Proposition 6.3.2.1 (Existence of Localizations). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. Then there exists an $\infty $-category $\operatorname{\mathcal{D}}$ and a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
6.3.2 Existence of Localizations
Our goal in this section is to prove the following:
Remark 6.3.2.2 (Uniqueness of Localizations). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. Proposition 6.3.2.1 asserts that there exists an $\infty $-category $\operatorname{\mathcal{D}}$ and a morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. In this case, for every $\infty $-category $\operatorname{\mathcal{E}}$, composition with $F$ induces a bijection (Proposition 6.3.1.13). In other words, the $\infty $-category $\operatorname{\mathcal{D}}$ corepresents the functor It follows that $\operatorname{\mathcal{D}}$ is uniquely determined (up to canonical isomorphism) as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$. We will sometimes emphasize this uniqueness by referring to $\operatorname{\mathcal{D}}$ as the localization of $\operatorname{\mathcal{C}}$ with respect to $W$, and denoting it by $\operatorname{\mathcal{C}}[W^{-1}]$. Beware that the localization $\operatorname{\mathcal{C}}[W^{-1}]$ is not well-defined up to isomorphism as a simplicial set: in fact, any equivalent $\infty $-category can also be regarded as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Remark 6.3.1.20).
\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ) \]
\[ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }} \rightarrow \operatorname{Set}\quad \quad \operatorname{\mathcal{E}}\mapsto \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ). \]
Warning 6.3.2.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ be an $\infty $-category. We have now given two different definitions for the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$:
According to Notation 6.3.1.1, $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ denotes the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those diagrams $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ which carry each edge of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$.
By the convention of Remark 6.3.2.2, $\operatorname{\mathcal{C}}[W^{-1}]$ denotes an $\infty $-category equipped with a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. We can then consider the $\infty $-category of functors from $\operatorname{\mathcal{C}}[W^{-1}]$ to $\operatorname{\mathcal{E}}$, which we will temporarily denote by $\operatorname{Fun}'( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$.
Beware that these $\infty $-categories are usually not isomorphic. However, they are equivalent: if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, then composition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}'( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ (Proposition 6.3.1.13). Note that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ does not depend on any auxiliary choices: it is well-defined up to equality as a simplicial subset of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. By contrast, the $\infty $-category $\operatorname{Fun}'( \operatorname{\mathcal{C}}[W^{-1}],\operatorname{\mathcal{E}})$ depends on the choice of the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ (and is therefore well-defined up to equivalence, but not up to isomorphism).
Our proof of Proposition 6.3.2.1 will make use of the following:
Proposition 6.3.2.4. Suppose we are given a diagram of simplicial sets where $F_0$ exhibits $\operatorname{\mathcal{D}}_0$ as a localization of $\operatorname{\mathcal{C}}_0$ with respect to some collection of edges $W_0$. Then ( 6.11 ) is a categorical pushout square if and only if $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W = T(W_0)$.
6.11
\begin{equation} \begin{gathered}\label{equation:localization-and-categorical-pushout-easy} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [r]^{F_0} \ar [d]^-{T} & \operatorname{\mathcal{D}}_0 \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^{F} & \operatorname{\mathcal{D}}, } \end{gathered} \end{equation}
Proof. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. For any diagram $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, the composition $(G \circ F)|_{\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{E}}$ factors through $F_0$, and therefore carries each edge of $W_0$ to an isomorphism in $\operatorname{\mathcal{E}}$. It follows that $G \circ F$ carries each edge of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$. We therefore have a commutative diagram of $\infty $-categories
6.12
\begin{equation} \begin{gathered}\label{equation:localization-and-categorical-pushout-easy2} \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \ar [r]^{\circ F} \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}}) \ar [r] \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [d]^{ \circ T} \\ \operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}}) \ar [r]^{ \circ F_0} & \operatorname{Fun}( \operatorname{\mathcal{D}}_0[W_0^{-1}], \operatorname{\mathcal{E}}) \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}}). } \end{gathered} \end{equation}
The right side of (6.12) is a pullback square in which the horizontal maps are isofibrations (Remark 6.3.1.8), and therefore a categorical pullback square (Corollary 4.5.2.27). By assumption, precomposition with $F_0$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_0[W_0^{-1}], \operatorname{\mathcal{E}})$. Applying Propositions 4.5.2.18 and 4.5.2.21, we conclude that the outer rectangle is a categorical pullback square if and only if precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$. The desired result now follows by allowing the $\infty $-category $\operatorname{\mathcal{E}}$ to vary. $\square$
Using Proposition 4.5.2.21, we can give a very explicit construction localizations $\operatorname{\mathcal{C}}[W^{-1}]$ when $W$ is a collection of edges in “general position”.
