9.1.1 Local Objects
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $W$. Recall that $F$ is a reflective localization if it admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. In this case, the functor $G$ is automatically fully faithful (Proposition 6.3.3.6), and its essential image is a reflective subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ (Corollary 6.2.2.17). In this situation, we can extract the subcategory $\operatorname{\mathcal{C}}'$ directly from $W$.
Definition 9.1.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We say that an object $C \in \operatorname{\mathcal{C}}$ is $w$-local if precomposition with the homotopy class $[w]$ induces a homotopy equivalence of mapping spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,C) \xrightarrow {\circ [w]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)$. We say that $C$ is $w$-colocal if postcomposition with $[w]$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \xrightarrow {[w] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$.
If $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, we say that an object $C \in \operatorname{\mathcal{C}}$ is $W$-local if it is $w$-local for each $w \in W$. Similarly, we say that $C$ is $W$-colocal if it is $w$-colocal for each $w \in W$.
Example 9.1.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $w: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$. Then every object of $\operatorname{\mathcal{C}}$ is $w$-local.
Variant 9.1.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $w: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and let $C \in \operatorname{\mathcal{C}}$ be an object which is $w$-local. Then $C$ is $w'$-local, for any morphism $w': X' \rightarrow Y'$ which is a retract of $w$ (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).
Proposition 9.1.1.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collections of morphisms $W$, and let $C$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The object $C$ is $W$-local, in the sense of Definition 9.1.1.1.
- $(2)$
For every object $C' \in \operatorname{\mathcal{C}}$, the functor $F$ induces a homotopy equivalence of mapping spaces $\theta _{C',C}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C',C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C'), F(C) )$.
Proof.
Fix an uncountable regular cardinal $\kappa $ for which both $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\kappa $-small. Precomposition with $F$ determines a functor $F^{\ast }: \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. It follows from Proposition 7.6.7.13 that the functor $F^{\ast }$ admits a left adjoint $F_{!}$ (given by left Kan extension along $F^{\operatorname{op}}$). Let $h^{\operatorname{\mathcal{C}}}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ and $h^{\operatorname{\mathcal{D}}}_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively, so that the diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{F} \ar [r]^-{h^{\operatorname{\mathcal{C}}}_{\bullet } } & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [d]^{ F_{!} } \\ \operatorname{\mathcal{D}}\ar [r]^-{ h^{\operatorname{\mathcal{D}}}_{\bullet } } & \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) } \]
commutes up to isomorphism (Example 8.4.4.5), where the horizontal maps are fully faithful (Theorem 8.3.3.13). It follows that, for every pair of objects $C', C \in \operatorname{\mathcal{C}}$, we can identify $\theta _{C',C}$ with the comparison map
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h^{\operatorname{\mathcal{C}}}_{C'}, h^{\operatorname{\mathcal{C}}}_{C} ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( F_{!} h^{\operatorname{\mathcal{C}}}_{C'}, F_{!} h^{\operatorname{\mathcal{C}}}_{C} ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h^{\operatorname{\mathcal{C}}}_{C'}, F^{\ast } F_{!} h^{\operatorname{\mathcal{C}}}_{C} ) \end{eqnarray*}
given by precomposition with the unit $u: h^{\operatorname{\mathcal{C}}}_{C} \rightarrow F^{\ast } F_{!} h^{\operatorname{\mathcal{C}}}_{C}$. Combining this observation with Proposition 8.3.1.1, we see that condition $(2)$ can be restated as follows:
- $(2')$
The unit map $u: h^{\operatorname{\mathcal{C}}}_{C} \rightarrow F^{\ast } F_{!} h^{\operatorname{\mathcal{C}}}_{C}$ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
Our assumption that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ guarantees that the pullback functor $F^{\ast }$ is fully faithful. Using Remark 6.2.2.18, we see that $u$ is an isomorphism if and only the representable functor $h_{C}^{\operatorname{\mathcal{C}}}$ belongs to the essential image of $F^{\ast }$: that is, the collection of functors $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ which carry every morphism of $W$ to an isomorphism in $\operatorname{\mathcal{S}}^{< \kappa }$. This is a reformulation of $(1)$.
$\square$
Corollary 9.1.1.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $W$. Suppose that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Then $G$ is fully faithful, and the essential image of $G$ is spanned by the collection of $W$-local objects of $\operatorname{\mathcal{C}}$.
Proof.
