# Kerodon

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### 9.1.4 The Small Object Argument

In §3.1.7, we showed that every simplicial set $X$ admits an anodyne morphism $X \hookrightarrow Q$, where $Q$ is a Kan complex (Corollary 3.1.7.2). The proof is easy to describe: if $X$ is not a Kan complex, then there is some horn $\sigma _0: \Lambda ^{n}_{i} \rightarrow X$ which cannot be extended to an $n$-simplex of $X$. This defect can be remedied by replacing $X$ by the pushout $\Delta ^ n \coprod _{ \Lambda ^{n}_{i} } X$. The desired Kan complex $Q$ is obtained by a (possibly transfinite) iteration of this procedure. A similar strategy can be used to prove many related results (see for example Exercise 3.1.7.11, Proposition 4.1.3.2, and Proposition 4.2.4.8). Following Quillen (), we will refer to this proof strategy as the small object argument. Our goal in this section is to formalize a version of this argument in the $\infty$-categorical setting. First, we need a bit of terminology.

Definition 9.1.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $W'$ denote the collection of those morphisms $w': X' \rightarrow Y'$ for which there exists a pushout square

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

where $w \in W$. We say that a morphism $f$ of $\operatorname{\mathcal{C}}$ is a transfinite pushout of morphisms of $W$ if it is a transfinite composition of morphisms of $W'$, in the sense of Definition 9.1.2.1.

Remark 9.1.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which is a transfinite pushout of morphisms which belong to $W$. Then:

• If an object $C \in \operatorname{\mathcal{C}}$ is weakly $W$-local, then it is weakly $f$-local.

• If an object $C \in \operatorname{\mathcal{C}}$ is $W$-local, then it is $f$-local.

The first assertion follows from Propositions 9.1.3.14 and 9.1.3.17; the second follows from Remark 9.1.1.10 and Variant 9.1.3.18.

We can now formulate the main result of this section:

Theorem 9.1.4.3 (The Small Object Argument). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that:

• The $\infty$-category $\operatorname{\mathcal{C}}$ is locally small and admits small colimits.

• The collection $W$ is small.

• For each morphism $w: X \rightarrow Y$ which belongs to $W$, the object $X \in \operatorname{\mathcal{C}}$ is $\kappa$-compact for some small cardinal $\kappa$ (see Definition ).

For every object $C \in \operatorname{\mathcal{C}}$, there exists a morphism $f: C \rightarrow C'$ where $C'$ is $W$-local and $f$ is a transfinite pushout of morphisms of $W$.

Warning 9.1.4.4. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then the morphism $f: C \rightarrow C'$ of Theorem 9.1.4.3 can be chosen to depend functorially on $C$. Beware that this is generally not possible if $\operatorname{\mathcal{C}}$ is an $\infty$-category (see Example 9.1.4.5).

Example 9.1.4.5. Fix an integer $n \geq 0$. Let $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ be the $\infty$-category of spaces and let $W$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ given by the inclusion maps $\{ \operatorname{Ex}^{\infty }( \operatorname{\partial \Delta }^{m} ) \hookrightarrow \operatorname{Ex}^{\infty }( \Delta ^ m ) \} _{0 \leq m \leq n}$. Then an object $X \in \operatorname{\mathcal{C}}$ weakly $W$-local if and only if it is $n$-connective (see Definition 3.2.4.5). In this case, Theorem 9.1.4.3 asserts that every Kan complex $X$ admits a morphism $f: X \rightarrow Y$, where $Y$ is an $n$-connective Kan complex which can be obtained from $X$ by attaching cells of dimension $\leq n$. Beware that, if $n > 0$, then $Y$ cannot be chosen to depend functorially on $X$.

Corollary 9.1.4.6. 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. Assume that:

• The $\infty$-category $\operatorname{\mathcal{C}}$ is locally small and admits small colimits.

• The collection $W$ is small.

• For each morphism $w: X \rightarrow Y$ which belongs to $W$, the objects $X$ and $Y$ are $\kappa$-compact for some small infinite cardinal $\kappa$.

Then $\operatorname{\mathcal{C}}'$ is a reflective localization of $\operatorname{\mathcal{C}}$.

Proof. For each morphism $w: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, choose a morphism $\delta _{w}: Y \coprod _{X} Y \rightarrow Y$ which is a relative codiagonal of $w$ (see Variant 7.6.3.15). Note that, if $X$ and $Y$ are $\kappa$-compact for some infinite cardinal $\kappa$, then the pushout $Y \coprod _{X} Y$ is also $\kappa$-compact (Proposition ). Let $W'$ be the smallest collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $W$ and is closed under the construction $w \mapsto \gamma _{w}$. By virtue of Remark 9.1.1.12, an object of $\operatorname{\mathcal{C}}$ is $W$-local if and only if it is $W'$-local. We may therefore replace $W$ by $W'$ and thereby reduce to proving Corollary 9.1.4.6 in the special case where $W$ is closed under the formation of relative codiagonals.