Corollary 6.3.2.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{D}}$ be the quotient of $\operatorname{\mathcal{C}}$ obtained by collapsing each edge $w \in W$, so that we have a pushout diagram of simplicial sets If $\iota $ is a monomorphism of simplicial sets, then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.
6.13
\begin{equation} \begin{gathered}\label{equation:easy-localization} \xymatrix@C =50pt@R=50pt{ \coprod _{w \in W} \Delta ^1 \ar [r]^{\iota } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \coprod _{w \in W} \Delta ^0 \ar [r] & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}
Proof. The assumption that $\iota $ is a monomorphism guarantees that the diagram (6.13) is a categorical pushout square (Example 4.5.4.12). By virtue of Proposition 6.3.2.4, it will suffice to show that the left vertical map of (6.13) exhibits $\operatorname{\mathcal{D}}_0 = \coprod _{w \in W} \Delta ^0$ as a localization of $\operatorname{\mathcal{C}}_0 = \coprod _{w \in W} \Delta ^1$ with respect to the collection of nondegenerate edges of $\operatorname{\mathcal{C}}_0$. This follows by combining Example 6.3.1.14 with Remark 6.3.1.15. $\square$
Example 6.3.2.6 (Contracting an Edge). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $w: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{C}}$ where $X \neq Y$. Let $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}/ \Delta ^1$ be the simplicial set obtained from $\operatorname{\mathcal{C}}$ by collapsing the edge $w$. Then the quotient map $\operatorname{\mathcal{C}}\twoheadrightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W = \{ w\} $.
To prove Proposition 6.3.2.1 in general, we will need a homotopy-invariant replacement for the operation of collapsing edges.
Corollary 6.3.2.7. Let $Q$ denote the quotient of the simplicial set $\Delta ^3$ obtained by collapsing the edges $e = \operatorname{N}_{\bullet }( \{ 0 < 2 \} )$ and $e' = \operatorname{N}_{\bullet }( \{ 1 < 3 \} )$, so that we have a pushout diagram Then the projection map $Q \rightarrow \Delta ^0$ is a categorical equivalence of simplicial sets.
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \amalg \operatorname{N}_{\bullet }( \{ 1 < 3 \} ) \ar [r] \ar [d] & \Delta ^3 \ar [d] \\ \Delta ^0 \amalg \Delta ^0 \ar [r] & Q. } \]
Proof. It follows from Corollary 6.3.2.5 that the quotient map $\Delta ^3 \twoheadrightarrow Q$ exhibits $Q$ as a localization of $\Delta ^3$ with respect to $W = \{ e, e' \} $. It will therefore suffice to show that the projection map $q: \Delta ^3 \rightarrow \Delta ^0$ has the same property (Corollary 6.3.1.20). Note that $q$ is a weak homotopy equivalence, and therefore exhibits $\Delta ^0$ as a localization of $\Delta ^3$ with respect to the collection $W'$ of all edges of $\Delta ^3$. To complete the proof, it will suffice to show that these localizations are the same: that is, a functor of $\infty $-categories $F: \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ carries each edge of $W$ to an isomorphism if and only if it carries each edge of $W'$ to an isomorphism. This is a restatement of the two-out-of-six property for isomorphisms (Proposition 5.4.6.5). $\square$
Corollary 6.3.2.8. Let $Q$ be the quotient of $\Delta ^3$ described in Corollary 6.3.2.7. Then the morphism exhibits $Q$ as a localization of $\Delta ^1$ with respect to $\{ \operatorname{id}_{\Delta ^1} \} $.
\[ e: \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \hookrightarrow \Delta ^3 \twoheadrightarrow Q \]
Proof. By virtue of Corollary 6.3.2.7, this is equivalent to the statement that the projection map $\Delta ^1 \twoheadrightarrow \Delta ^0$ exhibits $\Delta ^0$ as a localization of $\Delta ^1$ with respect to $\{ \operatorname{id}_{\Delta ^1} \} $ (Remark 6.3.1.20), which follows from Example 6.3.1.14. $\square$
Corollary 6.3.2.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, and form a pushout diagram of simplicial sets where $e: \Delta ^1 \rightarrow Q$ is the morphism described in Corollary 6.3.2.8. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.
