# Kerodon

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### 9.1.3 Weakly Local Objects

Let $\operatorname{\mathcal{C}}$ be a category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. By definition, an object $C \in \operatorname{\mathcal{C}}$ is $W$-local (in the sense of Definition 9.1.1.1) if, for every morphism $f: X \rightarrow C$ in $\operatorname{\mathcal{C}}$ and every morphism $w: X \rightarrow Y$ which belongs to $W$, there is a unique morphism $g: Y \rightarrow C$ satisfying $g \circ w = f$, as indicated in the diagram

$\xymatrix@C =50pt@R=50pt{ & Y \ar@ {-->}[dr]^{g} & \\ X \ar [ur]^{w} \ar [rr]^{f} & & C. }$

It will sometimes be useful to consider the following weaker condition:

Definition 9.1.3.1. Let $\operatorname{\mathcal{C}}$ be a 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 weakly $w$-local if, for every morphism $f: X \rightarrow C$ of $\operatorname{\mathcal{C}}$, there exists a morphism $g: Y \rightarrow C$ satisfying $g \circ w = f$. If $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, we say that $C$ is weakly $W$-local if it is weakly $w$-local for each $w \in W$.

Example 9.1.3.2 (Kan Complexes). Let $\operatorname{\mathcal{C}}= \operatorname{Set_{\Delta }}$ be the category of simplicial sets and let $W$ be the collection of all horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, where $n > 0$ and $0 \leq i \leq n$. Then a simplicial set is weakly $W$-local if and only if it is a Kan complex.

Example 9.1.3.3 ($\infty$-Categories). Let $\operatorname{\mathcal{C}}= \operatorname{Set_{\Delta }}$ be the category of simplicial sets and let $W$ be the collection of all inner horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, where $0 < i < n$. Then a simplicial set is weakly $W$-local if and only if it is an $\infty$-category.

Example 9.1.3.4 (Contractible Kan Complexes). Let $\operatorname{\mathcal{C}}= \operatorname{Set_{\Delta }}$ be the category of simplicial sets and let $W$ be the collection of inclusion maps $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^{n}$. Then a simplicial set is weakly $W$-local if and only if it is a contractible Kan complex.

Definition 9.1.3.1 has an obvious counterpart in the setting of $\infty$-categories:

Definition 9.1.3.5. 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 weakly $w$-local if, for every morphism $f: X \rightarrow C$, there exists a $2$-simplex with boundary indicated in the diagram

$\xymatrix@C =50pt@R=50pt{ & Y \ar@ {-->}[dr] & \\ X \ar [ur]^{w} \ar [rr]^{f} & & C. }$

If $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, we say that an object $C \in \operatorname{\mathcal{C}}$ is weakly $W$-local if it is weakly $w$-local for each $w \in W$.

Example 9.1.3.6. Let $\operatorname{\mathcal{C}}$ be a category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Then an object $C \in \operatorname{\mathcal{C}}$ is weakly $W$-local (in the sense of Definition 9.1.3.1) if and only if it is weakly $W$-local when regarded as an object of the $\infty$-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 9.1.3.5).

Remark 9.1.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. It follows from Proposition 1.4.4.2 that an object $C \in \operatorname{\mathcal{C}}$ is weakly $W$-local if and only if, for every morphism $w: X \rightarrow Y$ which belongs to $W$, composition with the homotopy class $[w]$ induces a surjection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,C)$. In other words, the object $C$ is weakly $W$-local (in the sense of Definition 9.1.3.5) if and only if it is weakly $[W]$-local when regarded as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of Definition 9.1.3.1). Here $[W] = \{ [w]: w \in W \}$ denotes the collection of all homotopy classes of morphisms which belong to $W$.

Example 9.1.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. If an object $C \in \operatorname{\mathcal{C}}$ is $W$-local (in the sense of Definition 9.1.1.1), then it is weakly $W$-local.

Example 9.1.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $w: X \rightarrow Y$ a morphism of $\operatorname{\mathcal{C}}$ which admits a left homotopy inverse $r: Y \rightarrow X$. Then every object $C \in \operatorname{\mathcal{C}}$ is weakly $w$-local. In particular, if $w$ is an isomorphism, then every object of $\operatorname{\mathcal{C}}$ is weakly $w$-local.

