Kerodon

Proof. The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate. We next show that $(3)$ implies $(4)$. Let $\sigma $ be an $n$-simplex of $S$, and suppose that the projection map $\pi : \Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ exhibits $\Delta ^ n$ as a localization of $\Delta ^{n} \times _{S} X$ with respect to some collection of edges $W$. Since $\Delta ^ n$ is an $\infty $-category in which every isomorphism is an identity morphism, the diagram $\pi $ must carry each edge of $W$ to a degenerate edge of $\Delta ^ n$: that is, we have $W \subseteq W_{\Delta ^ n}$. Applying Corollary 6.3.1.22, we deduce that $\pi $ also exhibits $\Delta ^ n$ as a localization of $\Delta ^ n \times _{S} X$ with respect to $W_{\Delta ^{n}}$.

We now complete the proof by showing that $(4)$ implies $(1)$. Let us say that a simplicial set $T$ is good if, for every morphism $T \rightarrow S$, the projection map $T \times _{S} X \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to $W_{T}$. Assume that condition $(4)$ is satisfied, so that every standard simplex $\Delta ^{m}$ is good. We wish to show that every simplicial set $T$ is good. Using Proposition 6.3.4.1, we see that the collection of good simplicial sets is closed under filtered colimits; we may therefore assume without loss of generality that $T$ is finite. If $T = \emptyset $, the result is obvious. We may therefore assume that $T$ has dimension $n$ for some integer $n \geq 0$. We proceed by induction on $n$ and on the number of nondegenerate $n$-simplices of $T$. Fix a nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow T$. Using Proposition 1.1.4.12, we see that there is a pushout square of simplicial sets

where $T'$ is a simplicial set of dimension $\leq n$ having fewer nondegenerate $n$-simplices than $T$. By virtue of Proposition 6.3.4.2, to show that $T$ is good, it will suffice to show that the simplicial sets $\Delta ^ n$, $\operatorname{\partial \Delta }^ n$, and $T'$ are good. In the first case this follows from assumption $(4)$, and in the remaining cases it follows from our inductive hypothesis. $\square$

Proof. Let $W$ be the collection of all edges $w$ of $X$ such that $f(w)$ is degenerate in $S$. Since our hypothesis is stable under the formation of pullbacks, it will suffice to show that $f$ exhibits $S$ as a localization of $X$ with respect to $W$. Fix an $\infty $-category $\operatorname{\mathcal{C}}$; we wish to show that composition with $f$ induces a bijection

The injectivity of $\alpha $ follows immediately from the existence of the section $u$. To prove surjectivity, it will suffice to show that for every object $g \in \operatorname{Fun}( X[W^{-1}], \operatorname{\mathcal{C}})$ is isomorphic to $g \circ u \circ f$. Since $u \circ f$ and $\operatorname{id}_{X}$ belong to the same connected component of $\operatorname{Fun}_{/S}(X,X)$, it suffices to observe that postcomposition with $g$ carries every edge of $\operatorname{Fun}_{/S}(X,X)$ to an isomorphism in the $\infty $-category $\operatorname{Fun}( X, \operatorname{\mathcal{C}})$. $\square$

Proof. Let $\sigma : \Delta ^ n \rightarrow S$ be an $n$-simplex of $S$; we wish to show that $\sigma $ can be lifted to an $n$-simplex of $X$. Assume otherwise, so that the inclusion map $\operatorname{\partial \Delta }^{n} \times _{S} X \hookrightarrow \Delta ^ n \times _{S} X$ is an isomorphism. We have a commutative diagram of simplicial sets

where the vertical maps are weak homotopy equivalences (see Remarks 6.3.6.7 and 6.3.6.5). It follows that the inclusion $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$ is also a weak homotopy equivalence, which is a contradiction (since the relative homology group $\mathrm{H}_{n}( \Delta ^ n, \operatorname{\partial \Delta }^ n; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}$ is nonzero). $\square$

