# Kerodon

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### 9.1.2 Digression: Transfinite Composition

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $X: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ be a functor, which we display as a diagram

$X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \rightarrow \cdots$

Suppose that $X$ can be extended to a colimit diagram $\overline{X} : \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, carrying the cone point to an object $Y = \varinjlim (X)$. In this case, we can evaluate $\overline{X}$ on the edge $\{ 0 \} ^{\triangleright } \subseteq \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0})^{\triangleright }$ to obtain a morphism $g: X(0) \rightarrow Y$. Heuristically, we can think of the morphism $g$ as an “infinite composition” $\cdots f_{4} \circ f_{3} \circ f_{2} \circ f_{1} \circ f_0$. Our goal in this section is to extend this heuristic to more general well-ordered diagrams.

In what follows, we assume that the reader is familiar with the theory of ordinals (see §5.4.1 for a review). For every ordinal $\alpha$, let $\mathrm{Ord}_{\leq \alpha }$ denote the linearly ordered set of ordinals which are less than or equal to $\alpha$, and let $\mathrm{Ord}_{< \alpha }$ denote the subset consisting of ordinals which are strictly smaller than $\alpha$.

Definition 9.1.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that a morphism $f$ of $\operatorname{\mathcal{C}}$ is a transfinite composition of morphisms of $W$ if there exists an ordinal $\alpha$ and a functor $F: \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \rightarrow \operatorname{\mathcal{C}}$ with the following properties:

$(a)$

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

$(b)$

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

$(c)$

The morphism $F(0) \rightarrow F(\alpha )$ coincides with $f$.

In this case, we will say that $F$ exhibits $f$ as a transfinite composition of morphisms of $W$.

We say that $W$ is closed under transfinite composition if it contains every morphism which is a transfinite composition of morphisms of $W$.

Remark 9.1.2.2. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Then a morphism $f$ of $\operatorname{\mathcal{C}}$ is a transfinite composition of morphisms belonging to $W$ (in the sense of Definition 1.4.4.10) if and only if the corresponding morphism of the $\infty$-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a transfinite composition of morphisms belonging to $W$ (in the sense of Definition 9.1.2.1).

Variant 9.1.2.3. Let $A$ be a well-ordered set and let $\alpha$ denote its order type. Then there is a unique order-preserving bijection $\mathrm{Ord}_{ < \alpha } \simeq A$, which determines an isomorphism of simplicial sets $u: \operatorname{N}_{\bullet }( \mathrm{Ord}_{ \leq \alpha } ) \simeq \operatorname{N}_{\bullet }(A)^{\triangleright }$. If $\operatorname{\mathcal{C}}$ is an $\infty$-category containing a morphism $f$ and a collection of morphisms $W$, we will say that a diagram $F: \operatorname{N}_{\bullet }(A)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ exhibits $f$ as a transfinite composition of morphisms of $W$ if the composition $\operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \alpha } ) \simeq \operatorname{N}_{\bullet }(A)^{\triangleright } \xrightarrow {F} \operatorname{\mathcal{C}}$ exhibits $f$ as a transfinite composition of morphisms of $W$, in the sense of Definition 9.1.2.1.

Example 9.1.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Then every identity morphism of $\operatorname{\mathcal{C}}$ is a transfinite composition of morphisms of $W$ (take $\alpha = 0$ in Definition 9.1.2.1). In particular, if $W$ is closed under transfinite composition, then it contains every identity morphism of $\operatorname{\mathcal{C}}$.

Example 9.1.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Then every morphism of $W$ is a transfinite composition of morphisms of $W$ (take $\alpha = 1$ in Definition 9.1.2.1).

Example 9.1.2.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which contains a pair of composable morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$. Then any composition of $f$ with $g$ is a transfinite composition of morphisms of $W$ (take $\alpha = 2$ in Definition 1.4.4.10). In particular, if $W$ is closed under transfinite composition, then it is closed under composition.

Example 9.1.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$. Let $F$ denote the composite map $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )^{\triangleright } \twoheadrightarrow (\Delta ^{0})^{\triangleright } \simeq \Delta ^1 \xrightarrow {f} \operatorname{\mathcal{C}}$, which we display informally as a diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [drr]^-{f} \ar [r]^{\operatorname{id}} & X \ar [dr]^-{f} \ar [r]^{\operatorname{id}} & X \ar [d]^-{f} \ar [r]^{\operatorname{id}} & X \ar [dl]^-{f} \ar [r]^{\operatorname{id}} & \cdots \\ & & Y. & & }$

Since the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ is contractible (Example 3.2.4.2), the functor $F$ is a colimit diagram (Corollary 7.2.3.5), and therefore exhibits $f$ as a transfinite composition of morphisms belonging to the singleton $\{ \operatorname{id}_{X} \}$.

Remark 9.1.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is closed under transfinite composition. Combining Examples 9.1.2.4 and 9.1.2.7, we deduce that $W$ contains every isomorphism of $\operatorname{\mathcal{C}}$.

Remark 9.1.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is closed under transfinite composition. Then $W$ is closed under isomorphism: that if, if $f$ and $g$ are morphisms of $\operatorname{\mathcal{C}}$ which are isomorphic (as objects of the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$), then $f$ belongs to $W$ if and only if $g$ belongs to $W$. This follows by combining Example 9.1.2.6 with Remark 9.1.2.8. In particular, the condition that a morphism $f$ of $\operatorname{\mathcal{C}}$ belongs to $W$ depends only on the homotopy class $[f]$.

