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5.2.1 Localizations of $\infty $-Categories

We begin by introducing some terminology.

Notation 5.2.1.1. 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 let $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those morphisms $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ that carry each edge of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$.

Remark 5.2.1.2. In the context of Notation 5.2.1.1, we will usually be interested in the situation where the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category (as suggested by the notation). However, it will be technically convenient to allow more general simplicial sets as well.

Example 5.2.1.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of degenerate edges of $\operatorname{\mathcal{C}}$. Then, for every $\infty $-category $\operatorname{\mathcal{E}}$, we have $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.

Example 5.2.1.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{E}}$ is a Kan complex, then $\operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (see Proposition 1.3.6.11).

Example 5.2.1.5. Let $W = \{ \operatorname{id}_{\Delta ^1} \} $ consist of the single nondegenerate edge of the standard $1$-simplex $\Delta ^1$. For every $\infty $-category $\operatorname{\mathcal{E}}$, $\operatorname{Fun}(\Delta ^1[W^{-1}], \operatorname{\mathcal{E}})$ is the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}})$ spanned by the isomorphisms in $\operatorname{\mathcal{E}}$ (Example 4.4.1.13).

Example 5.2.1.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category (Definition 1.2.5.1). Let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, let $[W]$ denote the collection of morphisms in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ which belong to the image of $W$, and let $F: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ be a functor of ordinary categories which exhibits $\operatorname{\mathcal{D}}$ as a strict localization of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with respect to $[W]$ (Definition 5.2.0.1). If $\operatorname{\mathcal{E}}$ is an ordinary category, then we have a canonical isomorphism of simplicial sets

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) ) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ). \]

Remark 5.2.1.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. For every $\infty $-category $\operatorname{\mathcal{E}}$, the canonical isomorphism $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ) \simeq \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}))$ restricts to an isomorphism of full subcategories

\[ \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ) \simeq \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) ). \]

This follows immediately from the criterion of Theorem 4.4.4.4.

Remark 5.2.1.8. 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. Then the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is replete. That is, if $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ are isomorphic objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, then $F$ carries edges of $W$ to isomorphisms in $\operatorname{\mathcal{E}}$ if and only if $F'$ carries edges of $W$ to isomorphisms in $\operatorname{\mathcal{E}}$ (see Example 4.4.1.13).

Definition 5.2.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. We say that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every $\infty $-category $\operatorname{\mathcal{E}}$, the precomposition map $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is fully faithful, and its essential image is the full subcategory $\operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.

Remark 5.2.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. If $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to a collection of edges $W$, then, for every $\infty $-category $\operatorname{\mathcal{E}}$ and every morphism $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, the composite map $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ carries each element of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$. In particular, if $\operatorname{\mathcal{D}}$ itself is an $\infty $-category, then $F$ carries each element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}$.

Example 5.2.1.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $W$ be a collection of degenerate edges of $\operatorname{\mathcal{C}}$. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if and only if it is a categorical equivalence of simplicial sets (see Proposition 4.5.2.8).

Proposition 5.2.1.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ (Definition 5.2.1.9).

$(2)$

For every $\infty $-category $\operatorname{\mathcal{E}}$, the functor $\operatorname{Fun}( \operatorname{\mathcal{D}},\operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ factors through the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ and induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$.

$(3)$

For every $\infty $-category $\operatorname{\mathcal{E}}$, the functor $\operatorname{Fun}( \operatorname{\mathcal{D}},\operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ factors through the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ and induces a homotopy equivalence of Kan complexes $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq }$.

$(4)$

For every $\infty $-category $\operatorname{\mathcal{E}}$, the functor $\operatorname{Fun}( \operatorname{\mathcal{D}},\operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ factors through the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ and induces a bijection of sets $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq } )$.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Corollary 5.1.2.19 (and the repleteness of the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$). The implication $(2) \Rightarrow (3)$ follows from Remark 4.5.1.19 and the implication $(3) \Rightarrow (4)$ from Remark 3.1.5.4. We will complete the proof by showing that $(4) \Rightarrow (2)$. Assume that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfies condition $(4)$, and let $\operatorname{\mathcal{E}}$ be an $\infty $-category; we wish to show that the precomposition functor $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories. For this, it will suffice to show that for every simplicial set $\operatorname{\mathcal{B}}$, the induced map

\[ \theta : \pi _0( \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } )) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) )^{\simeq } ) \]

is a bijection. Using Remark 5.2.1.7, we can identify $\theta $ with the map

\[ \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}) )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{Fun}(\operatorname{\mathcal{B}}, \cal \operatorname{\mathcal{E}}) )^{\simeq } ), \]

