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9.1.4 The Small Object Argument

In ยง3.1.7, 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 ([MR0223432]), 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 weakly $W$-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 3.5.1.1). 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 7.6.3.19). 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 4.7.1.34), we can choose an ordinal $\alpha $ and a bijection $\ell : S \rightarrow \mathrm{Ord}_{ < \alpha }$. Let $Q$ denote the disjoint union $(S \times [1]) \coprod \mathrm{Ord}_{\leq \alpha }$. 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 [1]) \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 $\operatorname{N}_{\bullet }( (S \times [1]) \coprod \mathrm{Ord}_{ < \beta } )$ (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 $S \times [1]$ 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). $\square$