Kerodon

Proof. We have a commutative diagram of sets

where the square on the right is a pullback. Our assumption that $C$ is weakly $w$-local guarantees that the bottom horizontal map is surjective, so that the upper horizontal map on the right is also surjective. Since Proposition 9.5 is a pushout square, the horizontal map on the upper left is also surjective (Warning 7.6.2.3). It follows that the composite map $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y', C) \xrightarrow { \circ [w'] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X',C)$ is also surjective. $\square$

Proof. Fix an object $C \in \operatorname{\mathcal{C}}$, and let $h_{C}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ denote the functor represented by $C$. We wish to show that, for every morphism $w: X \rightarrow Y$ which belongs to $W$, the image $h_{C}(w)$ is a homotopy equivalence of Kan complexes. By virtue of Remark 3.5.1.19, it will suffice to show that $h_{C}(w)$ is $n$-connective for every integer $n \geq 0$. The proof proceeds by induction on $n$. In the case $n = 0$, we wish to show that the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)$ is surjective on connected components, which follows from our assumption that $C$ is weakly $W$-local. Let us therefore assume that $n > 0$. Using the criterion of Corollary 3.5.1.29 (together with Exercise 7.6.3.13), we are reduced to proving the $(n-1)$-connectivity of the relative diagonal of $h_{C}(w)$ (formed in the $\infty $-category $\operatorname{\mathcal{S}}$). Since the functor $h_{C}$ preserves limits (Proposition 7.4.1.18), we can identify the relative diagonal of $h_{C}(w)$ with $h_{C}( \gamma _{X/Y} )$, where $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$ denotes a relative codiagonal of $w$. By assumption, we can arrange that $\gamma _{X/Y}$ is also contained in $W$, so the desired result follows from our inductive hypothesis. $\square$

Proof. Let $U: \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}$ be the projection map and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which can be written as a transfinite composition of morphisms of $\operatorname{\mathcal{C}}$. Suppose we are given a morphism $X \rightarrow C$ in $\operatorname{\mathcal{C}}$, which we identify with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/C}$ satisfying $U( \widetilde{X} ) = X$. We wish to show that there is a morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ of $\operatorname{\mathcal{C}}_{/C}$ satisfying $U( \widetilde{f} ) = f$.

Choose an ordinal $\alpha $ and a functor $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits $f$ as a transfinite composition of morphisms of $W$. Let $Q$ be the collection of all ordered pairs $(\beta , \widetilde{F}_{\leq \beta } )$, where $\beta \leq \alpha $ is an ordinal and $\widetilde{F}_{\leq \beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) \rightarrow \operatorname{\mathcal{C}}_{/C}$ is a functor satisfying $\widetilde{F}_{\leq \beta }(0) = \widetilde{X}$ and $U \circ \widetilde{F}_{\leq \beta } = F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$. We regard $Q$ as a partially ordered set, where $(\beta , \widetilde{F}_{\leq \beta } ) \leq (\beta ', \widetilde{F}'|_{\leq \beta '})$ if $\beta \leq \beta '$ and $\widetilde{F}_{\leq \beta } = \widetilde{F}'_{\leq \beta '} |_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$.

We first claim that $Q$ satisfies the hypotheses of Zorn's lemma. Let $Q_0 \subseteq Q$ be a linearly ordered subset of $Q$; we wish to show that $Q_0$ admits an upper bound. If $Q_0$ is empty, we can take this upper bound be the pair $(0, \widetilde{F}_{\leq 0} )$, where $\widetilde{F}_{\leq 0}$ is the constant functor taking the value $\widetilde{X}$. Without loss of generality, we may assume that $Q_0$ does not contain a maximal element (otherwise, there is nothing to prove). Write $Q_0 = \{ ( \beta _ i, \widetilde{F}_{\leq \beta _ i} ) \} _{i \in I}$ and let $\beta \leq \alpha $ be the supremum of the set $\{ \beta _ i \} _{i \in I}$. The functors $\widetilde{F}_{\leq \beta _ i}$ can then be amalgamated to a single functor $\widetilde{F}_{< \beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \beta } ) \rightarrow \operatorname{\mathcal{C}}_{/C}$. To find an upper bound for $Q_0$, it will suffice to show that the lifting problem

admits a solution. This follows immediately from our assumption that $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Applying Zorn's lemma, we deduce that $Q$ contains a maximal element $(\beta , \widetilde{F}_{\leq \beta } )$. To complete the proof, it will suffice to show that $\beta = \alpha $; we can then take $\widetilde{f}$ to be obtained by applying the functor $\widetilde{F}$ to the edge of $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } )$ given by the pair $(0, \alpha )$. Assume otherwise, let $F_{\leq \beta }$ denote the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } )}$, and let $\operatorname{\mathcal{D}}$ denote the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ F_{\leq \beta } / }$. Then $F_{\leq \beta +1}$ and $\widetilde{F}_{\leq \beta }$ can be identified with objects $D,D' \in \operatorname{\mathcal{D}}$, and the maximality of $(\beta , \widetilde{F}_{\leq \beta } )$ guarantees that $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' ) = \emptyset $. Since the inclusion map $\{ \beta \} \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } )$ is right anodyne (Corollary 4.6.7.24), the restriction map $V: \operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{ F_{\leq \beta } / } \rightarrow \operatorname{\mathcal{C}}_{ F(\beta ) / }$ is a trivial Kan fibration (Corollary 4.3.6.14). It follows that the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{ F(\beta ) / } }( V(D), V(D') )$ is also empty: that is, there is no $2$-simplex of $\operatorname{\mathcal{C}}$ with boundary indicated in the diagram

This contradicts our assumption that the morphism $F(\beta ) \rightarrow F(\beta +1)$ belongs to $W$. $\square$

Proof. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $C$ is $w$-local. Then $W$ contains all identity morphisms, is closed under composition (Remark 9.2.1.11), and is closed under the formation of colimits in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ which are preserved by the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ (Remark 9.2.1.9). Applying Proposition 9.2.2.10 (and Remark 9.2.2.11), we conclude that $W$ is closed under transfinite composition. $\square$