Kerodon

Proof. Since $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, it is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ spanned by the $(\kappa ,\lambda )$-compact objects. Applying Proposition 9.3.1.16, we conclude that $\operatorname{\mathcal{E}}$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the full subcategory $\operatorname{\mathcal{E}}_0 = U^{-1}(\operatorname{\mathcal{C}}_0 )$. It follows that $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compactly generated, and that an object of $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compact if and only if it is a retract of an object of $\operatorname{\mathcal{E}}_0$ (Lemma 9.4.1.18). We conclude by observing that the full subcategory $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ is closed under the formation of retracts, since it is the inverse image of a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ which is closed under retracts (Remark 9.2.5.18). $\square$

Proof. It follows from Corollary 7.1.4.22 that the $\infty $-category $\operatorname{\mathcal{C}}_{/F}$ admits $\lambda $-small $\kappa $-filtered colimits which are preserved by the right fibration $U: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}$, so the desired result is a special case of Proposition 9.4.3.1. $\square$

Proof. Apply Theorem 9.4.3.4 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal. $\square$

Proof of Theorem 9.4.3.4. Apply Corollary 9.4.3.6 in the special case $\kappa = \aleph _0$. $\square$

Proof. It follows from Proposition 7.1.8.3 that the diagonal functor is $(\kappa ,\lambda )$-finitary. Our assumption that $F$ carries each vertex of $K$ to a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$ guarantees that $F$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ (Proposition 9.2.8.9). Applying Theorem 9.4.3.4, we conclude that the oriented fiber product $\{ F\} \vec{\times }_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, and that an object of $\{ F\} \vec{\times }_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact if and only if its image in $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact. The desired result now follows from the equivalence $\operatorname{\mathcal{C}}_{F/} \hookrightarrow \{ F\} \vec{\times }_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ supplied by Theorem 4.6.4.19. $\square$

Proof. We proceed by induction on the number of elements of $Q$. If $Q$ is empty, the result is clear. Otherwise, we can choose a maximal element $q \in Q$. Set

so that Example 4.6.4.10 supplies an equivalence of $\infty $-categories

Proof. Apply Theorem 9.4.3.4 in the special case $\operatorname{\mathcal{C}}= \Delta ^0$. $\square$

Proof. For each $i \in I$, let $\operatorname{\mathcal{C}}^{0}_{i}$ be the full subcategory of $\operatorname{\mathcal{C}}_{i}$ spanned by the $(\kappa ,\lambda )$-compact objects. Set $\operatorname{\mathcal{C}}^{0} = \prod _{i \in I} \operatorname{\mathcal{C}}^{0}_{i}$, which we regard as a full subcategory of $\operatorname{\mathcal{C}}$. Since $I$ is $\kappa $-small, Proposition 9.2.8.6 guarantees that every object of $\operatorname{\mathcal{C}}^{0}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{C}}$. To show that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, it will suffice to show that every object $C \in \operatorname{\mathcal{C}}$ can be realized as the colimit of $\lambda $-small $\kappa $-filtered diagram in $\operatorname{\mathcal{C}}^{0}$.

Write $C = \{ C_ i \} _{i \in I}$. For each $i \in I$, the assumption $\kappa \trianglelefteq \lambda $ guarantees that $C_ i$ can be realized as the colimit of a diagram $F_{i}: \operatorname{N}_{\bullet }(A_ i) \rightarrow \operatorname{\mathcal{C}}^{0}_{i}$, where $(A_ i, \leq )$ is a $\lambda $-small $\kappa $-directed partially ordered set (Remark 9.4.1.10). Set $A = \prod _{i \in I} A_ i$, so that the collection $\{ F_ i \} _{i \in I}$ determines a product functor $F: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}^{0}$. Beware that $A$ need not be $\lambda $-small. However, the disjoint union $X = \coprod _{i \in I} A_ i$ is $\lambda $-small. Applying our assumption $\kappa \trianglelefteq \lambda $, we deduce that there is a collection of $\kappa $-small subsets $\{ X_ j \subseteq X \} _{j \in J}$ indexed by a $\lambda $-small set $J$, having the property that every $\kappa $-small subset of $X$ is contained in some $X_ j$. For every pair of indices $i \in I$, $j \in J$, the intersection $A_ i \cap X_ j$ is $\kappa $-small, and therefore admits an upper bound $b_{i,j} \in A_{i}$. For each $j \in J$, we can regard the tuple $\{ b_{i,j} \} _{i \in I}$ as an element $b_ j \in A$. Then $B = \{ b_ j \} _{j \in J}$ is a $\lambda $-small subset of $A$. By construction, every $\kappa $-small subset of $A$ has an upper bound in $B$, so that $B$ is $\kappa $-directed. Moreover, for each $i \in I$, the composite map $B \hookrightarrow A \twoheadrightarrow A_{i}$ satisfies the criterion of Example 9.1.4.10, and therefore induces a right cofinal functor $\operatorname{N}_{\bullet }(B) \rightarrow \operatorname{N}_{\bullet }(A_ i)$. Applying Corollary 7.2.2.11, we deduce that $C_ i$ is a colimit of the diagram

