If $X$ is n-skeletal, then isomorphisms hold.

Question

How do I prove that if $X$ is $n$-skeletal, then $|X|\cong \Delta^\bullet\otimes _{\Delta^{op}} X_\bullet\cong \Delta^\bullet\otimes _{\Delta^{op}}Lan X_{\leq n}\cong\Delta^\bullet_{\leq n}\otimes _{\Delta^{op}_{\leq n}} X_{\leq n}$?

Answer 1

The point is that you have a commutative triangle

$$\left((\Delta_{\leq n}\stackrel{i}{\to} \Delta\stackrel{\Delta^\bullet}{\to}\mathrm{Top}\right)=\left(\Delta_{\leq n}\stackrel{\Delta^\bullet_{\leq n}}{\to}\mathrm{Top}\right).$$ The functors of tensoring with $\Delta^\bullet$ or $\Delta^\bullet_{\leq n}$ are just left Kan extensions, and so by the commutative triangle above the left Kan extension of a presheaf on $\Delta_{\leq n}$ along $\Delta^\bullet_{\leq n}$ canonically coincides with the left Kan extension along $\Delta^\bullet$ of its left Kan extension along the canonical inclusion $i$.