Remark 9.1.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing a $2$-simplex

$\xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z. }$

If an object $C \in \operatorname{\mathcal{C}}$ is weakly $u$-local and weakly $v$-local, then it is weakly $w$-local. Conversely, if $C$ is weakly $w$-local, then it is weakly $u$-local.

Remark 9.1.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $C \in \operatorname{\mathcal{C}}$ be an object which factors as a product of some collection of objects $\{ C_ i \} _{i \in I}$ (see Definition 7.6.1.3). If each $C_ i$ is weakly $W$-local, then $C$ is weakly $W$-local. In particular, any final object of $\operatorname{\mathcal{C}}$ is weakly $W$-local.

Remark 9.1.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $C \in \operatorname{\mathcal{C}}$ be an object. If $C$ is weakly $W$-local, then any retract of $C$ is also weakly $W$-local. In particular, the condition that $C$ is weakly $W$-local depends only on the isomorphism class of $C$.

Variant 9.1.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $w: X \rightarrow Y$ and $w': X' \rightarrow Y'$ be morphisms of $\operatorname{\mathcal{C}}$, and suppose that $w'$ is a retract of $w$ (in the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$). If an object $C \in \operatorname{\mathcal{C}}$ is weakly $w$-local, then it is also weakly $w'$-local. In particular, if we regard the object $C \in \operatorname{\mathcal{C}}$ is fixed, then the condition that $C$ is $w$-local depends only on the isomorphism class of $w$ (as an object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).

Proposition 9.1.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category containing a pushout diagram

9.3
$$\begin{gathered}\label{proposition:pushout-of-semiorthogonal} \xymatrix@C =50pt@R=50pt{ X \ar [d]^{w} \ar [r] & X' \ar [d]^{w'} \\ Y \ar [r] & Y'. } \end{gathered}$$

If an object $C \in \operatorname{\mathcal{C}}$ is weakly $w$-local, then it is also weakly $w'$-local.

Proof. We have a commutative diagram of sets

$\xymatrix@C =30pt@R=50pt{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y', C ) \ar [r] & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X', C) \times _{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, C ) } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \ar [r] \ar [d] & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X', C) \ar [d] \\ & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,C) \ar [r]^-{ \circ [w] } & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, C), }$

where the square on the right is a pullback. Our assumption that $C$ is weakly $w$-local guarantees that the bottom horizontal map is surjective, so that the upper horizontal map on the right is also surjective. Since Proposition 9.3 is a pushout square, the horizontal map on the upper left is also surjective (Warning 7.6.3.3). It follows that the composite map $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y', C) \xrightarrow { \circ [w'] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X',C)$ is also surjective. $\square$

Example 9.1.3.8 admits the following partial converse:

Proposition 9.1.3.15. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Suppose that every morphism $w: X \rightarrow Y$ of $W$ admits a relative codiagonal $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$ which also belongs to $W$ (Variant 7.6.3.19). Then an object $C \in \operatorname{\mathcal{C}}$ is $W$-local if and only if it is weakly $W$-local.

Proof. Fix an object $C \in \operatorname{\mathcal{C}}$, and let $h_{C}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ denote the functor represented by $C$. We wish to show that, for every morphism $w: X \rightarrow Y$ which belongs to $W$, the image $h_{C}(w)$ is a homotopy equivalence of Kan complexes. By virtue of Remark 3.5.1.19, it will suffice to show that $h_{C}(w)$ is $n$-connective for every integer $n \geq 0$. The proof proceeds by induction on $n$. In the case $n = 0$, we wish to show that the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)$ is surjective on connected components, which follows from our assumption that $C$ is weakly $W$-local. Let us therefore assume that $n > 0$. Using the criterion of Corollary 3.5.1.29 (together with Exercise 7.6.4.13), we are reduced to proving the $(n-1)$-connectivity of the relative diagonal of $h_{C}(w)$ (formed in the $\infty$-category $\operatorname{\mathcal{S}}$). Since the functor $h_{C}$ preserves limits (Proposition 7.4.5.17), we can identify the relative diagonal of $h_{C}(w)$ with $h_{C}( \gamma _{X/Y} )$, where $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$ denotes a relative codiagonal of $w$. By assumption, we can arrange that $\gamma _{X/Y}$ is also contained in $W$, so the desired result follows from our inductive hypothesis. $\square$

