Proposition 3.44. (Iterated delooping principle)
For every \(n \geq 0\), there is an equivalence
\[\bB^n\colon E_n\Grp(T) \;\rightleftarrows\; T^{\geq n}_* \noloc \bOmega^n.\]
Proof
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'
\[\bB\colon \Grp(T) \rightleftarrows T_*^{\geq 1} \noloc \bOmega\]
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 \[\bB \colon E_n\Grp(T^{\geq k-1}) \; \rightleftarrows \; E_{n-1}\Grp(T_*^{\geq k})\noloc \bOmega\]
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: \[\bB^n \colon E_n\Grp(T) \simeq E_{n-1}\Grp(T_*^{\geq 1}) \simeq \dots \simeq E_1\Grp(T_*^{\geq n-1}) \simeq E_0\Grp(T_*^{\geq n}) = T_*^{\geq n} \noloc \bOmega^n.\]