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6.3.6 Universal Localizations

The formation of localizations is generally not compatible with fiber products. If

$\xymatrix { X' \ar [r] \ar [d]^{f'} & X \ar [d]^{f} \\ S' \ar [r] & S }$

is a pullback diagram of simplicial sets where the morphism $f$ exhibits $S$ as a localization of $X$ (with respect to some collection of edges of $X$), then the morphism $f'$ need not have the same property. To address this point, it will be convenient to introduce a more restrictive notion of localization.

Definition 6.3.6.1. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We will say that $f$ is universally localizing if, for every morphism of simplicial sets $S' \rightarrow S$, the projection map $S' \times _{S} X \rightarrow S'$ exhibits $S'$ as a localization of $S' \times _{S} X$ with respect to some collection of edges $W$.

If $f: X \rightarrow S$ is a universally localizing morphism of simplicial sets, then it exhibits $S$ as a localization of $X$ with respect to some collection of edges $W$. It is possible to be more precise: we can take $W$ to be the collection of edges of $X$ having degenerate image in $S$.

Proposition 6.3.6.2. Let $f: X \rightarrow S$ be a morphism of simplicial sets. For every morphism $T \rightarrow S$, let $W_{T}$ denote the collection of all edges $w = (w_ T,w_ X)$ of the fiber product $T \times _{S} X$ for which $w_ T$ is a degenerate edge of $T$. The following conditions are equivalent:

$(1)$

For every morphism of simplicial sets $T \rightarrow S$, the projection map $T \times _{S} X \rightarrow T$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to $W_{T}$.

$(2)$

The morphism $f$ is universally localizing, in the sense of Definition 6.3.6.1.

$(3)$

For every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ exhibits $\Delta ^ n$ as a localization of $\Delta ^{n} \times _{S} X$ with respect to some collection of edges of $\Delta ^ n \times _{S} X$.

$(4)$

For every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ exhibits $\Delta ^ n$ as a localization of $\Delta ^{n} \times _{S} X$ with respect to $W_{\Delta ^ n}$.

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 6.3.1.20, 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 1.1.3.13, we see that there is a pushout square of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \Delta ^ n \ar [d]^{\sigma } \\ T' \ar [r] & T, }$

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. $\square$

Corollary 6.3.6.3. Let $f: X \rightarrow S$ be a universally localizing morphism of simplicial sets, and let $W$ be the collection of edges $w$ of $X$ for which $f(w)$ is a degenerate edge of $S$. Then $f$ exhibits $S$ as a localization of $X$ with respect to $W$.

Remark 6.3.6.4. Let $f: X \rightarrow S$ be a universally localizing morphism of simplicial sets. Then $f$ is a weak homotopy equivalence (see Remark 6.3.1.14).

Remark 6.3.6.5. Let $X$ be a simplicial set. Then the projection map $X \rightarrow \Delta ^0$ is universally localizing if and only if $X$ is weakly contractible. This follows by combining Propositions 6.3.1.18 and 6.3.5.1.

Remark 6.3.6.6. Suppose we are given a pullback diagram of simplicial sets

$\xymatrix { X' \ar [r] \ar [d]^{f'} & X \ar [d]^{f} \\ S' \ar [r] & S. }$

If $f$ is universally localizing, then $f'$ is universally localizing.

Proposition 6.3.6.7. Let $f: X \rightarrow S$ be a universally localizing morphism of simplicial sets. Then $f$ is surjective.

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

$\xymatrix { \operatorname{\partial \Delta }^{n} \times _{S} X \ar [r]^{\sim } \ar [d] & \Delta ^ n \times _{S} X \ar [d] \\ \operatorname{\partial \Delta }^{n} \ar [r] & \Delta ^ n, }$

where the vertical maps are weak homotopy equivalences (see Remarks 6.3.6.6 and 6.3.6.4). 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). $\square$

Proposition 6.3.6.8. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be universally localizing morphisms of simplicial sets. Then the composition $(g \circ f): X \rightarrow Z$ is universally localizing.

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 6.3.6.7 (and Remark 6.3.6.6) 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 6.3.6.2, 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 6.3.1.19, we conclude that $\pi = g' \circ f'$ exhibits $Z'$ as the localization of $X'$ with respect to $W_0 \cup W = W$, as desired. $\square$

Corollary 6.3.6.9. Let $f: X \rightarrow S$ be a universally localizing morphism of simplicial sets, and let $K$ be a weakly contractible simplicial set. Then the composite map $X \times K \rightarrow X \xrightarrow {f} S$ is universally localizing.

Proof. By virtue of Proposition 6.3.6.8, it will suffice to show that the projection map $X \times K \rightarrow X$ is universally localizing. Using Remark 6.3.6.6, we can reduce to the case $X = \Delta ^0$, in which case the desired result follows from Remark 6.3.6.5. $\square$

Proposition 6.3.6.10. The collection of universally localizing morphisms is closed under the formation of filtered colimits (when regarded as a full subcategory of the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$).

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 6.3.6.2 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 $W$. $\square$

Proposition 6.3.6.11. Suppose we are given a commutative diagram of simplicial sets

6.5
$$\begin{gathered}\label{equation:universal-localization-pushout} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [dr]^{ f_{01} } \ar [rr] \ar [dd] & & X_{0} \ar [dd] \ar [dr]^{ f_{0} } & \\ & S_{01} \ar [rr] \ar [dd] & & S_{0} \ar [dd] \\ X_{1} \ar [rr] \ar [dr] & & X \ar [dr]^{f} & \\ & S_{1} \ar [rr] & & S } \end{gathered}$$

with the following properties:

$(a)$

The front and back faces

$\xymatrix { S_{01} \ar [r] \ar [d] & S_0 \ar [d] & X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ S_1 \ar [r] & S & X_1 \ar [r] & X }$

are pushout squares.

$(b)$

The morphisms $S_{01} \rightarrow S_{0}$ and $X_{01} \rightarrow X_{0}$ are monomorphisms.

$(c)$

The morphisms $f_{01}$, $f_{0}$, and $f_{1}$ are universally localizing.

Then the morphism $f$ is universally localizing.

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 6.3.6.11 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 6.3.6.2, 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 (6.5) are categorical pushout squares (Proposition 4.5.3.7). Applying Proposition 6.3.4.2, we conclude that $f$ exhibits $S$ as a localization of $X$ with respect to $W$. $\square$