Example 8.5.6.7 (041F)

Example 8.5.6.7. For any $\infty $-category $\operatorname{\mathcal{C}}$, we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dl] \ar [dr]^{\delta } & \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \ar [rr] & & \operatorname{End}^{\mathrm{idm}}_{\operatorname{\mathcal{C}}}, } \]

where the vertical maps are the diagonal embeddings. If $\operatorname{\mathcal{C}}$ is a Kan complex, then the left vertical map is a homotopy equivalence of Kan complexes (since the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ is weakly contractible; see Remark 8.5.3.4). In this case, Proposition 8.5.6.4 reduces to the assertion that the diagonal map

\[ \delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{End}^{\mathrm{idm}}_{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) \quad \quad X \mapsto (X, \operatorname{id}_{X} ) \]

has a left homotopy inverse. This is clear: the map $\delta $ has a left inverse in the category of simplicial sets, given by evaluation at the vertex of $\Delta ^1 / \operatorname{\partial \Delta }^1$. Beware that $\delta $ is usually not a homotopy equivalence, since the simplicial set $\Delta ^1 / \operatorname{\partial \Delta }^1$ is not contractible.