$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 10.2.6.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, and let $\overline{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\mathrm{min}} ) \rightarrow \operatorname{\mathcal{C}}$ be a splitting of $X_{\bullet }$ (in the sense of Definition 10.2.6.3). For every integer $n$, the following conditions are equivalent:
- $(1)$
The functor $\overline{X}$ is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )$ of Notation 10.2.6.16.
- $(2)$
The augmented simplicial object $X_{\bullet }$ is $n$-coskeletal: that is, it is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq n})^{\operatorname{op}}$ (Definition 10.2.5.10).
Proof.
For each integer $k \geq -1$, we will show that the following conditions are equivalent:
- $(1_ k)$
The functor $\overline{X}$ is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )^{\operatorname{op}}$ at the object $[k+1]$.
- $(2_ k)$
The functor $X_{\bullet }$ is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq n})^{\operatorname{op}}$ at the object $[k]$.
Let $(\operatorname{{\bf \Delta }}_{+}^{\leq n})_{ / [k] }$ denote the fiber product $(\operatorname{{\bf \Delta }}_{+})_{ / [k] } \times _{ \operatorname{{\bf \Delta }}_{+} } \operatorname{{\bf \Delta }}_{+}^{\leq n}$, and define $(\operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\leq n+1} )_{ / [k+1] }$ similarly. By virtue of Corollary 7.2.2.3, it will suffice to show that the concatentation functor $C_{+}$ induces a right cofinal functor
\[ G: \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{+}^{\leq n})_{ / [k] } ) \rightarrow \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{ \mathrm{min}}^{\leq n+1} )_{ / [k+1] } ). \]
This follows from Corollary 7.2.3.7, since the functor $G$ admits a left adjoint $F$ (which carries a morphism $\alpha : [m] \rightarrow [k+1]$ of $\operatorname{{\bf \Delta }}_{\mathrm{min}}$ to the nondecreasing function
\[ \{ k \in [m]: \alpha (k) > 0 \} \xrightarrow {\alpha } \{ 1 < 2 < \cdots < k \} \simeq [k]. \]
$\square$