Kerodon

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Example 2.4.2.3 (Delooping). Let $M_{\bullet }$ be a simplicial monoid. We let $BM_{\bullet }$ denote the simplicial category having a single object $X$, with $\operatorname{Hom}_{BM}(X,X)_{\bullet } = M_{\bullet }$ and the composition law is given by the monoid structure on $M_{\bullet }$. We will refer to $BM_{\bullet }$ as the delooping of the simplicial monoid $M_{\bullet }$. Note that the construction $M_{\bullet } \mapsto BM_{\bullet }$ induces an equivalence of categories

$\{ \text{Simplicial Monoids} \} \simeq \{ \text{Simplicial Categories \operatorname{\mathcal{C}} with \operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X\} } \} .$