Exercise 9.1.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a relative codiagonal $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$. Show that an object $C \in \operatorname{\mathcal{C}}$ is $w$-local if and only if it is both $\gamma _{X/Y}$-local and weakly $w$-local.

We have the following $\infty$-categorical counterpart of Proposition 1.5.4.11:

Proposition 9.1.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of morphisms $w: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ such that $C$ is weakly $w$-local. Then $W$ is closed under transfinite composition (see Definition 9.1.2.1).

Proof. Let $U: \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}$ be the projection map and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which can be written as a transfinite composition of morphisms of $\operatorname{\mathcal{C}}$. Suppose we are given a morphism $X \rightarrow C$ in $\operatorname{\mathcal{C}}$, which we identify with an object $\widetilde{X} \in \operatorname{\mathcal{C}}_{/C}$ satisfying $U( \widetilde{X} ) = X$. We wish to show that there is a morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ of $\operatorname{\mathcal{C}}_{/C}$ satisfying $U( \widetilde{f} ) = f$.

Choose an ordinal $\alpha$ and a functor $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits $f$ as a transfinite composition of morphisms of $W$. Let $Q$ be the collection of all ordered pairs $(\beta , \widetilde{F}_{\leq \beta } )$, where $\beta \leq \alpha$ is an ordinal and $\widetilde{F}_{\leq \beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) \rightarrow \operatorname{\mathcal{C}}_{/C}$ is a functor satisfying $\widetilde{F}_{\leq \beta }(0) = \widetilde{X}$ and $U \circ \widetilde{F}_{\leq \beta } = F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$. We regard $Q$ as a partially ordered set, where $(\beta , \widetilde{F}_{\leq \beta } ) \leq (\beta ', \widetilde{F}'|_{\leq \beta '})$ if $\beta \leq \beta '$ and $\widetilde{F}_{\leq \beta } = \widetilde{F}'_{\leq \beta '} |_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$.

We first claim that $Q$ satisfies the hypotheses of Zorn's lemma. Let $Q_0 \subseteq Q$ be a linearly ordered subset of $Q$; we wish to show that $Q_0$ admits an upper bound. If $Q_0$ is empty, we can take this upper bound be the pair $(0, \widetilde{F}_{\leq 0} )$, where $\widetilde{F}_{\leq 0}$ is the constant functor taking the value $\widetilde{X}$. Without loss of generality, we may assume that $Q_0$ does not contain a maximal element (otherwise, there is nothing to prove). Write $Q_0 = \{ ( \beta _ i, \widetilde{F}_{\leq \beta _ i} ) \} _{i \in I}$ and let $\beta \leq \alpha$ be the supremum of the set $\{ \beta _ i \} _{i \in I}$. The functors $\widetilde{F}_{\leq \beta _ i}$ can then be amalgamated to a single functor $\widetilde{F}_{< \beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \beta } ) \rightarrow \operatorname{\mathcal{C}}_{/C}$. To find an upper bound for $Q_0$, it will suffice to show that the lifting problem

$\xymatrix@C =50pt@R=50pt{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \beta } ) \ar [r]^-{ \widetilde{F}_{< \beta } } \ar [d] & \operatorname{\mathcal{C}}_{/C} \ar [d]^{U} \\ \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \beta } ) \ar [r]^-{ F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) } } & \operatorname{\mathcal{C}}}$

