$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Variant 10.2.6.20. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is equipped with a functor $T: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The functor $T$ admits a right Kan extension $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$.
- $(2)$
The composite functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq n})^{\operatorname{op}} \xrightarrow { C_{+} } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} )^{\operatorname{op}} \xrightarrow {T} \operatorname{\mathcal{C}} \]
can be extended to an $n$-coskeletal augmented simplicial object of $\operatorname{\mathcal{C}}$.
Proof.
We maintain the notations from the proof of Lemma 10.2.6.19. By virtue of Corollary 7.3.5.8, it will suffice to show that for every integer $k \geq -1$, the following conditions are equivalent:
- $(1_ k)$
The diagram
\[ \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1})_{ /[k+1]} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} ) \xrightarrow {T} \operatorname{\mathcal{C}}^{\operatorname{op}} \]
admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.
- $(2_ k)$
The diagram
\[ \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{+}^{\leq n})_{ / [k] } ) \xrightarrow {G} \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1})_{ /[k+1]} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\leq n+1} ) \xrightarrow {T} \operatorname{\mathcal{C}}^{\operatorname{op}} \]
admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.
As in the proof of Lemma 10.2.6.19, the functor $G$ is right cofinal, so the equivalence of $(1_ k)$ and $(2_ k)$ is a special case of Corollary 7.2.2.10.
$\square$