Why is the universal $n$-gerbe universal? (HTT, 7.2.2.26)

Question

Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to X$ of $\mathcal{X}$ is called an $n$-gerbe banded by $A$ if $f$ is both $n$-truncated and $n$-connective, and there is a (preferred) isomorphism $\pi_n(f)\cong A\times \widetilde{X}$ of group objects of $\operatorname{Disc}(\mathcal{X}_{/\widetilde{X}})$. (For safety, I will take $n\geq 2$.) For example, the base point $u:1\to K(A,n+1)$ and the projection $K(A,n)\to 1$ are both $n$-gerbes banded by $A$.

In $\S$ 7.2.2 of Higher Topos Theory, Lurie constructs an $\infty$-category $\operatorname{Gerb}_n^A(\mathcal{X})$ of $n$-gerbes in $\mathcal{X}$ banded by $A$, and claims that the object $u:1\to K(A,n+1)$ is terminal in $\operatorname{Gerb}_n^A(\mathcal{X})$. I do not understand his argument, and I need someone's help.

His argument goes as follows:

Let $f:\widetilde{X}\to X$ be an $n$-gerbe banded by $A$ (equipped with a preferred isomrphism $\pi_n(f)\cong A\times \widetilde{X}$). We wish to show that $\operatorname{Map}_{\operatorname{Gerb}_n^A(\mathcal{X})}(f,u)$ is contractible. He then says that "this assertion is local on $X$," so we may assume that $f$ is the trivial $n$-gerbe $X\times K(A,n)\to X$. Under this reduction, the claim has already been proved in a separate lemma (7.2.2.25), and so we are done.

Here is what I do not understand. What does he mean when he says that the assertion is local? It suggests a picture in which every $n$-gerbe banded by $A$ is sort of a colimit of trivial gerbes. This is true, say, when $\mathcal{X}=\mathcal{S}_{/K}$ is the $\infty$-topos of spaces over a fixed space $K$; but I am completely lost for general $\mathcal{X}$. Can anyone help me? Thanks in advance.

Answer 1

(Turning my comments into an answer.)

The relevant claim is an assertion about a gerbe $f : \widetilde X \to X$. (Namely, the assertion that the mapping space $\mathrm{Map}_{\mathrm{Gerb}^A_n(\mathcal X)}(f,u)$ is contractible.) Lurie writes that this assertion "is local on $X$". I believe what is meant is that if $p : X' \to X$ is an arbitrary effective epimorphism, and the assertion holds for the pullback $p^* f : X' \times_X \widetilde X \to X'$ of the gerbe $f$ along $p$, then the assertion holds for $f$.

Admitting that the assertion is indeed local on $X$ in this sense, we reduce to the case of a trivial gerbe as follows. For $p$, we take the effective epimorphism $f : \widetilde X \to X$. This is assumed to be $n$-connective, so it is a fortiori an effective epimorphism. I claim that the pullback $f^* f: \widetilde X \times_X \widetilde X \to \widetilde X$ is the trivial gerbe $\widetilde X \times K(A,n)\to \widetilde X$. By Proposition 7.2.2.12 (3), applied to the slice $\mathcal X / \widetilde X$, any gerbe with a section is trivial, so it suffices to produce a section. We are done as the diagonal map $\widetilde X \to \widetilde X \times_X \widetilde X$ is a section.