Kerodon

Example 3.3.1.12. Let $M$ be a nonunital monoid and let $M^{+} = M \cup \{ e\} $ denote the monoid obtained from $M$ by adjoining a unit element (Remark 1.3.2.11). Let $B_{\bullet }( M^{+} )$ denote the classifying simplicial set of Construction 1.3.2.5, and let $B_{\bullet }(M)$ be the semisimplicial set introduced in Variant 1.3.2.12. The inclusion map $M \hookrightarrow M^{+}$ induces a monomorphism of semisimplicial sets $\iota : B_{\bullet }M \hookrightarrow B_{\bullet }(M^{+} )$, whose image consists of the nondegenerate simplices of $B_{\bullet }(M^{+})$. It follows that the simplicial set $B_{\bullet }(M^{+} )$ is braced and that $\iota $ extends to an isomorphism of simplicial sets $(B_{\bullet } M)^{+} \xrightarrow {\sim } B_{\bullet }(M^{+})$ (Corollary 3.3.1.11).