The assertion that $G$ is fully faithful follows from Proposition 6.3.3.6. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be the unit of an adjunction between $F$ and $G$. Then an object $C \in \operatorname{\mathcal{C}}$ belongs to the essential image of $G$ if and only if the morphism $\eta _{C}: C \rightarrow (G \circ F)(C)$ is an isomorphism. This is equivalent to the requirement that, for every object $B \in \operatorname{\mathcal{C}}$, composition with $\eta _{C}$ induces a homotopy equivalence of mapping spaces $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, (G \circ F)(C) )$. We conclude by observing that $\theta $ factors as a composition
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B,C) \xrightarrow { F } \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(B), F(C) ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, (G \circ F)(C) ) \]
where the second map is the homotopy equivalence of Proposition 6.2.1.17.
$\square$
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. Beware that, in general, $\operatorname{\mathcal{C}}'$ is not a reflective subcategory of $\operatorname{\mathcal{C}}$. To ensure this, we need some additional assumptions on $W$.
Definition 9.1.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $W$ is localizing if the following conditions are satisfied:
- $(1)$
Every isomorphism of $\operatorname{\mathcal{C}}$ is contained in $W$.
- $(2)$
The collection of morphisms $W$ satisfies the two-out-of-three property. That is, for every $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]
in $\operatorname{\mathcal{C}}$, if any two of the morphisms $u$, $v$, and $w$ belongs to $W$, then so does the third.
- $(3)$
For every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $w: X \rightarrow Y$ which belongs to $W$, where the object $Y$ is $W$-local.
We will say that $W$ is colocalizing if it satisfies conditions $(1)$ and $(2)$, together with the following dual version of $(3)$:
- $(3')$
For every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $w: Y \rightarrow X$ which belongs to $W$, where the object $Y$ is $W$-colocal.
Proposition 9.1.1.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a reflective localization functor, and let $W$ be the collection of all morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $F(w)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $W$ is localizing.
Proof.
Conditions $(1)$ and $(2)$ of Definition 9.1.1.15 follow immediately from the definitions (and do not require any assumptions on $F$). We will verify condition $(3)$. Since $F$ is a reflective localization functor, it admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For every object $Y \in \operatorname{\mathcal{D}}$, the image $G(Y) \in \operatorname{\mathcal{C}}$ is $W$-local (Corollary 9.1.1.14). In particular, if $X$ is an object of $\operatorname{\mathcal{C}}$, then $(G \circ F)(X)$ is a $W$-local object of $\operatorname{\mathcal{C}}$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be the unit of an adjunction between $F$ and $G$. To complete the proof, it will suffice to show that the unit map $\eta _{X}: X \rightarrow (G \circ F)(X)$ belongs to $W$: that is, that $F(\eta _{X})$ is an isomorphism in $\operatorname{\mathcal{D}}$. Since the functor $G$ is fully faithful, this follows from Remark 6.3.3.5.
$\square$
Corollary 9.1.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a left adjoint to the inclusion functor $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$. Let $W$ be the collection of all morphisms $w: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ for which $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$. Then:
- $(1)$
The collection $W$ is localizing (Definition 9.1.1.15).
- $(2)$
Every object of $\operatorname{\mathcal{C}}'$ is $W$-local (Definition 9.1.1.1).
- $(3)$
If $\operatorname{\mathcal{C}}'$ is replete, then every $W$-local object of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}'$.
Proof.
Combine Proposition 9.1.1.16 with Corollary 9.1.1.14.
$\square$
We now prove the converse of Corollary 9.1.1.17: every localizing collection of morphisms of an $\infty $-category $\operatorname{\mathcal{C}}$ can be obtained from a reflective localization of $\operatorname{\mathcal{C}}$.
Proposition 9.1.1.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a localizing collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}'$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. Then:
- $(1)$
The full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective (Definition 6.2.2.1).
- $(2)$
The inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$.
- $(3)$
A morphism $w$ of $\operatorname{\mathcal{C}}$ is contained in $W$ if and only if $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$.
- $(4)$
The functor $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.
Proof.
Let $X$ be an object of $\operatorname{\mathcal{C}}$. Our assumption that $W$ is localizing guarantees that there exists a morphism $w_ X: X \rightarrow X'$ which belongs to $W$, where $X' \in \operatorname{\mathcal{C}}'$. By definition, every object $C \in \operatorname{\mathcal{C}}'$ is $W$-local, so composition with $w_ X$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X',C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, C)$. It follows that $w_ X$ exhibits $X'$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, in the sense of Definition 6.2.2.1. Assertion $(1)$ follows by allowing the object $X$ to vary. The implication $(1) \Rightarrow (2)$ follows from Proposition 6.2.2.15, and the implication $(3) \Rightarrow (4)$ from Example 6.3.3.7.
