Kerodon

Proof. Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which is a transfinite composition of morphisms of $W$; we wish to show that $f \in W$. 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 $W$. For each $\beta \leq \alpha $, the functor $F$ carries the ordered pair $(0 \leq \beta )$ to a morphism $f_{\beta }: F(0) \rightarrow F(\beta )$. We will complete the proof by showing that each the morphisms $f_{\beta }$ belongs to $W$. The proof proceeds by transfinite induction on $\beta $. If $\beta = 0$, then $f_{\beta } = \operatorname{id}_{X}$ and the desired result follows from assumption $(1)$. If $\beta $ is a nonzero limit ordinal, then the desired result follows from assumption $(3)$. Remark 9.1.1.9. It will therefore suffice to treat the case where $\beta = \gamma +1$ is a successor ordinal. In this case, the desired result follows by applying assumption $(2)$ to the diagram

since $f_{\gamma }$ belongs to $W$ by virtue of our inductive hypothesis. $\square$

Proof. Note that we can identify $K(A,B)$ with the (filtered) colimit of the simplicial subsets $K(A, B')$, where $B'$ ranges over the collection of all subsets of $B$ which are obtained from $A$ by adjoining finitely many elements. Since the collection of categorical equivalences is stable under the formation of filtered colimits (Corollary 4.5.7.2), it will suffice to prove Lemma 9.1.2.17 in the special case where $B \setminus A$ is finite.

Let $A_0 \subseteq A$ be the collection of elements which have the form $b_{-}$ or $b_{+}$, where $b$ is an element of $B \setminus A$. Note that, if $A' \subseteq A$ is a subset which contains $A_0$ and we set $B' = A' \cup (B \setminus A)$, then the pair $(A', B')$ also satisfies condition $(\ast )$. Moreover, we have $K(A',B') = K(A,B) \cap \operatorname{N}_{\bullet }(B')$. It follows that $K(A,B)$ can be written as a filtered colimit of simplicial subsets $K(A',B')$, where $A'$ ranges over finite subsets of $A$ which contain $A_0$. Applying Corollary 4.5.7.2 again, we are reduced to proving Lemma 9.1.2.17 under the additional assumption that $A$ is finite. We will also assume that $A$ is nonempty (otherwise, $B$ is empty and therefore is nothing to prove).

If $B = \emptyset $, there is nothing to prove. We may therefore assume without loss of generality that $B = [n] = \{ 0 < 1 < \cdots n \} $ for some nonnegative integer $n$, so that $\operatorname{N}_{\bullet }(B)$ can be identified with the standard $n$-simplex $\Delta ^ n$. Note that the simplicial subset $K(A,B) \subseteq \Delta ^ n$ contains the spine $\operatorname{Spine}[n]$ of Example 1.5.7.7. The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.5.7.7), and therefore a categorical equivalence. It will therefore suffice to show that the inclusion map $\operatorname{Spine}[n] \hookrightarrow K(A,B)$ is also a categorical equivalence. In fact, we will show that it is inner anodyne.

Write $A = \{ a_0 < a_1 < \cdots < a_ m \} $, where $a_0 = 0$ and $a_ m = n$. Then $\operatorname{N}_{\bullet }(A)$ is the image of a nondegenerate $m$-simplex $\sigma : \Delta ^{m} \rightarrow \Delta ^{n}$, given by $\sigma (i) = a_ i$. Let $K' \subseteq \Delta ^ n$ denote the simplicial subset consisting of simplices which satisfy condition $(1)$: more concretely, $K'$ is the union of the images of nondegenerate simplices

Note that the inverse image $\sigma ^{-1}( K' )$ identifies with the spine $\operatorname{Spine}[m]$, so that we have a pushout diagram

Since the inclusion map $\operatorname{Spine}[m] \hookrightarrow \Delta ^{m}$ is inner anodyne (Example 1.5.7.7), it follows that the inclusion $K' \hookrightarrow K(A,B)$ is also inner anodyne. We are therefore reduced to showing that the inclusion map $\operatorname{Spine}[n] \hookrightarrow K'$ is inner anodyne. This follows from the observation that we also have a pushout diagram

where the left vertical map is inner anodyne by virtue of Example 1.5.7.7. $\square$

Proof of Proposition 9.1.2.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\overline{W}$ be the collection of morphisms which can be written as transfinite compositions of morphisms belonging to $W$. Suppose we are given a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits the underlying map $f: F(0) \rightarrow F(\alpha )$ as a transfinite composition of morphisms of $\overline{W}$. We wish to show that $f$ also belongs to $\overline{W}$.

