$ for each $\alpha$. Appropriate usage of second condition in the definition of gerbe defines an outer isomorphism of sheaves on $U_{\alpha\beta}$; namely

$$\lambda_{\alpha\beta}:\underline{\text{Aut}}(a_\beta)|_{U_{\alpha\beta}}\rightarrow \underline{\text{Aut}}(a_\alpha)|_{U_{\alpha\beta}}.$$

Given a gerbe $\mathcal{G}$ on $X$; the collection of sheaves of groupoids $\underline{\text{Aut}}(a_\alpha)$ and outer automorphisms $\{\lambda_{\alpha\beta}\}$ is called the band of the gerbe of $X$.

I understand the definition and some immediate remarks about bands. I could not get the main idea behind the association of a band for a gerbe. Above definition is from the notes Introduction to the language of gerbe and stacks by Ieke Moerdijk. I have had a look at the notes on 1-gerbes and 2-gerbes by Lawrence Breen.

Question :

1. How should one think about the band of a gerbe?
2. It was mentioned in another question that, band of a gerbe was a misguided attempt. Still there are many notes that talks about band of a gerbe. So, is the present day definition different from that of the definition of Giraud? What was the motivation for the change? Is there a better definition or usage or understanding of the notion of a band of a gerbe since then?

Answer 1

if $X$ is a connected topological space without a chosen base point, then what is $\pi_1(X)$? The good answer is of course that it's a groupoid. [Actually, let's also assume that $\pi_n(X)=0$ for $n\ge 2$]

But let's say that that we disallow that good answer, and that we want something more like a group...

Then the next best answer is that it's a group well defined up to isomorphism, where the isomorphism is itself well defined up to inner automorphism.

Now, sheaffify the above story, and you get what the band of a gerbe is.

A sheaf of connected spaces with $\pi_n=0$ for $n\ge 2$ is the same thing as a gerbe. Taking $\pi_1$ (in the "next best answer" way) yields the band of the gerbe.

Answer 2

Like for sheaf of groups, you can try to describe the patching data required to glue sheaves of groupoids ("stacks"). A grebe is a sheaf of groupoids, locally equivalent to a sheaf of the form $BG$ for a sheaf of groups $G$. However, since $Hom(BG,BH)$ and $Hom(G,H)$ are not the same, gluing these local $BG$-s together require different descent datum than what you need in order to glue the $G$-s thenselves. In some sense, it is "easier" to glue the classifying groupoids. In fact, $Hom(BG,BH)\cong Hom(G,H)//H$, where here the quotient is the orbit groupoid. It has $\pi_0Hom(BG,BH) = Hom(G,H)/H$ and $\pi_1(Hom(BG,BH),f)=Z_H(Im(f))$, the centralizer of the image of $f$. The gluing data for a grebe takes into account both $\pi_0$ and $\pi_1$. When trying to glue the local $BG$-s in pairs, we need to choose a homotopy class of isomorphisms between their restrictions to the intersection. Namely, for $U_1$ and $U_2$ we need a class in $\pi_0(Hom(BG_1,BG_2))$. But now the compatibility of the choice become an extra structure rather than a property, and the collection of choices of compatibilities of this identifications at triple intersections becomes a torsor for $\pi_1(Hom(BG_1,BG_3))$ where $BG_i$ the the local stack at the open set $U_i$ for $i=1,2,3$. Hence, essentially, the $\pi_1$ controls the degrees of freedom in choosing how the gluing data is compatible. Of course, now this data has to strictly satisfy compatibility on quadraple intersections.

From this description of the descent data for a stack, you immediately see that the band is just what happen when we ignore the $\pi_1$-part. Equivalently, it describes how the local $BG$-s glue when considered as objects of the 1-category of groupoids, where we identify naturally isomorphic functors. While the glued object is not very meaningful, it still gives some partial information about the stack itself, more or less like the components of a top. space gives information about the space itself.

Answer 3

I will start by explaining the easiest possible case of bundle gerbes, when the band A (alias structure group) is an abelian Lie group.

A bundle n-gerbe with band A over a smooth manifold M is a principal bundle over M with its structure ∞-group being the Lie ∞-group B^n(A).

Here B^n(A) can be described concretely as a simplicial presheaf S↦UΓ(C^∞(S,A)), where S ranges over the cartesian site (smooth manifolds diffeomorphic to R^n), and Γ: Ch→sAb is the Dold–Kan functor and U: sAb→sSet is the forgetful functor.

Bundles over M with structure group B^n(A) can be described as (derived) maps y(M)→B^{n+1}(A). Here B^{n+1}(A) was defined above and y(M) denotes the (restricted) Yoneda embedding of M, i.e., the simplicial presheaf S↦C^∞(S,M), where the set C^∞(S,M) is turned into a discrete simplicial set.

Such derived maps can be computed by cofibrantly replacing y(M) and fibrantly replacing B^{n+1}(A) in (say) the local projective model structure on simplicial presheaves on the cartesian site. However, B^{n+1}(A) is an objectwise Kan complex and satisfies the homotopy descent property, so is already fibrant in the local projective model structure. A cofibrant resolution of y(M) can be written down as the Čech nerve of a good open cover U of the smooth manifold M. This is a simplicial presheaf whose presheaf of n-simplices is the coproduct of representables U_{i_0}∩⋯∩U_{i_n}, where i ranges over all (n+1)-tuples such that the above intersection is nonempty.

Now computing the derived mapping spaces using the above resolutions produces the classical Čech complex for bundle gerbes.

For nonabelian gerbes the above formal setup works equally well, with the proviso that the Lie ∞-group B^n(A) can now be replaced by any Lie ∞-group G, and instead of B^{n+1}(A) we use B(G), obtained by using (say) the Dwyer–Kan classifying space functor for simplicial groups objectwise on the values of the presheaf G.

There is also a nonabelian analog of the functor Γ, which takes as an input a crossed module or a crossed complex and produces a simplicial group, which one can use to construct G.

One can then compute the derived mapping space in a similar manner, the only difference being is that we can no longer convert it to a chain complex because G need not be abelian.

So the answer to question (1) is that one thinks of the band of an (abelian) bundle n-gerbe as giving rise to the structure Lie ∞-group B^n(A) and for question (2) the modern approach dictates that we simply use simplicial presheaves (or any equivalent formalism, such as quasicategorical ∞-sheaves), possibly passing from crossed complexes to presheaves of simplicial groups, if desired.