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If $K$ is a simplicial set, we can extend it to a bisimplicial set $K^\prime: \Delta^{op} \to sSet$ by setting $K^\prime_n = (K_n)_\delta$ where $(-)_\delta$ means taking a discrete simplicial set. There is a weak equivalence $hocolim K^\prime \to K$.

If K is a finite-dimensional simplicial set, it is isomorphic to some $n$-skeleton, say $sk_n K \cong K$. The same is true for $K^\prime$. Denote by $\Delta^{op}_{\leq n}$ the full subcategory of $\Delta^{op}$ on the objects $[0],\ldots,[n]$ and by $i: \Delta^{op}_{\leq n} \to \Delta^{op}$ the inclusion. My question:

Is the map $hocolim \left( i^\ast (K^\prime) : \Delta^{op}_{\leq n} \to sSets \right) \to K$ a weak equivalence?

Edit: added some $op$'s

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    $\begingroup$ Yes. The point is that left adjoints compose, and $\mathrm{sk}_n \cong i_! i^*$. $\endgroup$
    – Zhen Lin
    Jul 19, 2014 at 7:56
  • $\begingroup$ Thank you, Zhen, but could you elaborate on how this helps? $\endgroup$ Jul 19, 2014 at 12:31
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    $\begingroup$ Taking homotopy colimits is also a left adjoint (at least, if you work at the level of $(\infty, 1)$-categories). Let $p$ be the unique functor $\Delta \to \mathbb{1}$. Then $\operatorname{colim} \cong p_!$, so $\operatorname{colim} \operatorname{sk}_n \cong p_! i_! i^* \cong \operatorname{colim} i^*$. $\endgroup$
    – Zhen Lin
    Jul 19, 2014 at 12:45
  • $\begingroup$ Ok thanks, I think this transfers easily to a model categorical approach. All the functors should be left Quillen with respect to the respective projective structures, thus I can start with a projectively cofibrant replacement to compute the homotopy colimit. $\endgroup$ Jul 19, 2014 at 23:15

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