For each ordinal $\beta < \alpha $, let $u_{\beta }: F(\beta ) \rightarrow F(\beta +1)$ denote the morphism of $\operatorname{\mathcal{C}}$ obtained by evaluating $F$ on the pair $(\beta , \beta +1)$. By assumption, $u_{\beta }$ belongs to $\overline{W}$. We can therefore choose a well-ordered set $(B(\beta ), \leq _{\beta } )$ and a diagram $G_{\beta }: \operatorname{N}_{\bullet }( B(\beta ))^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which exhibits $u_{\beta }$ as a transfinite composition of morphisms of $W$, in the sense of Variant 9.1.2.3. Since $W$ contains all isomorphisms in $\operatorname{\mathcal{C}}$, we can assume without loss of generality that each $B(\beta )$ is nonempty (see Examples 9.1.2.4 and 9.1.2.5), and therefore contains a smallest element $a_{\beta }$. Let $a_{\alpha }$ be an auxiliary symbol, set $B(\alpha )= \{ a_{\alpha } \} $, and let $B$ denote the disjoint union $\coprod _{ \beta \leq \alpha } B(\beta )$. Given elements $b \in B(\beta )$ and $b' \in B(\beta ')$, we write $b \leq b'$ if either $\beta < \beta '$, or $\beta = \beta '$ and $b \leq _{\beta } b'$. Set $A = \{ a_{\beta } \} _{\beta \leq \alpha } \subseteq B$. The construction $\beta \mapsto a_{\beta }$ determines an order-preserving bijection $\mathrm{Ord}_{\leq \alpha } \xrightarrow {\sim } A$, so that the diagram $F$ can be identified with a functor from $\operatorname{N}_{\bullet }(A)$ to $\operatorname{\mathcal{C}}$. For each $\beta < \alpha $, let us identify $G_{\beta }$ with a functor from $\operatorname{N}_{\bullet }( B(\beta ) \cup \{ a_{ \beta +1} \} )$ to $\operatorname{\mathcal{C}}$. Then the functors $F$ and $\{ G_{\beta } \} _{\beta < \alpha }$ determine a morphism of simplicial sets $H_0: K(A,B) \rightarrow \operatorname{\mathcal{C}}$, where $K(A,B) \subseteq \operatorname{N}_{\bullet }(B)$ is the simplicial subset appearing in the statement of Lemma 9.1.2.17. Since the inclusion map $K(A,B) \hookrightarrow \operatorname{N}_{\bullet }(B)$ is a categorical equivalence of simplicial sets, we can extend $H_0$ to a diagram $H: \operatorname{N}_{\bullet }( B ) \rightarrow \operatorname{\mathcal{C}}$.

Note that the linear ordering on $B$ is a well-ordering, with largest element $a_{\alpha }$. We claim that $H$ exhibits $f$ as a transfinite composition of morphisms of $W$, in the sense of Variant 9.1.2.3. It follows immediately from the construction that $H$ carries the pair $(a_0 \leq a_{\alpha } )$ to the morphism $f$. Moreover, if an element $b \in B$ has an immediate predecessor $b' \in B$, then there is a (unique) ordinal $\beta < \alpha $ such that both $b$ and $b'$ belong to $B(\beta ) \cup \{ a_{\beta +1} \} $; our assumption on $G_{\beta }$ then guarantees that the morphism $H(b') \rightarrow H(b)$ belongs to $W$. To complete the proof, it will suffice to show that if $b \neq a_0$ is an element of $B$ which does not have an immediate predecessor, then the restriction $H|_{ \operatorname{N}_{\bullet }( B_{\leq b} ) }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Note that $b$ belongs to $B(\beta )$ for some unique ordinal $0 \leq \beta \leq \alpha $. We consider three cases:

• Suppose that $b$ is not equal to $a_{\beta }$. In this case, the inclusion map $B(\beta )_{ < b } \hookrightarrow B_{< b }$ is cofinal (in the sense of Definition 4.7.1.26), and therefore induces a right cofinal morphism of simplicial sets $\operatorname{N}_{\bullet }( B(\beta )_{< b} ) \hookrightarrow \operatorname{N}_{\bullet }( B_{7.2.3.4). Using Corollary 7.2.2.2, we are reduced to showing that the diagram $H|_{ \operatorname{N}_{\bullet }( B(\beta )_{\leq b} )}$ is a colimit diagram in $\operatorname{\mathcal{C}}$, which follows from our assumption on $G_{\beta }$.

• Suppose that $b = a_{\beta }$ and that $\beta = \gamma +1$ is a successor ordinal. In this case, the inclusion map $B_{ \gamma } \hookrightarrow B_{< b}$ is cofinal, and therefore induces a right cofinal morphism of simplicial sets $\operatorname{N}_{\bullet }( B_{\beta } ) \hookrightarrow \operatorname{N}_{\bullet }( B_{< b })$. The desired result now follows again Corollary 7.2.2.2, since the restriction $H|_{ \operatorname{N}_{\bullet }( B_{\gamma } \cup \{ b\} ) }$ can be identified with $G_{\gamma }$ and is therefore a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

• Suppose that $b = a_{\beta }$, where $\beta $ is a nonzero limit ordinal. In this case, the inclusion map $A_{ < b } \hookrightarrow B_{ < b }$ is cofinal and therefore induces a right cofinal morphism of simplicial sets $\operatorname{N}_{\bullet }( A_{7.2.2.2, we are reduced to showing that the restriction $H|_{ \operatorname{N}_{\bullet }( A_{\leq b} ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. We conclude by observing that this restriction identifies with $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$.

