Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 3.1.3.10. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r]^{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^{g} \ar@ {-->}[ur]^{ \overline{f} } & S. } \]

Then:

  • If $S \simeq \Delta ^{0}$ is a final object of the category of simplicial sets, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}_{A/}(B,X)$ (as simplicial subsets of $\operatorname{Fun}(B,X)$).

  • If $A \simeq \emptyset $ is an initial object of the category of simplicial sets, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}_{/S}(B,X)$ (as simplicial subsets of $\operatorname{Fun}(B,X)$).

  • If $S \simeq \Delta ^{0}$ and $A \simeq \emptyset $ are final and initial objects, respectively, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}(B,X)$.