6.14
\begin{equation} \begin{gathered}\label{equation:make-general-localization} \xymatrix@C =50pt@R=50pt{ \coprod _{w \in W} \Delta ^1 \ar [r] \ar [d]^{\coprod _{w \in W} e} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \coprod _{w \in W} Q \ar [r] & \operatorname{\mathcal{D}}, } \end{gathered} \end{equation}
Proof. Note that $e$ is a monomorphism of simplicial sets: that is, it corresponds to an edge of $Q$ where the source and target are distinct. It follows that the diagram (6.14) is a categorical pushout square (Example 4.5.4.12). By virtue of Proposition 6.3.2.4, it will suffice to show that the left vertical map of (6.14) exhibits $\operatorname{\mathcal{D}}_0 = \coprod _{w \in W} Q$ as a localization of $\operatorname{\mathcal{C}}_0 = \coprod _{w \in W} \Delta ^1$ with respect to the collection of nondegenerate edges of $\operatorname{\mathcal{C}}_0$. This follows by combining Corollary 6.3.2.8 with Remark 6.3.1.15. $\square$
Remark 6.3.2.10 (Sizes of Localization). In the situation of Corollary 6.3.2.9, suppose that the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small, for some infinite cardinal $\kappa $. Then the simplicial set $\operatorname{\mathcal{D}}$ is also $\kappa $-small. This follows from the observation that $Q$ is a finite simplicial set (in fact, $Q$ has exactly eleven nondegenerate simplices).
Proposition 6.3.2.1 is a consequence of the following more precise assertion:
Proposition 6.3.2.11. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. Then there exists an $\infty $-category $\operatorname{\mathcal{C}}[W^{-1}]$ and a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Moreover, we can arrange that the assignment $(\operatorname{\mathcal{C}},W) \mapsto ( \operatorname{\mathcal{C}}[W^{-1}], F )$ is functorial and commutes with filtered colimits.
Proof. Combine Corollary 6.3.2.9 with Proposition 4.1.3.2. $\square$
Corollary 6.3.2.12. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets which is given as the colimit (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$) of a filtered diagram of morphisms $\{ G_{\alpha }: \operatorname{\mathcal{D}}_{\alpha } \rightarrow \operatorname{\mathcal{E}}_{\alpha } \} $. Assume that:
Each morphism $G_{\alpha }$ exhibits $\operatorname{\mathcal{E}}_{\alpha }$ as a localization of $\operatorname{\mathcal{D}}_{\alpha }$ with respect to some collection of edges $S_{\alpha }$.
Each of the transition maps $\operatorname{\mathcal{D}}_{\alpha } \rightarrow \operatorname{\mathcal{E}}_{\beta }$ of the diagram carries $S_{\alpha }$ into $S_{\beta }$.
Let us regard $S = \varinjlim S_{\alpha }$ as a collection of edges of the simplicial set $\operatorname{\mathcal{D}}$. Then $G$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{D}}$ with respect to $S$.
Proof. For every simplicial set $\operatorname{\mathcal{C}}$ equipped with a set of edges $W$, let $F_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ be as in Proposition 6.3.2.11. For each index $\alpha $, let $T_{\alpha }$ denote the image of $S_{\alpha }$ in $\operatorname{\mathcal{E}}_{\alpha }$. We then have a commutative diagram of simplicial sets
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{D}}_{\alpha } \ar [r]^{G_{\alpha }} \ar [d]^{ F_{\operatorname{\mathcal{D}}_{\alpha }}} & \operatorname{\mathcal{E}}_{\alpha } \ar [d]^{ F_{\operatorname{\mathcal{E}}_{\alpha }} } \\ \operatorname{\mathcal{D}}_{\alpha }[ S_{\alpha }^{-1} ] \ar [r] & \operatorname{\mathcal{E}}_{\alpha }[ T_{\alpha }^{-1} ], } \]
depending functorially on $\alpha $. Our assumption that $G_{\alpha }$ exhibits $\operatorname{\mathcal{E}}_{\alpha }$ as a localization of $\operatorname{\mathcal{D}}_{\alpha }$ with respect to $S_{\alpha }$ guarantees that the lower horizontal and right vertical maps are categorical equivalences of simplicial sets. Setting $U = G(T)$ and passing to the colimit over $\alpha $, we obtain a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{D}}\ar [r]^{G} \ar [d]^{F_{\operatorname{\mathcal{D}}}} & \operatorname{\mathcal{E}}\ar [d]^{ F_{\operatorname{\mathcal{E}}} } \\ \operatorname{\mathcal{D}}[ S^{-1} ] \ar [r] & \operatorname{\mathcal{E}}[ T^{-1} ] } \]
where the lower horizontal and right vertical maps are categorical equivalences (Corollary 4.5.7.2). It follows that $G$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{D}}$ with respect to $S$. $\square$
Proposition 6.3.2.13. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then any localization of $\operatorname{\mathcal{C}}$ is also essentially $\kappa $-small.
Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is $\kappa $-small. If $\operatorname{\mathcal{C}}[W^{-1}]$ is a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $W$, then there is a categorical equivalence of simplicial sets $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$, where $\operatorname{\mathcal{D}}$ is the localization of Corollary 6.3.2.9. Since $\operatorname{\mathcal{D}}$ is $\kappa $-small (Remark 6.3.2.10), it follows that $\operatorname{\mathcal{C}}[W^{-1}]$ is essentially $\kappa $-small (see Proposition 4.7.5.5). $\square$