Kerodon

Proof of Proposition 9.2.4.1. Apply Theorem 9.2.4.3 in the special case $\kappa = \lambda = \aleph _0$. $\square$

Proof. Recall that every ordinal $\alpha $ can be written uniquely as a sum $\alpha _0 + n$, where $\alpha _0$ is a limit ordinal and $n$ is a nonnegative integer. Let us say that $\alpha $ is even if $n$ is an even integer, and that $\alpha $ is odd if $n$ is an odd integer. For every ordinal $\beta \leq \kappa $, let $\mathrm{Ord}_{< \beta }^{+}$ denote the linearly ordered set of even ordinals which are smaller than $\beta $, and let $\mathrm{Ord}_{< \beta }^{-}$ denote the linearly ordered set of odd ordinals which are smaller than $\beta $. We will construct a commutative diagram of $\infty $-categories

Assuming this is possible, our assumption that $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are $\kappa $-sequentially cocomplete guarantees that the diagrams $T^{-}$ and $T^{+}$ admit colimits $C_{-} = \varinjlim ( T^{-} )$ and $C_{+} = \varinjlim ( T^{+} )$ in the $\infty $-categories $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$, respectively. Since the functors $F_{-}$ preserves $\kappa $-sequential colimits, the object $F_{-}( C_{-} ) \in \operatorname{\mathcal{C}}$ is a colimit of the diagram $F_{-} \circ T^{-} = T|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{<\kappa }^{-} ) }$. Similarly, $F_{+}( C_{+} )$ is a colimit of the diagram $F_{+} \circ T^{+} = T|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{<\kappa }^{+} ) }$. Corollary 7.2.3.4 guarantees that the inclusion maps $\operatorname{N}_{\bullet }( \mathrm{Ord}_{<\kappa }^{-} ) \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{<\kappa } ) \hookleftarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{<\kappa }^{+} )$ are right cofinal, so that $F_{-}( C_{-} )$ and $F_{+}(C_{+})$ can also be viewed as colimits of the diagram $T$ (Corollary 7.2.2.11). Using Proposition 7.1.1.12, we can choose an isomorphism $u: F_{-}(C_{-}) \rightarrow F_{+}(C_{+})$ in the $\infty $-category $\operatorname{\mathcal{C}}$. It follows that the triple $(C_{-}, C_{+}, u)$ can be viewed as an object of the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$.

It remains to construct the diagram (9.7). Let $P$ denote the collection of all tuples $(\alpha , T_{< \alpha }, T^{-}_{< \alpha }, T^{+}_{< \alpha } )$ where $\alpha \leq \kappa $ is an ordinal and

are functors satisfying $F_{-} \circ T_{< \alpha }^{-} = T_{< \alpha }|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \alpha }^{-} ) }$ and $F_{+} \circ T_{< \alpha }^{+} = T_{< \alpha }|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \alpha }^{+} ) }$. We regard $P$ as a partially ordered set, where $(\alpha , T_{< \alpha }, T^{-}_{< \alpha }, T^{+}_{< \alpha } ) \leq (\beta , T_{< \beta }, T^{-}_{< \beta }, T^{+}_{< \beta } )$ if $\alpha \leq \beta $ and the functors $T_{< \alpha }$, $T^{-}_{< \alpha }$, and $T^{+}_{< \alpha }$ are the restrictions of $T_{< \beta }$, $T^{-}_{< \beta }$, and $T^{+}_{< \beta }$, respectively. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma, and therefore contains a maximal element $(\gamma , T_{< \gamma }, T^{-}_{< \gamma }, T^{+}_{< \gamma } )$. To complete the proof, it will suffice to show that $\gamma = \kappa $. Assume otherwise. For simplicity, we will assume that $\gamma $ is even (the case where $\gamma $ is odd can be handled by a similar argument). To obtain a contradiction, it will suffice to show that $T_{< \gamma }$ and $T^{+}_{< \gamma }$ can be extended to functors

satisfying $F_{+} \circ T^{+}_{\leq \gamma } = T_{\leq \gamma }|_{ \operatorname{N}_{\bullet }( \mathrm{Ord}_{\leq \gamma }^{+} )}$. Unwinding the definitions, we must show that the $\infty $-category

is nonempty. We distinguish two cases:

