6.2.3 Local Objects
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In this section, we discuss the classification of reflective subcategories $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. Our main goal is to show that if $\operatorname{\mathcal{C}}'$ is replete, then it is determined by the collection of $\operatorname{\mathcal{C}}'$-local equivalences in $\operatorname{\mathcal{C}}$ (Corollary 6.2.3.13). We begin by introducing some terminology.
Definition 6.2.3.1. 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 6.2.3.2. 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.
Proposition 6.2.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $W$ be the collection of $\operatorname{\mathcal{C}}'$-local equivalences in $\operatorname{\mathcal{C}}$. Then an object $C \in \operatorname{\mathcal{C}}$ is $W$-local if and only if it is isomorphic to an object of $\operatorname{\mathcal{C}}'$. In particular, if $\operatorname{\mathcal{C}}'$ is replete, then coincides with the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects.
Proof.
Let $C$ be an object of $\operatorname{\mathcal{C}}$. Assume that $C$ is $W$-local; we wish to show that $C$ is isomorphic to an object of $\operatorname{\mathcal{C}}'$ (the converse follows from Remarks 6.2.3.5 and 6.2.3.4). Since $\operatorname{\mathcal{C}}'$ is reflective, we can choose a $\operatorname{\mathcal{C}}'$-local equivalence $w: C \rightarrow C'$, where $C'$ belongs to $\operatorname{\mathcal{C}}'$. Since $C'$ is also $W$-local, Remark 6.2.3.7 guarantees that $w$ is an isomorphism, so that $C$ is isomorphic to the object $C' \in \operatorname{\mathcal{C}}'$.
$\square$
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 6.2.3.9. 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 $W$.
- $(2)$
Suppose we are given a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $v$ is contained in $W$. Then $u$ is contained in $W$ if and only if $w$ is contained in $W$.
- $(3)$
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.
Example 6.2.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $W$ be the collection of all $\operatorname{\mathcal{C}}'$-local equivalences in $\operatorname{\mathcal{C}}$. Then $W$ automatically satisfies conditions $(1)$ and $(2)$ of Definition 6.2.3.9 (Remarks 6.2.2.4 and 6.2.2.5). If the full subcategory $\operatorname{\mathcal{C}}'$ is reflective, then $W$ is localizing. Similarly, if $\operatorname{\mathcal{C}}'$ is a coreflective subcategory of $\operatorname{\mathcal{C}}$, then the collection of $\operatorname{\mathcal{C}}'$-colocal equivalences is colocalizing.
Lemma 6.2.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}'$ be a full subcategory, and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which satisfies the following conditions:
- $(0)$
Every morphism $u: X \rightarrow Y$ which belongs to $W$ is a $\operatorname{\mathcal{C}}'$-local equivalence.
- $(1)$
Every isomorphism of $\operatorname{\mathcal{C}}$ is contained in $W$.
- $(2_{-})$
Suppose we are given a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $v$ belongs to $W$. Then $u$ belongs to $W$ if and only if $w$ belongs to $W$.
- $(3)$
For every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $u: X \rightarrow Y$ which belongs to $W$, where the object $Y$ is contained in $\operatorname{\mathcal{C}}'$.
Then $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{E}}$ and $W$ is the collection of $\operatorname{\mathcal{C}}'$-local equivalences.
Proof.
It follows immediately from conditions $(0)$ and $(3)$ that the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective. Let $u: X \rightarrow Y$ be a $\operatorname{\mathcal{C}}'$-local equivalence; we wish to show that $u$ is contained in $W$ (the converse follows from $(0)$). Applying condition $(0)$, we can choose morphisms $v_{X}: X \rightarrow X'$ and $v_{Y}: Y \rightarrow Y'$ which belong to $W$, where $X'$ and $Y'$ are contained in the subcategory $\operatorname{\mathcal{C}}'$. Then $v_{X}$ is a $\operatorname{\mathcal{C}}'$-local equivalence, so there exists a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ X \ar [r]^{u} \ar [d]^{v_ X} & Y \ar [d]^{v_{Y} } \\ X' \ar [r]^{u'} & Y' } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $u$ is a $\operatorname{\mathcal{C}}'$-local equivalence, it follows that $u'$ is also a $\operatorname{\mathcal{C}}'$-local equivalence, and therefore an isomorphism (Remark 6.2.2.4). It follows $(1)$ that the morphism $u'$ belongs to $W$. Applying condition $(2_{-})$ twice, we conclude that $u$ also belongs to $W$.
$\square$
Proposition 6.2.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(a)$
Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects. Then $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{C}}$ and $W$ is the collection of $\operatorname{\mathcal{C}}'$-local equivalences.
- $(b)$
There exists a reflective subcategory $\operatorname{\mathcal{C}}'' \subseteq \operatorname{\mathcal{C}}$ such that $W$ is the collection of $\operatorname{\mathcal{C}}''$-local equivalences.
- $(c)$
The collection of morphisms $W$ is localizing (Definition 6.2.3.9).
- $(d)$
The collection of morphisms $W$ satisfies conditions $(1)$, $(2_{-})$, and $(3)$ of Lemma 6.2.3.11.
Proof.
The implications $(a) \Rightarrow (b)$ and $(c) \Rightarrow (d)$ are trivial, the implication $(b) \Rightarrow (c)$ follows from Example 6.2.3.10, and the implication $(d) \Rightarrow (a)$ follows from Lemma 6.2.3.11.
$\square$
Corollary 6.2.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then there is a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Localizing collections of morphisms of $\operatorname{\mathcal{C}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Reflective replete subcategories of $\operatorname{\mathcal{C}}$} \} , } \]
which assigns to each localizing collection $W$ the full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ spanned by the $W$-local objects.
Proof.
Combine Propositions 6.2.3.12 and 6.2.3.8 (the inverse bijection carries a replete full subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ to the collection of $\operatorname{\mathcal{C}}'$-local equivalences).
$\square$