Remark 11.6.0.45 (02VU)

Remark 11.6.0.45. The contents of this tag are now at Example 4.6.6.8.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. The slice diagonal morphism $\delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ carries each $n$-simplex of $\operatorname{\mathcal{C}}_{/F}$ to an $n$-simplex $\sigma $ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $, which we can identify with a map $\Delta ^0 \diamond K \rightarrow \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})$. It is not difficult to see that this map factors (uniquely) through the comparison map $c: \Delta ^0 \diamond K \twoheadrightarrow K^{\triangleleft }$ of Notation 4.5.8.3, and can therefore also be viewed as an $n$-simplex of the simplicial set $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $. Consequently, $\delta _{/F}$ factors as a composition

\[ \operatorname{\mathcal{C}}_{/F} \xrightarrow { \delta '_{/F} } \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} \xrightarrow {\iota } \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} , \]

where $\iota $ is a monomorphism of simplicial sets given by precomposition with $c$. Since $c$ is a categorical equivalence of simplicial sets (Theorem 4.5.8.8), the functor $\iota $ is an equivalence of $\infty $-categories: this follows by applying Corollary 4.5.2.32 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}}) \ar [rr]^{ \circ c} \ar [dr] & & \operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{Fun}(K, \operatorname{\mathcal{C}}), & } \]

since the vertical maps are isofibrations (Corollary 4.4.5.3). It follows from Theorem 4.6.4.17 that the functor

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

is also an equivalence of $\infty $-categories. Similarly, the coslice diagonal morphism $\delta _{F/}$ factors through an equivalence of $\infty $-categories

\[ \delta '_{F/}: \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}). \]