Kerodon

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Example 8.5.8.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then a morphism $\operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ can be identified with a triple $(X,e, \sigma )$, where $X$ is an object of $\operatorname{\mathcal{C}}$, $e: X \rightarrow X$ is an endomorphism of $X$, and $\sigma$ is a $2$-simplex of $\operatorname{\mathcal{C}}$ with boundary indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{e} & \\ X \ar [ur]^{e} \ar [rr]^{e} & & X, }$

so that $\sigma$ witnesses the identity $[e] = [e] \circ [e]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.