Proof. Choose a diagram $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ which exhibits $f$ as a transfinite composition of morphisms of $W$. For every pair of ordinals $0 \leq \gamma \leq \beta \leq \alpha $, let $u_{\beta , \gamma }: F(\beta ) \rightarrow F(\gamma )$ denote the morphism of $\operatorname{\mathcal{C}}$ obtained by evaluating $F$ on the edge $(\gamma \leq \beta )$ of $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } )$. We write $\beta \sim \gamma $ if, for every ordinal $\lambda $ satisfying $\gamma \leq \lambda \leq \beta $, the morphisms $u_{\beta , \lambda }$ and $u_{\lambda , \gamma }$ are both isomorphisms. It is not difficult to see that this is an equivalence relation on the set $\mathrm{Ord}_{\leq \alpha }$. For every ordinal $\beta \leq \alpha $, the equivalence class of $\beta $ contains a smallest element which we will denote by $\beta _{-}$ (since $\mathrm{Ord}_{\leq \alpha }$ is well-ordered), and a largest element which we will denote by $\beta _{+}$ (since the collection of isomorphisms is closed under transfinite composition; see Corollary 9.1.2.12).

Choose a subset $A \subseteq \mathrm{Ord}_{\leq \alpha }$ which contains exactly one representative of each $\sim $-equivalence class. Our assumption that $f = u_{\alpha ,0}$ is not an isomorphism guarantees that $0$ and $\alpha $ belong to different equivalence classes; we can therefore arrange that both $0$ and $\alpha $ are contained in $A$. We will complete the proof by showing that the diagram $F|_{ \operatorname{N}_{\bullet }(A) }$ exhibits $f$ as a transfinite composition of morphisms of $W$ which are not isomorphisms (in the sense of Variant 9.1.2.3).

For any pair of ordinals $\gamma < \beta $ which belong to $A$, we have inequalities $\gamma \leq \gamma _{+} < \beta _{-} \leq \beta $. Then $u_{\beta , \gamma }$ factors as a composition

where the maps on the left and right are isomorphisms. In particular, $u_{ \beta , \gamma }$ is isomorphic to $u_{ \beta _{-}, \gamma _{+} }$ as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$. If $\beta $ is an immediate successor of $\gamma $ in $A$, then $\beta _{-}$ is an immediate successor of $\gamma _{+}$ in $\mathrm{Ord}_{\leq \alpha }$. Our assumption on $F$ then guarantees that $u_{ \beta _{-}, \gamma _{+} }$ is contained in $W$. Since $W$ is closed under isomorphism, it follows that $u_{\beta , \gamma }$ is also contained in $W$. Moreover, $u_{\beta , \gamma }$ cannot be an isomorphism (otherwise we would have $\beta \sim \gamma $, contradicting our assumption that $A$ contains exactly one representative of each equivalence class).

For each element $\beta \in A$, set $A_{\leq \beta } = \{ \gamma \in A: \gamma \leq \beta \} $ and $A_{ < \beta } = \{ \gamma \in A: \gamma < \beta \} $. To complete the proof, it will suffice to show that if $\beta \neq 0$ is not the immediate successor of another element of $A$, then the restriction $F|_{ \operatorname{N}_{\bullet }( A_{\leq \beta } ) }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $u_{\beta , \beta _{-} }$ is an isomorphism, it will suffice to show that $G = F|_{ \operatorname{N}_{\bullet }( A_{ < \beta } \cup \{ \beta _{-} \} )}$ is a colimit diagram (Corollary 7.1.2.14). Our assumption that $\beta $ has no immediate predecessor in $A$ guarantees that $\beta _{-}$ is a limit ordinal and that $A_{ < \beta }$ is a cofinal subset of in $\mathrm{Ord}_{< \beta _{-} }$. It follows that the inclusion map $\operatorname{N}_{\bullet }( A_{< \beta } ) \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \beta } )$ is right cofinal (Corollary 7.2.3.4). The desired result now follows from Corollary 7.2.2.2, since the restriction $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta _{-} } ) }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. $\square$

Proof. Assume that $f$ belongs to the transfinite closure of $W$; we will show that $f$ is either an isomorphism or a transfinite composition of morphisms of $W$ (the converse is clear, since the transfinite closure of $W$ contains all isomorphisms: see Remark 9.1.2.8). Let $W^{+}$ be the union of $W$ with the collection of all identity morphisms of $\operatorname{\mathcal{C}}$. Applying Proposition 9.1.2.16, we see that $f$ is a transfinite composition of morphisms of $W^{+}$. If $f$ is not an isomorphism, then Proposition 9.1.2.18 guarantees that $f$ is a transfinite composition of morphisms which belong to $W^{+}$ and are not isomorphism, and therefore belong to $W$. $\square$