# Kerodon

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Remark 5.3.2.12. Let $n \geq 1$ and let $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets which we denote by

$X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n).$

Let $\mathscr {F}': [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the constant diagram taking the value $X(0)$. Let $\mathscr {F}_0 \subseteq \mathscr {F}$ be the subfunctor given by the diagram

$\emptyset \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n),$

and define $\mathscr {F}'_0 \subseteq \mathscr {F}'$ similarly, so that we have a pushout diagram

$\xymatrix@R =50pt@C=50pt{ \mathscr {F}'_0 \ar [r] \ar [d] & \mathscr {F}_0 \ar [d] \\ \mathscr {F}' \ar [r] & \mathscr {F} }$

in the category $\operatorname{Fun}( [n], \operatorname{Set_{\Delta }})$. Applying Remark 5.3.2.8, we deduce that the induced diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}'_0) \ar [r] \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}_0) \ar [d] \\ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}') \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) }$

is also a pushout square. Using Example 5.3.2.4 and Remark 5.3.2.7, we can rewrite this diagram as

$\xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ) \times X(0) \ar [r] \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( X(1) \rightarrow \cdots \rightarrow X(n) ) \ar [d] \\ \Delta ^ n \times X(0) \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( X(0) \rightarrow \cdots \rightarrow X(n) ). }$