Fix an object $C \in \operatorname{\mathcal{C}}$. Using Theorem 9.1.4.3, we see that there exists a morphism $f: C \rightarrow C'$, where $C'$ is weakly $W$-local and $f$ is a transfinite pushout of morphisms which belong to $W$. Using Proposition 9.1.3.15, we see that $C'$ belongs to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $f$ exhibits $C'$ as a $\operatorname{\mathcal{C}}'$-reflection of $C$: that is, every object of $\operatorname{\mathcal{C}}'$ is $f$-local. This follows from Remark 9.1.4.2. $\square$

Our proof of Theorem 9.1.4.3 will require some preliminaries.

Lemma 9.1.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits small colimits, let $\{ w_ s: X_ s \rightarrow Y_ s \} _{s \in S}$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$ indexed by a set $S$, let $\{ e_{s}: X_{s} \rightarrow C \} _{ s \in S}$ be another collection of morphisms of $\operatorname{\mathcal{C}}$, and let $D$ be a colimit of the diagram

$(S \times \Delta ^1) \coprod _{ S \times \{ 0\} } S^{\triangleright } \xrightarrow { (\{ w_ s \} , \{ e_ s \} ) } \rightarrow \operatorname{\mathcal{C}}.$

Then the tautological map $u: C \rightarrow D$ is a transfinite pushout of morphisms belonging to $\{ w_ s \} _{s \in S}$.

Proof. Using the well-ordering theorem (Theorem 5.4.1.34), we can choose an ordinal $\alpha$ and a bijection $\ell : S \rightarrow \mathrm{Ord}_{ < \alpha }$. Let $Q$ denote the disjoint union $(S \times ) \coprod \mathrm{Ord}_{\leq \alpha }$. For elements $q,q' \in Q$, we write $q \leq q'$ if (exactly) one of the following conditions holds:

• There exist an element $s \in S$ such that $q = (s,i)$ and $q' = (s, i' )$, where $i \leq i'$.

• We have $q = \beta$ and $q' = \beta '$ for ordinals $\beta , \beta ' \in \mathrm{Ord}_{\leq \alpha }$ satisfying $\beta \leq \beta '$ (for the usual ordering of $\mathrm{Ord}_{\leq \alpha }$).

• We have $q = (s,0)$ for some $s \in S$ and $q' = \beta$ for some $\beta \in \mathrm{Ord}_{\leq \alpha }$.

• We have $q = (s,1)$ and $q' = \beta$ for some $\beta \in \mathrm{Ord}_{\leq \alpha }$ satisfying $\ell (s) < \beta$.

Let $Q_0 = (S \times ) \coprod \{ 0 \}$, which we regard as a partially ordered subset of $Q$. By construction, the nerve $\operatorname{N}_{\bullet }( Q_{0} )$ can be identified with the pushout $(S \times \Delta ^1) \coprod _{ S \times \{ 0\} } S^{\triangleright }$. Consequently, the collections $\{ e_ s \} _{s \in S}$ and $\{ w_ s \} _{s \in S}$ determine a diagram $F_0: \operatorname{N}_{\bullet }( Q_{\leq 0 } ) \rightarrow \operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits small colimits, the diagram $F_0$ admits a left Kan extension $F: \operatorname{N}_{\bullet }( Q ) \rightarrow \operatorname{\mathcal{C}}$ (Proposition 7.6.7.13). Then $D = F( \alpha )$ is a colimit of the diagram $F_{0}$, and we can identify $u$ with the morphism obtained by evaluating the functor $F$ on the edge of $\operatorname{N}_{\bullet }( Q )$ given by the pair $(0 \leq \alpha )$. We will show that the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } )}$ exhibits $u$ as a transfinite pushout of morphisms belonging to $\{ w_ s \} _{s \in S}$.

We first claim that that if $\beta \leq \alpha$ is a nonzero limit ordinal, then the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \beta } ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Set $Q_{\leq \beta } = \{ q \in Q: q \leq \beta \}$ and $Q_{ < \beta } = \{ q \in Q: q < \beta \}$. Since the functor $F$ is left Kan extended from $\operatorname{N}_{\bullet }(Q_0)$, it is also left Kan extended from larger $\infty$-category $\operatorname{N}_{\bullet }( (S \times ) \coprod \mathrm{Ord}_{ < \beta } )$ (see Corollary 7.3.8.8). It follows that $F|_{ \operatorname{N}_{\bullet }( Q_{\leq \beta } ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. It will therefore suffice to show that $\operatorname{N}_{\bullet }(\iota )$ is right cofinal, where $\iota$ denotes the inclusion map $\mathrm{Ord}_{< \beta } \hookrightarrow Q_{ < \beta }$ (Corollary 7.2.2.2). This is a special case of Corollary 7.2.3.7, since $\iota$ admits a left adjoint (given on $S \times $ by the construction $(s,0) \mapsto 0$ and $(s,1) \mapsto \ell (s) + 1$).

