Corollary 5.2.4 (05ZI) · ScalingStacks

[05ZI]

Corollary 5.2.4. Let $\mathcal{C}$ be an $\infty$ -topos, let $k\geq 1$ be an integer, and let $X\in\mathcal{C}_{*}^{\left[k,2k-1\right]}$ . If $X$ admits an $H$ -structure, then it admits a unique $\infty$ -delooping.

[05ZJ]

Proof. By 5.2.2, the $\infty$ -operad $\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right)$ is either $\mathbb{E}_{0}$ or $\mathbb{E}_{\infty}$ . On the other hand, the existence of an $H$ -structure is equivalent to $\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(2\right)\neq\varnothing$ . Thus, $X$ admits an $H$ -structure if and only if $\operatorname{End}_{\mathcal{C}_{*}}^{\operatorname{\scriptsize{red}}}\left(X\right)\simeq\mathbb{E}_{\infty}$ if and only if $X$ admits a unique $\infty$ -delooping. ∎