Invoking Example 7.1.3.11, we see that $C = \{ C_ i \} _{i \in I}$ is a colimit of the $\lambda $-small $\kappa $-filtered diagram $\operatorname{N}_{\bullet }(B) \hookrightarrow \operatorname{N}_{\bullet }(A) \xrightarrow { \prod _{i \in I} F_ i } \operatorname{\mathcal{C}}^{0}$.

We complete the proof by observing that if the object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact, then it is a retract of $F(b)$ for some $b \in B$, and therefore also belongs to $\operatorname{\mathcal{C}}^{0}$. $\square$

Proof. Apply Proposition 9.4.3.13 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal. $\square$

Proof. If $K$ and $L$ are subsets of $\operatorname{\mathcal{E}}$, let us write $K \Subset L$ if $K$ is contained in $L$ and the following addition conditions are satisfied:

$(a)$

For $0 \leq i < n$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{ U } \\ \Delta ^ n \ar [r] \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}} \]

having the property that $\sigma _0$ factors through $K$ admits a solution, where the $n$-simplex $\sigma $ is contained in $L$.

$(b)$

For every finite simplicial subset $K_0 \subseteq K$, the composite map $K_0 \hookrightarrow L \hookrightarrow \operatorname{Ex}^{\infty }(L)$ is nullhomotopic.

If $K$ is $\lambda $-small, then the collection of finite simplicial subsets $K_0 \subseteq K$ and the collection of all lifting problems of the above form are both $\lambda $-small. We can then use our assumptions that $\operatorname{\mathcal{E}}$ is weakly contractible and that $U$ is a left fibration to choose a $\lambda $-small subset $L \subseteq \operatorname{\mathcal{E}}$ satisfying $K \Subset L$. Applying this observation repeatedly, we can choose a sequence of $\lambda $-small sets

The union $\operatorname{\mathcal{E}}' = \bigcup _{n \geq 0} K(n)$ then has the desired properties. $\square$

Proof of Theorem 9.4.3.4. Let $\kappa \leq \lambda $ be regular cardinals and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories which satisfy conditions $(a)$, $(b)$, and $(c)$ of Theorem 9.4.3.4. Let $\operatorname{\mathcal{C}}_{\pm }$ be the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ and let $\operatorname{\mathcal{C}}_{\pm }^{0}$ denote the full subcategory of $\operatorname{\mathcal{C}}_{\pm }$ spanned by those objects $(C_{-}, C_{+}, u)$ where $C_{-}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{-}$ and $C_{+}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{+}$. Proposition 7.1.9.4 guarantees that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-cocomplete. Let $\operatorname{\mathcal{C}}_{\pm }^{1} \subseteq \operatorname{\mathcal{C}}_{\pm }$ be the smallest full subcategory which contains $\operatorname{\mathcal{C}}_{\pm }^{0}$ and is closed under $\lambda $-small $\kappa $-filtered colimits. Note that every object of $\operatorname{\mathcal{C}}_{\pm }^{0}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{C}}_{\pm }$ (Proposition 9.2.8.3), and therefore also as an object of $\operatorname{\mathcal{C}}_{\pm }^{1}$. Applying Proposition 9.3.2.3, we conclude that the inclusion functor $\operatorname{\mathcal{C}}_{\pm }^{0} \hookrightarrow \operatorname{\mathcal{C}}_{\pm }^{1}$ exhibits $\operatorname{\mathcal{C}}_{\pm }^{1}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_{\pm }^{0}$. It follows that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }^{1}$ is $(\kappa ,\lambda )$-compactly generated, and that an object of $\operatorname{\mathcal{C}}_{\pm }^{1}$ is $(\kappa ,\lambda )$-compact if and only if it a retract of an object of $\operatorname{\mathcal{C}}_{\pm }^{0}$ (Lemma 9.4.1.18). Since $\operatorname{\mathcal{C}}_{\pm }^{0}$ is closed under retracts (Remark 9.2.5.18), every such retract is already contained in $\operatorname{\mathcal{C}}_{\pm }^{0}$. We will complete the proof by showing that $\operatorname{\mathcal{C}}_{\pm }^{1} = \operatorname{\mathcal{C}}_{\pm }$.