Now suppose that $\beta = \gamma +1$ is a successor ordinal. Let $s \in S$ be the unique element satisfying $\ell (s) = \gamma$. We will complete the proof by showing that the morphism $F(\gamma ) \rightarrow F(\beta )$ in $\operatorname{\mathcal{C}}$ can be realized as a pushout of $w_ s$. More precisely, we will show that the functor $F$ carries the diagram

$\xymatrix@R =50pt@C=50pt{ (s,0) \ar [r] \ar [d] & (s,1) \ar [d] \\ \gamma \ar [r] & \beta }$

in $\operatorname{N}_{\bullet }( Q )$ to a pushout diagram in the $\infty$-category $\operatorname{\mathcal{C}}$. Arguing as above, we see that the restriction $F|_{ Q_{\leq \beta } }$ is a colimit diagram. Using Corollary 7.2.2.2 again, we are reduced to showing that $\operatorname{N}_{\bullet }( \iota )$ is right cofinal, where $\iota$ denotes the inclusion of partially ordered sets $\{ (s,1) > (s,0) < \beta \} \hookrightarrow Q_{< \beta }$. This again follows from Corollary 7.2.3.7, since $\iota$ admits a left adjoint given by the construction

$(q \in Q_{< \beta } ) \mapsto \begin{cases} q & \text{ if q = (s,0) or q = (s,1) } \\ \beta & \text{ otherwise. } \end{cases}$
$\square$

Lemma 9.1.4.8. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty$-category which admits small colimits, and let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$. For every object $C \in \operatorname{\mathcal{C}}$, there exists a morphism $u: C \rightarrow D$ which is a transfinite pushout of morphisms of $W$ with the following property: for every morphism $w: X \rightarrow Y$ which belongs to $W$ and every morphism $e: X \rightarrow C$, there exists a commutative diagram

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

in the $\infty$-category $\operatorname{\mathcal{C}}$.

Proof. For each morphism $w: X \rightarrow Y$ belonging to $W$, let $\{ f_{s}: X \rightarrow C\} _{ s \in S_{w} }$ be a set of representatives for the homotopy classes of morphisms from $X$ to $C$. Since $\operatorname{\mathcal{C}}$ is locally small, the collection $S_{w}$ is small. Our assumption that $W$ is small then guarantees that the disjoint union $S = \coprod _{w \in W} S_{w}$ is small. The desired result now follows by applying Lemma 9.1.4.7 to the collection of morphisms $\{ f_ s \} _{s \in S}$. $\square$

Lemma 9.1.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which admits small filtered colimits. Let $U$ be a collection of morphisms with the following property: for every object $D \in \operatorname{\mathcal{C}}$, there exists a morphism $u: D \rightarrow E$ which belongs to $U$. Then, for every object $C \in \operatorname{\mathcal{C}}$ and every (small) ordinal $\alpha$, there exists a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ satisfying the following conditions:

$(a)$

For every nonzero limit ordinal $\lambda \leq \alpha$, the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \lambda }) }$ is a colimit diagram.

$(b)$

For every ordinal $\gamma < \alpha$, the morphism $F(\gamma ) \rightarrow F(\gamma +1)$ belongs to $U$.

$(c)$

The object $F(0)$ coincides with $C$.

Proof. Let $Q$ denote the collection of all diagrams $F_{\beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) \rightarrow \operatorname{\mathcal{C}}$ satisfying conditions $(a)$, $(b)$, and $(c)$, where $\beta$ is an ordinal $\leq \alpha$. We regard $Q$ as a partially ordered set, where $F_{\beta } \leq F_{\beta '}$ if $\beta \leq \beta '$ and $F_{\beta } = F_{\beta '} |_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$. Note that $Q$ is nonempty: it has a least element given by the diagram $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq 0} ) \simeq \{ C\} \hookrightarrow \operatorname{\mathcal{C}}$ taking the value $C$. We claim that $Q$ satisfies the hypothesis of Zorn's lemma: that is, every linearly ordered set $Q' \subset Q$ admits an upper bound. Without loss of generality, we may assume that $Q'$ is nonempty and has no largest element. In this case, the elements of $Q'$ can be amalgamated to a diagram $F_{ < \lambda }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ < \lambda } ) \rightarrow \operatorname{\mathcal{C}}$, where $\lambda \leq \alpha$ is a nonzero limit ordinal. Our assumption on $\operatorname{\mathcal{C}}$ then guarantees that $F_{ < \lambda }$ can be extended to a colimit diagram

$F_{\lambda }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \lambda } ) \simeq \operatorname{N}_{\bullet }( \mathrm{Ord}_{ < \lambda } )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}.$

By construction, this diagram satisfies conditions $(a)$, $(b)$, and $(c)$, and is therefore an upper bound for $Q'$.