admits a solution. This follows immediately from our assumption that $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Applying Zorn's lemma, we deduce that $Q$ contains a maximal element $(\beta , \widetilde{F}_{\leq \beta } )$. To complete the proof, it will suffice to show that $\beta = \alpha$; we can then take $\widetilde{f}$ to be obtained by applying the functor $\widetilde{F}$ to the edge of $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } )$ given by the pair $(0, \alpha )$. Assume otherwise, let $F_{\leq \beta }$ denote the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } )}$, and let $\operatorname{\mathcal{D}}$ denote the coslice $\infty$-category $\operatorname{\mathcal{C}}_{ F_{\leq \beta } / }$. Then $F_{\leq \beta +1}$ and $\widetilde{F}_{\leq \beta }$ can be identified with objects $D,D' \in \operatorname{\mathcal{D}}$, and the maximality of $(\beta , \widetilde{F}_{\leq \beta } )$ guarantees that $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' ) = \emptyset$. Since the inclusion map $\{ \beta \} \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } )$ is right anodyne (Corollary 4.6.7.24), the restriction map $V: \operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{ F_{\leq \beta } / } \rightarrow \operatorname{\mathcal{C}}_{ F(\beta ) / }$ is a trivial Kan fibration (Corollary 4.3.6.13). It follows that the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{ F(\beta ) / } }( V(D), V(D') )$ is also empty: that is, there is no $2$-simplex of $\operatorname{\mathcal{C}}$ with boundary indicated in the diagram

$\xymatrix@C =50pt@R=50pt{ & F(\beta +1) \ar@ {-->}[dr] & \\ F(\beta ) \ar [ur] \ar [rr] & & C. }$

This contradicts our assumption that the morphism $F(\beta ) \rightarrow F(\beta +1)$ belongs to $W$. $\square$

Variant 9.1.3.18. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of morphisms $w: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ such that $C$ is $w$-local. Then $W$ is closed under transfinite composition.

Proof. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $C$ is $w$-local. Then $W$ contains all identity morphisms, is closed under composition (Remark 9.1.1.11), and is closed under the formation of colimits in the $\infty$-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ which are preserved by the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ (Remark 9.1.1.9). Applying Proposition 9.1.2.10 (and Remark 9.1.2.11), we conclude that $W$ is closed under transfinite composition. $\square$

We now introduce some terminology which is motivated by the preceding discussion.

Definition 9.1.3.19. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that $W$ is weakly saturated if it satisfies the following conditions:

$(1)$

The collection $W$ is closed under pushouts: that is, for every pushout diagram

$\xymatrix@C =50pt@R=50pt{ X \ar [d]^{w} \ar [r] & X' \ar [d]^{w'} \\ Y \ar [r] & Y' }$

in the $\infty$-category $\operatorname{\mathcal{C}}$, if $w$ belongs to $W$, then $w'$ also belongs to $W$.

$(2)$

The collection $W$ is closed under the formation of retracts (in the $\infty$-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).

$(3)$

The collection $W$ is closed under transfinite composition (Definition 9.1.2.1).

Remark 9.1.3.20. Let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ be the nerve of an ordinary category $\operatorname{\mathcal{C}}_0$. Then a collection of morphisms of $\operatorname{\mathcal{C}}$ is weakly saturated (in the sense of Definition 9.1.3.19) if and only if is weakly saturated when regarded as a collection of morphisms of $\operatorname{\mathcal{C}}_0$ (in the sense of Definition 1.5.4.12).

Example 9.1.3.21. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of all morphisms $w: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ such that $C$ is weakly $w$-local. Then $W$ is weakly saturated. This follows from Proposition 9.1.3.14, Variant 9.1.3.13, and Proposition 9.1.3.17.

Variant 9.1.3.22. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of all morphisms $w: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ such that $C$ is $w$-local. Then $W$ is weakly saturated. This follows from Remark 9.1.1.10, Remark 9.1.1.11, and Variant 9.1.3.18.

Remark 9.1.3.23. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Any intersection of weakly saturated collections of morphisms of $\operatorname{\mathcal{C}}$ is also weakly saturated. In particular, for any collection $W$ of morphisms of $\operatorname{\mathcal{C}}$, there is a smallest collection $\overline{W}$ which is weakly saturated and contains $W$. We will refer to $\overline{W}$ as the weakly saturated collection generated by $W$.