• Suppose that $\gamma $ is a limit ordinal. Then the inclusion map $\operatorname{N}_{\bullet }( \mathrm{Ord}^{+}_{< \gamma } ) \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \gamma } )$ is right cofinal (Corollary 7.2.3.4) and therefore right anodyne (Proposition 7.2.1.3). Applying Proposition 4.3.6.13, we deduce that the restriction map $\operatorname{\mathcal{C}}_{ T_{< \gamma } / } \rightarrow \operatorname{\mathcal{C}}_{ ( F_{+} \circ T^{+}_{< \gamma } )/}$ is a trivial Kan fibration. It follows that projection onto the second factor induces a trivial Kan fibration $\operatorname{\mathcal{E}}\rightarrow (\operatorname{\mathcal{C}}_{+})_{ T^{+}_{< \gamma } / }$. In particular, to show that $\operatorname{\mathcal{E}}$ is nonempty, it suffices to show that the coslice $\infty $-category $(\operatorname{\mathcal{C}}_{+})_{ T^{+}_{< \gamma } / }$ is nonempty. This follows from our assumption that $\operatorname{\mathcal{C}}_{+}$ is $\kappa $-filtered, since the simplicial set $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \gamma }^{+} )$ is $\kappa $-small.

• Suppose that $\gamma = \beta + 1$ is the successor of an odd ordinal $\beta $. Set $A = \mathrm{Ord}^{+}_{< \gamma } \cup \{ \beta \} $. Then the inclusion map $\operatorname{N}_{\bullet }(A) \hookrightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \gamma } )$ is right cofinal (Corollary 7.2.3.4) and therefore right anodyne (Proposition 7.2.1.3). Let $T' = T_{< \gamma } |_{ \operatorname{N}_{\bullet }(A)}$. Then the restriction map $\operatorname{\mathcal{C}}_{ T_{< \gamma } / } \rightarrow \operatorname{\mathcal{C}}_{T' / }$ is a trivial Kan fibration (Proposition 4.3.6.13), and therefore induces a trivial Kan fibration from $\operatorname{\mathcal{E}}$ to the $\infty $-category

  \[ \operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}_{T'/ } \times _{ \operatorname{\mathcal{C}}_{ ( F_{+} \circ T^{+}_{< \gamma } )/ } } (\operatorname{\mathcal{C}}_{+})_{ T^{+}_{< \gamma } /}. \]

  We will complete the proof by showing that $\operatorname{\mathcal{E}}'$ is weakly contractible (and therefore nonempty). This follows from Theorem 7.2.3.1, since $F_{+}$ is a right cofinal functor of $\kappa $-filtered $\infty $-categories and therefore induces a cofinal functor of coslice $\infty $-categories $(\operatorname{\mathcal{C}}_{+})_{ T^{+}_{< \gamma } /} \rightarrow \operatorname{\mathcal{C}}_{ ( F_{+} \circ T^{+}_{< \gamma } )/ }$ (Corollary 9.1.4.18), because the simplicial set $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \gamma }^{+} )$ is $\kappa $-small.

Proof of Theorem 9.2.4.3. Let $\kappa \leq \lambda $ be regular cardinals and let

be a categorical pullback diagram of $\infty $-categories which satisfies conditions $(1)$, $(2)$, and $(3)$ of Theorem 9.2.4.3, and let $E: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}$ be the functor given by the composition $F_{+} \circ E_{+} = F_{-} \circ E_{-}$. For every $\lambda $-small diagram $q: K \rightarrow \operatorname{\mathcal{C}}_{\pm }$, the induced diagram of coslice $\infty $-categories

is also a categorical pullback square (Corollary 4.6.4.22) satisfying conditions $(1)$, $(2)$, and $(3)$ of Theorem 9.2.4.3: condition $(1)$ follows from Proposition 9.1.1.17, condition $(2)$ from Corollary 9.1.4.18, and condition $(3)$ from Corollary 7.1.7.5. Applying Lemma 9.2.4.5, we deduce that the $\infty $-category $( \operatorname{\mathcal{C}}_{\pm } )_{q/}$ is nonempty. Allowing $q$ to vary, we conclude that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $\lambda $-filtered.

We will complete the proof by showing that the functor $E_{-}$ is right cofinal (the assertion that $E_{+}$ is right cofinal follows by a similar argument). By virtue of Theorem 9.1.4.5, it will suffice to show that for every object $X \in \operatorname{\mathcal{C}}_{\pm }$, the induced map of coslice $\infty $-categories $(\operatorname{\mathcal{C}}_{\pm })_{X/} \rightarrow (\operatorname{\mathcal{C}}_{-})_{ E_{-}(X) / }$ is weakly right cofinal. Arguing as above, we can replace (9.8) by the diagram of coslice $\infty $-categories

and thereby reduce to the problem of showing that that the functor $E_{-}$ is weakly cofinal. Fix an object $Y \in \operatorname{\mathcal{C}}_{-}$; we wish to show that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm } \times _{ \operatorname{\mathcal{C}}_{-} } (\operatorname{\mathcal{C}}_{-})_{Y/}$ is nonempty. This follows by applying Lemma 9.2.4.5 to the outer rectangle in the commutative diagram

note that the bottom horizontal composition is right cofinal by virtue of Corollary 9.1.4.16. $\square$