Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 5.4.3.10. The functor

\[ \operatorname{N}_{\bullet }( \mathbf{Group} ) \rightarrow \operatorname{\mathcal{S}}_{\ast } \quad \quad G \mapsto B_{\bullet } G \]

is fully faithful.

Proof. By virtue of Proposition 5.4.3.5 and Corollary 4.6.7.8, it will suffice to show that the construction $G \mapsto B_{\bullet }G$ determines a weakly fully faithful functor from $\mathbf{Group}$ (regarded as a constant simplicial category) to the simplicial category $\operatorname{Kan}_{\ast }$. In other words, we must show that for every pair of groups $G$ and $H$, the canonical map

\[ \theta : \{ \textnormal{Group homomorphisms from $G$ to $H$} \} \rightarrow \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( B_{\bullet } G, B_{\bullet } H)_{\bullet } \]

is a homotopy equivalence of Kan complexes. In fact, we claim that $\theta $ is an isomorphism of simplicial sets. Let $BG$ denote the category having a single object $X$ with automorphism group $G$, and let $BH$ denote the category having a single object $Y$ with automorphism group $H$. Proposition 1.4.3.3 then supplies an isomorphism

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Kan}_{\ast }}( B_{\bullet } G, B_{\bullet } H)_{\bullet } & = & \operatorname{Fun}( \operatorname{N}_{\bullet }(BG), \operatorname{N}_{\bullet }(BH) ) \times _{ \operatorname{N}_{\bullet }(BH) } \operatorname{N}_{\bullet }( \{ Y\} ) \\ & \simeq & \operatorname{N}_{\bullet }( \operatorname{Fun}(BG,BH) ) \times _{ \operatorname{N}_{\bullet }( BH ) } \operatorname{N}_{\bullet }( \{ Y \} ) \\ & \simeq & \operatorname{N}_{\bullet }( \operatorname{Fun}( BG, BH ) \times _{BH} \{ Y \} ). \end{eqnarray*}

Note that if $F,F': BG \rightarrow BH$ are functors and $\alpha : F \rightarrow F'$ is a natural transformation with the property that $\alpha _{X}: F(X) \rightarrow F'(X)$ is the identity morphism $\operatorname{id}_{Y}$, then the functors $F$ and $F'$ are equal and $\alpha $ is the identity transformation (since $X$ is the only object of the category $BG$). It follows that the fiber product category $\operatorname{Fun}(BG, BH) \times _{BH} \{ Y\} $ is discrete: that is, it has only identity morphisms. We conclude by observing that the set of objects of the category $\operatorname{Fun}(BG, BH) \times _{BH} \{ Y\} $ can be identified with the set of group homomorphisms from $G$ to $H$. $\square$