Let us say that an object $C_{-} \in \operatorname{\mathcal{C}}_{-}$ is good if, for every object $C_{+} \in \operatorname{\mathcal{C}}_{+}$ and every morphism $u: F_{-}(C_{-} ) \rightarrow F_{+}( C_{+} )$, the triple $( C_{-}, C_{+}, u)$ belongs to the full subcategory $\operatorname{\mathcal{C}}_{\pm }^{1} \subseteq \operatorname{\mathcal{C}}$. We wish to show that every object of $\operatorname{\mathcal{C}}_{-}$ is good. Since $\operatorname{\mathcal{C}}_{-}$ is $(\kappa ,\lambda )$-compactly generated, it will suffice to prove the following:

$(a)$

The collection of good objects of $\operatorname{\mathcal{C}}_{-}$ is closed under $\lambda $-small $\kappa $-filtered colimits.

$(b)$

Every $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{-}$ is good.

We first prove $(a)$. Suppose we are given a $\lambda $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{J}}$ and a colimit diagram $G_{-}: \operatorname{\mathcal{J}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{-}$ which carries each object of $\operatorname{\mathcal{J}}$ to a good object of $\operatorname{\mathcal{C}}_{-}$; we wish to show that the object $C_{-} = G_{-}(P)$ is also good, where $P$ denotes the cone point of $\operatorname{\mathcal{J}}^{\triangleright }$. Fix a morphism $u: F_{-}( C_{-} ) \rightarrow F_{+}(C_{+} )$ as above. Since the projection map $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \{ C_{+} \} \rightarrow \operatorname{\mathcal{C}}_{-}$ is a right fibration and the inclusion map $\{ P\} \hookrightarrow \operatorname{\mathcal{J}}^{\triangleright }$ is right anodyne (Example 4.3.7.11), we can lift $G_{-}$ to a diagram $G_{\pm }: \operatorname{\mathcal{J}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\pm }$ such that $G_{\pm }( P ) = (C_{-}, C_{+}, u)$ and the composition $\operatorname{\mathcal{J}}^{\triangleright } \xrightarrow {G_{\pm } } \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{+}$ is the constant functor taking the value $C_{+}$. Applying Lemma 7.1.9.3, we see that $G_{\pm }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$. By construction, the restriction $G_{\pm }|_{ \operatorname{\mathcal{J}}}$ factors through $\operatorname{\mathcal{C}}_{\pm }^{1}$, so that $G_{\pm }(P) = (C_{-}, C_{+}, u)$ also belongs to $\operatorname{\mathcal{C}}_{\pm }^{1}$.

We now prove $(b)$. For the remainder of the proof, we fix an object $C_{-} \in \operatorname{\mathcal{C}}_{-}$ which is $(\kappa ,\lambda )$-compact; we wish to show that $C_{-}$ is good. Let us say that an object $C_{+} \in \operatorname{\mathcal{C}}_{+}$ is good if, for every morphism $u: F_{-}( C_{-} ) \rightarrow F_{+}( C_{+} )$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the triple $C_{\pm } = ( C_{-}, C_{+}, u)$ is contained in $\operatorname{\mathcal{C}}_{\pm }^{1}$. Note that, if $C_{+}$ is $(\kappa ,\lambda )$-compact, then this condition is automatic (since $C_{\pm }$ is an object of the subcategory $\operatorname{\mathcal{C}}_{\pm }^{0} \subseteq \operatorname{\mathcal{C}}_{\pm }^{1}$). Using our assumption that $\operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-compactly generated, we are reduced to proving the following:

$(c)$

The collection of good objects of $\operatorname{\mathcal{C}}_{+}$ is closed under $\lambda $-small $\kappa $-filtered colimits.