It remains to prove $(3)$. Choose a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow L$ which exhibits $L$ as a $\operatorname{\mathcal{C}}'$-reflection functor (see Definition 6.2.2.12). For each object $X \in \operatorname{\mathcal{C}}$, the morphism $\eta _{X}: X \rightarrow L(X)$ exhibits $L(X)$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, and can therefore be obtained by composing $w_{X}$ with an isomorphism $X' \xrightarrow {\sim } L(X)$. Since $W$ contains all isomorphisms and is closed under composition, it follows that $\eta _{X}$ belongs to $W$.
For every morphism $w: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the natural transformation $\eta $ determines a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{w} \ar [d]^{\eta _ X} & Y \ar [d]^{\eta _ Y} \\ L(X) \ar [r]^-{ L(w) } & L(Y) } \]
where $\eta _ X$ and $\eta _ Y$ belong to $W$. Using the two-out-of-three property, we see that $w$ is contained in $W$ if and only if $L(w)$ is contained in $W$. Since $L(X)$ and $L(Y)$ are $W$-local, this is equivalent to the requirement that $L(w)$ is an isomorphism (Remark 9.1.1.5).
$\square$
Corollary 9.1.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a pushout diagram
9.2
\begin{equation} \begin{gathered}\label{equation:localizing-morphisms-closed-under-pushout} \xymatrix@C =50pt@R=50pt{ X \ar [r]^-{w} \ar [d] & Y \ar [d] \\ X' \ar [r]^-{w'} & Y' } \end{gathered} \end{equation}
in $\operatorname{\mathcal{C}}$. If $W$ is a localizing collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $w$, then it also contains $w'$.
Proof.
Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $W$-local objects and let $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a left adjoint to the inclusion. The assumption $w \in W$ guarantees that $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$ (Proposition 9.1.1.18). Since $L$ carries the (9.2) to a pushout diagram in the $\infty $-category $\operatorname{\mathcal{C}}'$ (Corollary 7.1.3.21), it follows that $L(w')$ is also an isomorphism (Corollary 7.6.3.24). Applying Proposition 9.1.1.18 again, we conclude that $w'$ belongs to $W$.
$\square$
Notation 9.1.1.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a localizing collection of morphisms of $\operatorname{\mathcal{C}}$. We will often write $\operatorname{\mathcal{C}}[W^{-1}]$ for the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. By virtue of Proposition 9.1.1.18, this is consistent with Remark 6.3.2.2: that is, we can regard $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. This convention is very convenient, since the full subcategory of $W$-local objects is uniquely determined by $\operatorname{\mathcal{C}}$ and $W$. However, it has the potential to create confusion in some situations: see Warning 9.1.1.23 below.
Corollary 9.1.1.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the construction $W \mapsto \operatorname{\mathcal{C}}[W^{-1}]$ determines a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Localizing collections of morphisms of $\operatorname{\mathcal{C}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Reflective replete subcategories of $\operatorname{\mathcal{C}}$} \} . } \]
Proof.
Combine Proposition 9.1.1.18 with Corollary 9.1.1.17.
$\square$
Proposition 9.1.1.18 has a counterpart for colocalizing collections of morphisms:
Variant 9.1.1.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is colocalizing, and let $\operatorname{\mathcal{C}}'$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-colocal objects. Then:
- $(1)$
The full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is coreflective.
- $(2)$
The inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ admits a right adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$.
- $(3)$
A morphism $w$ of $\operatorname{\mathcal{C}}$ is contained in $W$ if and only if $L(w)$ is an isomorphism in $\operatorname{\mathcal{C}}'$.
- $(4)$
The functor $L$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.
Warning 9.1.1.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is both localizing and colocalizing. In this case, Proposition 9.1.1.18 and Variant 9.1.1.22 provide two different concrete realizations of the localization $\operatorname{\mathcal{C}}[W^{-1}]$, given by the full subcategories $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}\supseteq \operatorname{\mathcal{C}}''$ spanned by the $W$-local and $W$-colocal objects of $\operatorname{\mathcal{C}}$, respectively. Note that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ are necessarily equivalent as abstract $\infty $-categories. More precisely, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ is a functor which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as the localization of $\operatorname{\mathcal{C}}$ with respect to $W$, then the restrictions
\[ \operatorname{\mathcal{C}}' \xrightarrow { F|_{\operatorname{\mathcal{C}}'} } \operatorname{\mathcal{C}}[W^{-1}] \xleftarrow { F|_{\operatorname{\mathcal{C}}''} } \operatorname{\mathcal{C}}'' \]
are equivalences of $\infty $-categories. Beware that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ usually do not coincide when regarded as subcategories of $\operatorname{\mathcal{C}}$. See Warning 6.3.3.12.