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Warning 4.3.5.11. Recall that, if $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between categories, then the slice category $\operatorname{\mathcal{C}}_{/F}$ can be defined as the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}},\operatorname{\mathcal{C}}) } \{ F\} $ (see Remark 4.3.1.11). In the setting of simplicial sets, out definition is somewhat different. Nevertheless, to any morphism of simplicial sets $F: K \rightarrow \operatorname{\mathcal{C}}$, one can associate a comparison map

\[ \delta _{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\{ 0\} \times K, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1 \times K, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 1\} \times K, \operatorname{\mathcal{C}}) } \{ F \} \]

which we will refer to as the coslice diagonal morphism (see Construction 4.6.4.13). This map has the following features:

  • When $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, the morphism $\delta _{/F}$ is an isomorphism of simplicial sets.

  • When $\operatorname{\mathcal{C}}$ is an $\infty $-category, the morphism $\delta _{/F}$ is an equivalence of $\infty $-categories (Theorem 4.6.4.17).

  • When $\operatorname{\mathcal{C}}= \operatorname{Sing}_{\bullet }(X)$ is the singular simplicial set of a topological space $X$, the morphism $\delta _{/F}$ does not coincide with the isomorphism constructed in Example 4.3.5.9 (however, they are naturally homotopic: see Example ).

  • The morphism $\delta _{/F}$ is usually not an isomorphism of simplicial sets (see Warning 4.3.3.31).