Kerodon

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

Construction 3.1.3.7. Let $B$ and $X$ be simplicial sets, and let $\operatorname{Fun}(B,X)$ be the simplicial set parametrizing morphisms from $B$ to $X$ (Construction 1.4.3.1).

  • Suppose we are given another simplicial set $A$ equipped with a pair of morphism $i: A \rightarrow B$ and $f: A \rightarrow X$. In this case, we let $\operatorname{Fun}_{A/}(B,X) \subseteq \operatorname{Fun}(B,X)$ denote the fiber of the precomposition morphism $\operatorname{Fun}(B,X) \xrightarrow {\circ i} \operatorname{Fun}(A,X)$ over the vertex $f \in \operatorname{Fun}(A,X)$.

  • Suppose we are given another simplicial set $S$ equipped with a pair of morphism $g: B \rightarrow S$ and $q: X \rightarrow S$. We let $\operatorname{Fun}_{/S}(B,X) \subseteq \operatorname{Fun}(B,X)$ denote the fiber of the postcomposition morphism $\operatorname{Fun}(B,X) \xrightarrow { q \circ } \operatorname{Fun}(B,S)$ over the vertex $g \in \operatorname{Fun}(B,S)$.

  • Suppose we are given a commutative diagram of simplicial sets

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

    In this case, we let $\operatorname{Fun}_{A/ \, /S}(B, X) \subseteq \operatorname{Fun}(B,X)$ denote the simplicial subset given by the intersection $\operatorname{Fun}_{A/}(B,X) \cap \operatorname{Fun}_{/S}(B,X)$.