Corollary 11.8.0.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms in $\operatorname{\mathcal{C}}$. Let $K$ be a simplicial set, and let $\widetilde{W}$ be the collection of morphisms in $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ consisting of those natural transformations $w: F \rightarrow F'$ with the property that, for each vertex $x \in K$, the image $w_{x}: F(x) \rightarrow F'(x)$ belongs to $W$. If $G: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram which carries each vertex of $K$ to a $W$-local object of $\operatorname{\mathcal{C}}$, then $G$ is $\widetilde{W}$-local when viewed as an object of $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ (see Definition 9.2.1.1)
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Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty $-category. Extend $G$ to a functor $\overline{G}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$. Since the restriction map $\operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is a trivial Kan fibration (Proposition 1.5.7.6), we can replace $K$ by $\operatorname{\mathcal{K}}$ (and $G$ by $\overline{G}$) and thereby reduce to proving Corollary 11.8.0.1 in the special case where $K = \operatorname{\mathcal{K}}$ is an $\infty $-category.
Let $w: F \rightarrow F'$ be a morphism in $\operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})$ which belongs to $\widetilde{W}$; we wish to show that precomposition with $w$ induces a homotopy equivalence of Kan complexes $\theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) }(F',G) \xrightarrow { \circ [w] } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) }( F, G )$. Let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, let $U: \operatorname{Tw}(\operatorname{\mathcal{K}}) \rightarrow \operatorname{\mathcal{S}}$ be the functor given by the composition
and define $U': \operatorname{Tw}(\operatorname{\mathcal{K}}) \rightarrow \operatorname{\mathcal{S}}$ similarly. The morphism $w$ induces a natural transformation $\overline{w}: U' \rightarrow U$, which carries each object $(f: X \rightarrow Y)$ of $\operatorname{Tw}(\operatorname{\mathcal{K}})$ to the morphism of Kan complexes
given by precomposition with the morphism $w_{X}: F'(X) \rightarrow F(X)$ of $\operatorname{\mathcal{C}}$. Since $w_{X}$ belongs to $W$ and the object $G(Y) \in \operatorname{\mathcal{C}}$ is $W$-local, it follows that $\overline{w}_{f}$ is a homotopy equivalence. Applying Theorem 4.4.4.4, we see that $\overline{w}$ is an isomorphism (in the functor $\infty $-category $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{K}}), \operatorname{\mathcal{S}})$). We conclude by observing that Example 8.3.4.22 supplies an identification of $\theta $ with the image of $\overline{w}$ under the limit functor $\varprojlim : \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{K}}), \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$. $\square$
Remark 11.8.0.2. Let $X$ and $Y$ be Kan complexes, and let $f,g: X \rightarrow Y$ be morphisms. Then $f$ and $g$ are homotopic as morphisms of simplicial sets (that is, they belong to the same connected component of the Kan complex $\operatorname{Fun}(X,Y)$) if and only if they are homotopic as morphisms in the $\infty $-category $\operatorname{\mathcal{S}}$ (Definition 1.4.3.1). Consequently, the category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Construction 3.1.5.10 can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}$ (this is a special case of Proposition 2.4.6.9).
Remark 11.8.0.3. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.5.10). Then $\mathrm{h} \mathit{\operatorname{Kan}}$ can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}$ of Construction In ยง
. If $q: X \rightarrow S$ is a left fibration of simplicial sets, then Construction 5.2.5.2 determines a covariant transport functor , we show that (up to equivalence) $T$ can be promoted to a map of simplicial sets $S \rightarrow \operatorname{\mathcal{S}}$. In other words, the formation of covariant transport morphisms $(e: s \rightarrow s') \mapsto (e_{!}: X_{s} \rightarrow X_{s'})$ is compatible with composition not only up to homotopy, but up to coherent homotopy.
Construction 11.8.0.4 (The $\infty $-Category of Spaces). Let $\operatorname{Kan}$ denote the category of Kan complexes. We view $\operatorname{Kan}$ as a simplicial category, with simplicial morphism sets given by the constructoin We let $\operatorname{\mathcal{S}}$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ (Definition 2.4.3.5). We will refer to $\operatorname{\mathcal{S}}$ as the $\infty $-category of spaces.