Example 9.1.2.6 admits a partial converse:

Proposition 9.1.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that:

$(1)$

Every identity morphism of $\operatorname{\mathcal{C}}$ belongs to $W$.

$(2)$

The collection $W$ is closed under composition.

$(3)$

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

Then $W$ is closed under transfinite composition.

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

$\xymatrix@R =50pt@C=50pt{ & F(\gamma ) \ar [dr] & \\ X \ar [ur]^{ f_{\gamma } } \ar [rr]^{ f_{\beta } } & & F(\beta ), }$

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

Remark 9.1.2.11. In the statement of Proposition 9.1.2.10, it is not necessary to assume that $W$ is closed under the formation of all colimits in the $\infty$-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. It suffices to consider colimits of diagrams indexed by $\operatorname{N}_{\bullet }( \mathrm{Ord}_{ < \beta } )$ where $\beta$ is a limit ordinal; moreover, we can further restrict our attention to limits which are preserved by the evaluation functors $\operatorname{ev}_{0}, \operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.

Corollary 9.1.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$. Then $W$ is closed under transfinite composition.

Definition 9.1.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. The transfinite closure of $W$ is the smallest collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $W$ and is closed under transfinite composition.

Example 9.1.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$. It follows from Corollary 9.1.2.12 and Example 9.1.2.7 that $W$ is the smallest collection of morphisms of $\operatorname{\mathcal{C}}$ which is closed under transfinite composition: that is, it is the transfinite closure of the empty set.

Warning 9.1.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$. If $f$ is a transfinite composition of morphisms in $W$, then it belongs to the transfinite closure of $W$. Beware that, if we strictly adhere to the terminology of Definition 9.1.2.1, then the converse need not be true. For example, if $W = \emptyset$ and $f$ is an isomorphism, then $f$ belongs to the transfinite closure of $W$ (Example 9.1.2.14). However, $f$ is a transfinite composition of morphisms in $W$ if and only if it is an identity morphism (Example 9.1.2.4).

We can rule out the pathological behavior described in Warning 9.1.2.15 adding a mild additional assumption.

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 all morphisms of $\operatorname{\mathcal{C}}$ which are transfinite compositions of morphisms belonging to $W$. If $W$ contains all identity morphisms, then $\overline{W}$ is closed under transfinite composition (and is therefore the transfinite closure of $W$).

Our proof of Proposition 9.1.2.16 will make use of the following:

Lemma 9.1.2.17. Let $B$ be a linearly ordered set and let $A \subseteq B$ be a subset which satisfies the following condition:

$(\ast )$

For every element $b \in B$, the set $\{ a \in A: a \leq b \}$ has a largest element $b_{-}$, and the set $\{ a \in A: b \leq a \}$ has a smallest element $b_{+}$.

Let $K(A,B) \subseteq \operatorname{N}_{\bullet }(B)$ be the simplicial subset whose $n$-simplices are given by tuples $(b_0 \leq b_1 \leq b_2 \leq \cdots \leq b_ n)$ which satisfy one of the following conditions:

• Each of the elements $b_{i}$ belongs to $A$.

• For every element $a \in A$, either $a \leq b_0$ or $a \geq b_ n$.

Then the inclusion map $\iota : K(A,B) \hookrightarrow \operatorname{N}_{\bullet }( B )$ is a categorical equivalence of simplicial sets.

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 1.4.7.7. The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.4.7.7), 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

$\tau _{i}: \Delta ^{ a_{i} - a_{i-1} } \rightarrow \Delta ^{n} \quad \quad k \mapsto k + a_{i-1}.$

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

$\xymatrix@R =50pt@C=50pt{ \operatorname{Spine}[m] \ar [r] \ar [d] & \Delta ^{m} \ar [d]^{\sigma } \\ K' \ar [r] & K(A,B). }$

Since the inclusion map $\operatorname{Spine}[m] \hookrightarrow \Delta ^{m}$ is inner anodyne (Example 1.4.7.7), 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

$\xymatrix@R =50pt@C=50pt{ \coprod _{1 \leq i \leq m} \operatorname{Spine}[ a_{i} - a_{i-1} ] \ar [r] \ar [d] & \operatorname{Spine}[n] \ar [d] \\ \coprod _{1 \leq i \leq m} \Delta ^{a_{i} - a_{i-1} } \ar [r]^{ \{ \tau _ i \} } & K', }$

where the left vertical map is inner anodyne by virtue of Example 1.4.7.7. $\square$

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 5.4.1.26), 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 $F|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \beta } ) }$.

$\square$

Proposition 9.1.2.18. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is closed under isomorphism, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$ which is a transfinite composition of morphisms of $W$. If $f$ is not an isomorphism, then it is a transfinite composition of morphisms of $W$ which are not isomorphisms.

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

$F( \gamma ) \xrightarrow { u_{ \gamma _{+}, \gamma } } F( \gamma _{+} ) \xrightarrow { u_{ \beta _{-}, \gamma _{+} } } F( \beta _{-} ) \xrightarrow { u_{ \beta , \beta _{-} } } F( \beta ),$

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 $\operatorname{\mathcal{C}}$. $\square$

Corollary 9.1.2.19. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is closed under isomorphism. Then a morphism $f$ of $\operatorname{\mathcal{C}}$ belongs to the transfinite closure of $W$ if and only if $f$ is either an isomorphism or a transfinite composition of morphisms of $W$.

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 $W$. $\square$