Gerbes

First observe that there is an equivalence \(T^{\geq 0}_* \simeq T_*\). That is, every pointed object in \(T\) is automatically \((-1)\)-connected. Indeed, any map \(x\colon * \to X\) is a section of \(X \to *\), so the latter is an effective epimorphism by Lemma 2.38.Consider now the `delooping equivalence' from Proposition 2.35. By passing to \(E_{n-1}\)-groups, this results in an equivalence \(E_n\Grp(T) \simeq E_{n-1}\Grp(T_*^{\geq 1})\). Since the equivalence is implemented by taking loops, Corollary 3.34 implies that it restricts to an equivalence for every \(k \geq 1\). Since \(\Grp(C_*) \simeq \Grp(C)\) for any \(C\), we may now paste together all these equivalences to obtain the result:

By Corollary 3.34, the equivalence \(\bOmega^n\colon T^{\geq n}_* \iso E_n\Grp(T)\) from Proposition 3.44 restricts to an equivalence for all \(n\), with inverse given by \(\bB^n\). But since \(T_{\leq 0}\) is a 1-category, we have \(E_n\Grp(T_{\leq 0}) \simeq \Ab(T_{\leq 0})\) for all \(n \geq 2\).

Since \(X\) is \((n-1)\)-connected, the functor \(X \times -\colon T \to T_{/X}\) induces an equivalence \(T_{\leq n-2} \iso (T_{/X})_{\leq n-2}\) on \((n-2)\)-truncated objects, by Corollary 3.35. In particular, the abelian group object \(\pi_n(X) \in \Ab((T_{/X})_{\leq 0})\) corresponds to an object \(A \in \Ab(T_{\leq 0})\) under this equivalence.

For the first claim, consider the two canonical maps \(\bB^nA \to (\bB^nA)^{S^n}\) and \(\Omega^n\bB^nA \to (\bB^nA)^{S^n}\). Using that \(\bB^nA \in \CGrp(T)\), this induces a map in \(T_{/\bB^nA}\). We need to show that this induces an isomorphism on \(0\)-truncations. Since \(n\geq1\), the map \(* \twoheadrightarrow \bB^nA\) is an effective epimorphism. Pullback along it is conservative by Lemma 2.29, so the claim may be checked after this base change. There the map becomes the identity on \(\Omega^n\bB^nA\).For the second claim, apply the equivalence \(\bOmega^n\colon \mathrm{EM}_n(T)\iso E_n\Grp(T_{\leq0})\) from Corollary 3.45. Under this equivalence, a band on \(X\) is precisely an isomorphism \(\bOmega^nX\cong A\). The anima of pairs consisting of an object \(A'\) and an isomorphism \(A'\cong A\) is contractible. Transporting this statement across the equivalence proves the claim, including uniqueness of the band-preserving pointed isomorphism.

First, note that the functor is well-defined. By Example 3.42, the map \(0\colon *\to \bB^{n+1}A\) is an \(n\)-gerbe. Its fiber is \(\bB^nA\), so Lemma 3.49 supplies it with a canonical band by \(A\), which is inherited by every pullback. The claim is that this is the universal \(n\)-gerbe banded by \(A\).To see this, consider an \(n\)-gerbe \(\widetilde{X} \to X\) banded by \(A\). There is a unique cartesian square of the form

Here the cartesian square is required to identify the given band on \(\widetilde X\) with the band pulled back from the universal gerbe. More formally, the homotopy fiber of the functor in the statement over \(\widetilde X\to X\) is the subanima of spanned by the band-compatible cartesian squares. We show that this subanima is contractible.Step 1: Suppose first that \(\widetilde{X} \to X\) admits a section \(s\colon X \to \widetilde{X}\). In the slice topos \(T_{/X}\), the section makes \(\widetilde X\) into a pointed \(n\)-gerbe. By Lemma 3.49, there is a unique band-preserving pointed isomorphism \(\widetilde X\iso X\times\bB^nA\).A band-compatible cartesian square with the universal gerbe is now the same as a map \(u\colon X\to\bB^{n+1}A\) together with a trivialization of its fiber \(P_u:=X\times_{\bB^{n+1}A}*\to X\). Such a trivialization is equivalently a section of \(P_u\to X\), or equivalently a nullhomotopy of \(u\). The anima of pairs consisting of a map \(u\colon X\to\bB^{n+1}A\) and a nullhomotopy of \(u\) is \(\Hom_T(X,*)\), which is contractible. The band compatibility is built into the unique pointed identification supplied by Lemma 3.49. This proves the claim in the pointed case.Step 2: We now prove the claim in general. Since \(n\geq1\), the map \(\widetilde X\to X\) is an effective epimorphism. We may therefore take \(U=\widetilde X\); the pullback \(\widetilde X\times_XU\to U\) admits the diagonal as a section. Let \(\check{C}(U/X)\) denote the Čech nerve, and write \(U_k:=\check{C}(U/X)_k\) and \(\widetilde U_k:=\widetilde X\times_XU_k\). Each map \(\widetilde U_k\to U_k\) is an \(n\)-gerbe banded by \(A\) admitting a section. Step 1 therefore shows that the anima of band-compatible cartesian squares is contractible for every \(k\). Objects, isomorphisms, and identifications of their bands satisfy descent along effective epimorphisms. Hence the anima of band-compatible cartesian squares from \(\widetilde X\to X\) to the universal gerbe is the limit of these contractible animae, and is therefore contractible. Thus every homotopy fiber of the functor in the statement is contractible, proving that it is an equivalence.