Fix a $\lambda $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$ and a colimit diagram $H: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{+}$ carrying each object of $\operatorname{\mathcal{K}}$ to a good object of $\operatorname{\mathcal{C}}_{+}$; we wish to show that $C_{+} = H(Q)$ is also good, where $Q \in \operatorname{\mathcal{K}}^{\triangleright }$ denotes the cone point. Set $\operatorname{\mathcal{E}}= \{ C_{-} \} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}$ and $\overline{\operatorname{\mathcal{E}}} = \{ C_{-} \} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}^{\triangleright }$, so that projection onto the second factors determines left fibrations

Fix an uncountable cardinal $\mu $ of cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small, so that the left fibration $\overline{U}$ admits a covariant transport representation given by the composition

Since the functor $F_{+}$ is $(\kappa ,\lambda )$-finitely and the object $F_{-}(C_{-} )$ is $(\kappa ,\lambda )$-compact, this composite functor is a colimit diagram in $\operatorname{\mathcal{S}}_{< \mu }$. Applying Corollary 7.4.3.14, we see that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is a weak homotopy equivalence.

Fix a morphism $u: F_{-}(C_{-} ) \rightarrow F_{+}(C_{+} )$ in the $\infty $-category $\operatorname{\mathcal{C}}$; we wish to show that the triple $C_{\pm } = (C_{-}, C_{+}, u)$ belongs to the full subcategory $\operatorname{\mathcal{C}}_{\pm }^{1} \subseteq \operatorname{\mathcal{C}}_{\pm }$. Let us regard the triple $(C_{-}, Q, u)$ as an object $\widetilde{Q} \in \overline{\operatorname{\mathcal{E}}}$ satisfying $\overline{U}( \widetilde{Q} ) = Q$. Since $\overline{U}$ is a left fibration, the induced map

is a Kan fibration (Proposition 4.3.7.2). Set $\operatorname{\mathcal{E}}_{ / \widetilde{Q} } = \operatorname{\mathcal{E}}\times _{ \overline{\operatorname{\mathcal{E}}} } \overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} }$ so that projection onto the second factor induces Kan fibration $V: \operatorname{\mathcal{E}}_{ / \widetilde{Q} } \rightarrow \operatorname{\mathcal{E}}$. Note that the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} }$ has a final object (Proposition 4.6.7.23) and is therefore weakly contractible (Corollary 4.6.7.26). It follows from Corollary 3.3.7.4 that the inclusion map $\operatorname{\mathcal{E}}_{ / \widetilde{Q} } \hookrightarrow \overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} }$ is also a weak homotopy equivalence, so that the $\infty $-category $\operatorname{\mathcal{E}}_{ / \widetilde{Q} }$ is weakly contractible. Applying Lemma 9.4.3.15 to the left fibration $(U \circ V): \operatorname{\mathcal{E}}_{ / \widetilde{Q} } \rightarrow \operatorname{\mathcal{K}}$, we can choose a weakly contractible $\lambda $-small subset $\operatorname{\mathcal{K}}' \subseteq \operatorname{\mathcal{E}}_{ / \widetilde{Q} }$ for which the restriction $(U \circ V)|_{ \operatorname{\mathcal{K}}' }: \operatorname{\mathcal{K}}' \rightarrow \operatorname{\mathcal{K}}$ is a left fibration. Applying Corollary 9.1.5.14 (and Remark 9.1.3.7), we conclude that $\operatorname{\mathcal{K}}'$ is a $\kappa $-filtered $\infty $-category and that the functor $T = (U \circ V)|_{ \operatorname{\mathcal{K}}' }$ is right cofinal. The composition

determines a diagram $H': \operatorname{\mathcal{K}}'^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\pm }$, whose image in $\operatorname{\mathcal{C}}_{-}$ is the constant diagram taking the value $C_{-}$ and whose image in $\operatorname{\mathcal{C}}_{+}$ is given by the composition $H \circ T^{\triangleright }$. Applying Corollary 7.2.2.3 and Lemma 7.1.9.3, we deduce that $H'$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ Our assumptionon $H$ guarantees that $H'|_{ \operatorname{\mathcal{K}}'}$ factors through the full subcategory $\operatorname{\mathcal{C}}^{1}_{\pm }$. Since $\operatorname{\mathcal{C}}^{1}_{\pm }$ is closed under $\lambda $-small $\kappa $-filtered colimits, we conclude that the object $C_{\pm } = (C_{-}, C_{+}, u)$ also belong to $\operatorname{\mathcal{C}}^{1}_{\pm }$. $\square$