which is bijective by virtue of assumption $(4)$. $\square$

Example 5.2.1.13. Let $W = \{ \operatorname{id}_{\Delta ^1} \} $ consist of the single nondegenerate edge of the standard $1$-simplex $\Delta ^1$. Then the projection map $\Delta ^1 \rightarrow \Delta ^0$ exhibits $\Delta ^0$ as a localization of $\Delta ^1$ with respect to $W$. To prove this, it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{E}}$, the construction $X \mapsto \operatorname{id}_{X}$ induces a bijection of sets $\pi _0( \operatorname{\mathcal{E}}^{\simeq } ) \rightarrow \pi _0( \operatorname{Isom}(\operatorname{\mathcal{E}})^{\simeq } )$. The injectivity of this map is clear (the diagonal embedding $\operatorname{\mathcal{E}}\rightarrow \operatorname{Isom}(\operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})$ has a left inverse, given by evaluation at either vertex of $\Delta ^1$). To establish surjectivity, it will suffice to show that every isomorphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{E}}$ is isomorphic, when viewed as an object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})$, to the identity morphism $\operatorname{id}_{X}$. This follows from the observation that there exists a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{\operatorname{id}_ X} \ar [r]^-{\operatorname{id}_{X}} & X \ar [d]^{f} \\ X \ar [r]^-{f} & Y } \]

in the $\infty $-category $\operatorname{\mathcal{E}}$, which we can view as an isomorphism from $\operatorname{id}_{X}$ to $f$ in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}})$ (see Theorem 4.4.4.4).

Remark 5.2.1.14. 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 a collection of edges $W$. Then, for every Kan complex $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

(see Example 5.2.1.4). It follows that $F$ is a weak homotopy equivalence of simplicial sets.

Remark 5.2.1.15. 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 a collection of edges $W$. Let $[W]$ denote the collection of morphisms in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ which belong to the image of $W$. Then the induced functor $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ exhibits the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ as a classical localization of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with respect to $[W]$, in the sense of Definition 5.2.0.6. This follows immediately from Example 5.2.1.6.

Example 5.2.1.16. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, which we identify with edges of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Let $F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with respect to $W$. Then the induced functor $\operatorname{\mathcal{C}}\simeq \mathrm{h} \mathit{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \xrightarrow { \mathrm{h} \mathit{F} } \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ exhibits the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ as a classical localization of $\operatorname{\mathcal{C}}$ with respect to $W$, in the sense of in the sense of Definition 5.2.0.6.

Remark 5.2.1.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets, and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. If any two of the following three conditions is satisfied, then so is the third:

  • The morphism $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

  • The morphism $G \circ F$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

  • The morphism $G$ is a categorical equivalence of simplicial sets.

Proposition 5.2.1.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category, and let $W$ be the collection of all edges of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • The morphism $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

  • The $\infty $-category $\operatorname{\mathcal{D}}$ is a Kan complex and $F$ is a weak homotopy equivalence of simplicial sets.

Proof. We first prove that $(2)$ implies $(1)$. Assume that $\operatorname{\mathcal{D}}$ is a Kan complex and that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a weak homotopy equivalence; we wish to show that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. By virtue of Proposition 5.2.1.12, it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{E}}$, composition with $F$ induces a homotopy equivalence of Kan complexes $\theta : \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})^{\simeq }$. Since $\operatorname{\mathcal{D}}$ is a Kan complex, Proposition 4.4.3.18 allows us to identify $\theta $ with the canonical map

\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}^{\simeq }) \xrightarrow { \circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}^{\simeq } ), \]

which is a homotopy equivalence by virtue of our assumption that $F$ is a weak homotopy equivalence.

We now show that $(1)$ implies $(2)$. Assume that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Invoking Remark 5.2.1.14, we deduce that $F$ is a weak homotopy equivalence. We wish to show that $\operatorname{\mathcal{D}}$ is a Kan complex. Choose a weak homotopy equivalence $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, where $\operatorname{\mathcal{E}}$ is a Kan complex (Corollary 3.1.6.2). Then the composite map $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is also a weak homotopy equivalence (Remark 3.1.5.16). Invoking the implication $(2) \Rightarrow (1)$, we conclude that $G \circ F$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. It follows from Remark 5.2.1.17 that $G$ is an equivalence of $\infty $-categories. Since $\operatorname{\mathcal{E}}$ is a Kan complex, it follows that the $\infty $-category $\operatorname{\mathcal{D}}$ is also a Kan complex (Remark 4.5.1.21). $\square$