# Kerodon

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Example 5.7.2.16. Let $\operatorname{\mathcal{E}}$ be a simplicial set, which we identify with the morphism of simplicial sets $\Delta ^{0} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$ taking the value $\operatorname{\mathcal{E}}$. Then the simplicial set $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ can be identified with the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$. In particular, Construction 4.6.7.3 supplies a comparison morphism

$\theta _{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}= \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{0}, \operatorname{\mathcal{E}})_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}}) = \int _{\Delta ^{0}} \operatorname{\mathcal{E}}.$

If $\operatorname{\mathcal{E}}$ is an $\infty$-category, then $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }}) }( \Delta ^{0}, \operatorname{\mathcal{E}})$ is also an $\infty$-category, and the comparison morphism $\rho$ is an equivalence of $\infty$-categories (Theorem 4.6.7.9). Beware that $\theta _{\operatorname{\mathcal{E}}}$ is generally not an isomorphism (though it is always a monomorphism which is bijective on simplices of dimension $\leq 1$). For example, Example 5.7.2.15 implies that $2$-simplices of $\int _{\Delta ^0} \operatorname{\mathcal{E}}$ can be identified with morphisms of simplicial sets $\rho : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{E}}$ for which the restriction $\rho |_{ \Delta ^1 \times \{ 0\} }$ is a degenerate edge of $\operatorname{\mathcal{E}}$, as indicated in the diagram

$\xymatrix@R =10pt@C=10pt{ X \ar [rrrr]^{\operatorname{id}_ X} \ar [dddd]_{u} \ar [ddddrrrr] & & & & X \ar [dddd]^{w} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ Y \ar [rrrr]_{v} & & & & Z. }$

The corresponding $2$-simplex of $\int _{\Delta ^{0}} \operatorname{\mathcal{E}}$ belongs to the image of $\theta _{\operatorname{\mathcal{E}}}$ if and only if $\sigma$ is a left-degenerate $2$-simplex of $\operatorname{\mathcal{E}}$ (in which case it is given by $\theta _{\operatorname{\mathcal{E}}}(\tau )$).