Proof. Suppose we are given a morphism of simplicial sets $Z' \rightarrow Z$. Set $X' = Z' \times _{Z} X$, and let $W$ be the collection of those edges $w$ of $X'$ having degenerate image in $Z'$. We will show that the projection map $\pi : X' \rightarrow Z'$ exhibits $Z'$ as a localization of $X'$ with respect to $W$. Set $Y' = Z' \times _{Z} Y$, so that $\pi $ factors as a composition $X' \xrightarrow {f'} Y' \xrightarrow {g'} Z'$. It follows from Proposition 6.3.6.9 (and Remark 6.3.6.7) that $f'$ is a surjection of simplicial sets. In particular, the image $f'(W)$ is the collection of all edges $u$ of $Y'$ having the property that $g'(u)$ is a degenerate edge of $Z'$.

Let $W_0 \subseteq W$ be the collection of those edges $w$ of $X'$ for which $f'(w)$ is a degenerate edge of $Y'$. Applying Proposition 6.3.6.3, we conclude that $f'$ exhibits $Y'$ as a localization of $X'$ with respect to $W_0$, and that $g'$ exhibits $Z'$ as a localization of $Y'$ with respect to $f'(W)$. Applying Proposition 6.3.1.21, we conclude that $\pi = g' \circ f'$ exhibits $Z'$ as the localization of $X'$ with respect to $W_0 \cup W = W$, as desired. $\square$

Proof. By virtue of Proposition 6.3.6.10, it will suffice to show that the projection map $X \times K \rightarrow X$ is universally localizing. Using Remark 6.3.6.7, we can reduce to the case $X = \Delta ^0$, in which case the desired result follows from Remark 6.3.6.6. $\square$

Proof. Suppose that $f: X \rightarrow S$ is a morphism of simplicial sets which can be realized as the colimit of a filtered diagram $\{ f_{\alpha }: X_{\alpha } \rightarrow S_{\alpha } \} $ in the category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$, where each $f_{\alpha }$ is universally localizing. We wish to show that $f$ is universally localizing. Fix a morphism of simplicial sets $T \rightarrow S$ and let $W$ be the collection of all edges $w = (w_ T, w_ X)$ of $T \times _{S} X$ for which $w_{T}$ is a degenerate edge of $T$. Note that the projection map $f_{T}: T \times _{S} X \rightarrow T$ can be realized as a filtered colimit of morphisms $f_{T,\alpha }: T \times _{S} X_{\alpha } \rightarrow T \times _{S} S_{\alpha }$. For each index $\alpha $, let $W_{\alpha }$ denote the collection of edges of $T \times _{S} X_{\alpha }$ having degenerate image in $T$. Since $f_{\alpha }$ is universally localizing, Proposition 6.3.6.3 guarantees that $f_{T,\alpha }$ exhibits $T \times _{S} S_{\alpha }$ as a localization of $T \times _{S} X_{\alpha }$ with respect to $W_{\alpha }$. Applying Proposition 6.3.4.1, we conclude that $f_{T}$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to $W$. $\square$

Proof. Fix a morphism of simplicial sets $T \rightarrow S$; we wish to show that the projection map $f_{T}: T \times _{S} X \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to some collection of morphisms $W$. Since the hypotheses of Proposition 6.3.6.13 are stable with respect to pullback, we may assume without loss of generality that $T = S$. Let $W_0$ be the collection of edges $w$ of $X_0$ having the property that $f_0(w)$ is a degenerate edge of $S_0$, and define $W_{1}$ and $W_{01}$ similarly. Combining assumption $(c)$ with Proposition 6.3.6.3, we conclude that the morphism $f_0$ (respectively $f_1$, $f_{01}$) exhibits the simplicial set $S_0$ (respectively $S_1$, $S_{01}$) as a localization of $X_{0}$ (respectively $X_1$, $X_{01}$) with respect to $W_0$ (respectively $W_{1}$, $W_{01}$). Let $W$ be the collection of edges of $X$ given by the union of the images of $W_0$ and $W_1$. Note that conditions $(a)$ and $(b)$ guarantee that the front and back faces of the diagram (6.15) are categorical pushout squares (Proposition 4.5.4.11). Applying Proposition 6.3.4.2, we conclude that $f$ exhibits $S$ as a localization of $X$ with respect to $W$. $\square$