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 6.3.1.19).
\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ) \]
\[ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }} \rightarrow \operatorname{Set}\quad \quad \operatorname{\mathcal{E}}\mapsto \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } ). \]
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 not identical. 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:
Lemma 6.3.2.4. Let $Q$ be a contractible Kan complex, let $e: \Delta ^1 \hookrightarrow Q$ be a monomorphism of simplicial sets, and let $W = \{ \operatorname{id}_{ \Delta ^1} \} $ consist of the single nondegenerate edge of $\Delta ^1$. Then, for any $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $e$ induces a trivial Kan fibration of simplicial sets
\[ \theta : \operatorname{Fun}(Q, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \Delta ^1[ W^{-1}], \operatorname{\mathcal{E}}) = \operatorname{Isom}(\operatorname{\mathcal{E}}). \]
Proof. Since $e$ is a monomorphism, Corollary 4.4.5.3 immediately implies that $\theta $ is an isofibration when regarded as a functor from $\operatorname{Fun}(Q,\operatorname{\mathcal{E}})$ to $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})$. Using the pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(Q, \operatorname{\mathcal{E}}) \ar [d]^{\theta } \ar [r] & \operatorname{Fun}(Q, \operatorname{\mathcal{E}}) \ar [d]^{\theta } \\ \operatorname{Isom}(\operatorname{\mathcal{E}}) \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}), } \]
we deduce that $\theta $ is also an isofibration when regarded as a functor from $\operatorname{Fun}(Q, \operatorname{\mathcal{E}})$ to $\operatorname{Isom}(\operatorname{\mathcal{E}})$. Consequently, to show that $\theta $ is a trivial Kan fibration, it will suffice to show that it is an equivalence of $\infty $-categories (Proposition 4.5.5.20). In other words, we are reduced to proving that the morphism $e$ exhibits $Q$ as a localization of $\Delta ^1$ with respect to $W$. Let $q: Q \rightarrow \Delta ^0$ denote the projection map. Since $Q$ is contractible, the morphism $q$ is an equivalence of $\infty $-categories. By virtue of Remark 6.3.1.19, we are reduced to proving that the composite map $\Delta ^1 \xrightarrow {e} Q \xrightarrow {q} \Delta ^0$ exhibits $\Delta ^0$ as a localization of $\Delta ^1$ with respect to $W$, which follows from Example 6.3.1.14. $\square$
We will deduce Proposition 6.3.2.1 from the following more precise result:
Proposition 6.3.2.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$ such that, for each $w \in W$, the image $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $F$ factors as a composition where $G$ exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ and $H$ is an inner fibration (so that $\operatorname{\mathcal{C}}[W^{-1}]$ is also an $\infty $-category). Moreover, this factorization can be chosen to depend functorially on the diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and the collection of edges $W$, in such a way that the construction $(F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}, W) \mapsto \operatorname{\mathcal{C}}[W^{-1}]$ commutes with filtered colimits.
\[ \operatorname{\mathcal{C}}\xrightarrow {G} \operatorname{\mathcal{C}}[W^{-1}] \xrightarrow {H} \operatorname{\mathcal{D}}, \]
Proof. For each element $w \in W$, the image $F(w)$ can be regarded as a morphism from $\Delta ^1$ to the core $\operatorname{\mathcal{D}}^{\simeq }$. By virtue of Proposition 3.1.7.1, we can (functorially) choose a factorization of this morphism as a composition
\[ \Delta ^1 \xrightarrow { i_{w} } Q_{w} \xrightarrow { q_{w} } \operatorname{\mathcal{D}}^{\simeq }, \]
where $i_{w}$ is anodyne and $q_{w}$ is a Kan fibration. Since $\operatorname{\mathcal{D}}^{\simeq }$ is a Kan complex, $Q_{w}$ is also a Kan complex, which is contractible by virtue of the fact that $i_{w}$ is anodyne. Form a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \coprod _{w \in W} \Delta ^1 \ar [r] \ar [d]^{\coprod _{w \in W} i_ w} & \operatorname{\mathcal{C}}\ar [d]^{i} \\ \coprod _{w \in W} Q_ w \ar [r] & \operatorname{\mathcal{C}}'. } \]
We first claim that $i: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. Note that if $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism of simplicial sets which factors through $\operatorname{\mathcal{C}}'$, then for each $w \in W$ the morphism $G(w)$ belongs to the image of a functor $Q_ w \rightarrow \operatorname{\mathcal{E}}$, and is therefore an isomorphism in $\operatorname{\mathcal{E}}$. It follows that composition with $i$ induces a functor $\theta : \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$, and we wish to show that $\theta $ is an equivalence of $\infty $-categories. This follows by inspecting the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [d] \\ \prod _{w \in W} \operatorname{Fun}(Q_ w,\operatorname{\mathcal{E}}) \ar [r]^-{\theta '} & \prod _{w \in W} \operatorname{Isom}(\operatorname{\mathcal{E}}) \ar [r] & \prod _{w \in W} \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}}). } \]
The outer rectangle is a pullback square by the definition of $\operatorname{\mathcal{C}}'$, and the right square is a pullback by the definition of $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$. It follows that the left square is also a pullback. Lemma 6.3.2.4 implies that $\theta '$ is a trivial Kan fibration, so that $\theta $ is also a trivial Kan fibration (hence an equivalence of $\infty $-categories by Proposition 4.5.3.11).