Applying Zorn's lemma, we deduce that $Q$ has a maximal element $F_{\beta }: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) \rightarrow \operatorname{\mathcal{C}}$. We will complete the proof by showing that $\beta < \alpha$. Assume otherwise, and set $X = F_{\beta }(\beta )$. By assumption, we can choose a morphism $u: X \rightarrow Y$ which belongs to $U$. Let us identify $u$ with an object $\widetilde{Y}$ of the coslice $\infty$-category $\operatorname{\mathcal{C}}_{X/}$. Since the inclusion map $\{ \beta + 1\} \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta + 1} )$ is right anodyne (Example 4.3.7.11), the restriction map $\operatorname{\mathcal{C}}_{ F_{\beta } / } \rightarrow \operatorname{\mathcal{C}}_{ X/ }$ is a trivial Kan fibration (Corollary 4.3.6.13). We can therefore lift $\widetilde{Y}$ to an object of the $\infty$-category $\operatorname{\mathcal{C}}_{ F_{\beta } / }$, which we can identify with an extension of $F_{\beta }$ to a diagram $F_{\beta +1}: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \beta +1 }) \rightarrow \operatorname{\mathcal{C}}$. By construction, this diagram carries the pair $(\beta , \beta +1)$ to the morphism $u$ of $\operatorname{\mathcal{C}}$. It follows that $F_{\beta +1}$ is also an element of $Q$, contradicting the maximality of $F_{\beta }$. $\square$

Proof of Theorem 9.1.4.3. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty$-category which admits small colimits, let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\kappa$ be a small regular cardinal having the property that for each morphism $w: X \rightarrow Y$ which belongs to $W$, the object $X$ is $\kappa$-compact. Fix an object $C \in \operatorname{\mathcal{C}}$; we wish to show that there exists a morphism $f: C \rightarrow C'$ where $C'$ is weakly $W$-local and $f$ is a transfinite pushout of morphisms belonging to $W$. Without loss of generality, we may assume that $C$ itself is not weakly $W$-local (otherwise, we can take $f = \operatorname{id}_{C}$).

Let $W'$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$ which are pushouts of morphisms of $W$, and let $\overline{W}$ denote the transfinite closure of $W'$ (Definition 9.1.2.13). Let $U \subseteq \overline{W}$ denote the subcollection consisting of those morphisms $u$ which satisfy the requirement of Lemma 9.1.4.8. Using Lemma 9.1.4.9 we deduce that there exists a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ where $F(0) = C$ and $F$ exhibits the induced map $F(0) \xrightarrow {f} F(\kappa )$ as a transfinite composition of morphisms of $U$.

We first claim that the object $C' = F(\kappa )$ is weakly $W$-local. Let $w: X \rightarrow Y$ be a morphism which belongs to $W$. We wish to show that every morphism $[ \overline{e} ]: X \rightarrow F(\kappa )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ factors through the homotopy class $[w]$. Since $F$ is a colimit diagram and the object $X$ is $\kappa$-compact, the morphism $[ \overline{e} ]$ factors as a composition $X \xrightarrow { [e] } F(\alpha ) \rightarrow F(\kappa )$ for some ordinal $\alpha < \kappa$ and some morphism $e: X \rightarrow F(\alpha )$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Since the transition map $F( \alpha ) \rightarrow F(\alpha +1)$ belongs to $U$, we can choose a commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [d] \ar [r]^{w} \ar [d]^{e} & Y \ar [d]^{e'} \\ F(\alpha ) \ar [r] & F(\alpha +1). }$

It follows that $[ \overline{e} ]$ factors as a composition $X \xrightarrow {[w]} Y \xrightarrow { [e'] } F(\alpha +1) \rightarrow F(\kappa )$.

To complete the proof, it will suffice to show that $f$ is a transfinite composition of morphisms belonging to $W'$. By construction, $f$ is a transfinite pushout of morphisms of $U \subseteq \overline{W}$, and therefore belongs to $\overline{W}$. This follows from Corollary 9.1.2.19, since $f$ is not an isomorphism (otherwise, the object $C$ would also be weakly $W$-local, contrary to our initial assumption). $\square$