Note that the morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and the collection of morphisms $\{ q_{w}: Q_ w \rightarrow \operatorname{\mathcal{D}}^{\simeq } \subseteq \operatorname{\mathcal{D}}\} _{w \in W}$ can be amalgamated to a single morphism of simplicial sets $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}$. Applying Proposition 4.1.3.2, we can (functorially) factor $F'$ as a composition $\operatorname{\mathcal{C}}' \xrightarrow {G'} \operatorname{\mathcal{C}}[W^{-1}] \xrightarrow {H} \operatorname{\mathcal{D}}$, where $G'$ is inner anodyne and $H$ is an inner fibration. We conclude by observing that the composite map $G = (G' \circ i): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, by virtue of Remark 6.3.1.19. $\square$
Proof of Proposition 6.3.2.1. Apply Proposition 6.3.2.5 in the special case $\operatorname{\mathcal{D}}= \Delta ^{0}$. $\square$
Variant 6.3.2.6. Let $\kappa $ be an uncountable cardinal, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of edges $W$ (Definition 6.3.1.9). If $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, then $\operatorname{\mathcal{D}}$ is essentially $\kappa $-small.
Proof. Without loss of generality, we may assume that $F$ is a monomorphism of simplicial sets. Choose a categorical equivalence of simplicial sets $u: \operatorname{\mathcal{C}}\rightarrow \overline{\operatorname{\mathcal{C}}}$, where $\overline{\operatorname{\mathcal{C}}}$ is $\kappa $-small, and form a pushout diagram of simplicial sets
6.11
\begin{equation} \begin{gathered}\label{equation:localization-essentially-small} \xymatrix { \operatorname{\mathcal{C}}\ar [r]^{u} \ar [d]^{F} & \overline{\operatorname{\mathcal{C}}} \ar [d]^{\overline{F}} \\ \operatorname{\mathcal{D}}\ar [r]^{v} & \overline{\operatorname{\mathcal{D}}} } \end{gathered} \end{equation}
Then (6.11) is a categorical pushout square (Example 4.5.4.12), so $v$ is also a categorical equivalence (Proposition 4.5.4.10). Moreover, the morphism $\overline{F}$ exhibits $\overline{D}$ as a localization of $\overline{\operatorname{\mathcal{C}}}$ with respect to $u(W)$ (Corollary 6.3.4.3). We may therefore replace $F$ by $\overline{F}$, and thereby reduce to proving Variant 6.3.2.6 in the special case where $\operatorname{\mathcal{C}}$ is $\kappa $-small. In particular, the set of edges $W$ is $\kappa $-small.
Let $Q$ be a contractible Kan complex which is equipped with a monomorphism $\Delta ^1 \hookrightarrow Q$ and has only countably many simplices. Form a pushout diagram of simplicial sets
\[ \xymatrix { \coprod _{w \in W} \Delta ^1 \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{G} \\ \coprod _{w \in W} Q \ar [r] & \operatorname{\mathcal{C}}', } \]
so that $\operatorname{\mathcal{C}}'$ is $\kappa $-small (Remark 4.7.4.6). It follows from Corollary 6.3.4.3 that the morphism $G$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Using Proposition 4.7.5.5, we can choose an inner anodyne morphism $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}''$, where $\operatorname{\mathcal{C}}''$ is a $\kappa $-small $\infty $-category. Then $\operatorname{\mathcal{C}}''$ is also a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, so Remark 6.3.2.2 supplies a categorical equivalence of simplicial sets $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}''$. It follows that $\operatorname{\mathcal{D}}$ is essentially $\kappa $-small, as desired. $\square$