Kerodon

Proof of Proposition 9.7.1.6. Since $\kappa < \lambda $ and $\lambda $ has exponential cofinality $\geq \kappa $, the collection of isomorphism classes of $\kappa $-small simplicial sets $K$ is $\lambda $-small (Proposition 4.9.4.20). Moreover, if the simplicial set $K$ is fixed, then the collection of isomorphism classes of diagrams $K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is $\lambda $-small (Remark 4.9.5.13). We may therefore assume without loss of generality that the index set $J$ is $\lambda $-small.

Let us say that an object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ is good if, for every object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, the morphism space $\operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) }( \mathscr {F}, \mathscr {G} )$ is essentially $\lambda $-small. Since $\operatorname{\mathcal{C}}$ is locally $\lambda $-small, every representable functor $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is good (see Proposition 8.3.1.3). Using Variant 7.4.1.15, we see that the collection of good objects is closed under $\kappa $-small colimits. For each $j \in J$, $q_{j}$ denote the restriction $\overline{q}_ j|_{ K_{j} }$ and let $\mathscr {F}_ j$ be a colimit of the diagram $(h \circ q_ j): K_{j} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$. Since the simplicial set $K_ j$ is $\kappa $-small, it follows that $\mathscr {F}_ j$ is both good and $(\kappa ,\lambda )$-compact. Let $Y_ j \in \operatorname{\mathcal{C}}$ denote the value of $\overline{q}_{j}$ at the cone point of $K_ j^{\triangleright }$, so that $h \circ \overline{q}_ j$ determines a morphism $w_{j}: \mathscr {F}_ j \rightarrow h_{ Y_ j }$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$.

Set $W = \{ w_ j \} _{j \in J}$. For every $\lambda $-cocomplete $\infty $-category $\operatorname{\mathcal{D}}$, let $\operatorname{Fun}_{Q}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carry each $\overline{q}_ j$ to a colimit diagram in $\operatorname{\mathcal{D}}$, and $\operatorname{Fun}^{\lambda -\mathrm{cocont}}_{Q}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }), \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }), \operatorname{\mathcal{D}})$ spanned by those $\lambda $-cocontinuous functors $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) \rightarrow \operatorname{\mathcal{D}}$ which carry each $h \circ \overline{q}_ j$ to a colimit diagram in $\operatorname{\mathcal{D}}$. Note that, if $F$ is $\lambda $-cocontinuous, then the latter condition is satisfied if and only if $F$ carries each $w_ j$ to an isomorphism in $\operatorname{\mathcal{D}}$. In the special case where $F$ is the functor represented by an object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, we conclude that $\mathscr {G}$ belongs to $\overline{\operatorname{\mathcal{C}}}$ if and only if it is $W$-local. Assertion $(1)$ now follows from Proposition 9.6.3.10. To prove $(2)$, we note that there is a commutative diagram

where the right side is a categorical pullback square and the left horizontal map is an equivalence of $\infty $-categories (Remark 9.6.3.11). It will therefore suffice to show that the horizontal map on the bottom right is an equivalence of $\infty $-categories, which follows from Theorem 8.4.3.2. $\square$

Proof. Using Corollary 8.3.3.17, we may assume that there exists a fully faithful $\mathbb {K}$-cocontinuous functor $\iota : \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$, where $\widehat{\operatorname{\mathcal{D}}}$ is a $\lambda $-cocomplete $\infty $-category (for example, we can take $\widehat{\operatorname{\mathcal{D}}} = \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}_{ < \mu } )^{\operatorname{op}}$, where $\mu $ is some uncountable regular cardinal of exponential cofinality $\geq \lambda $). Replacing $\operatorname{\mathcal{D}}$ by its essential image, we may assume that $\operatorname{\mathcal{D}}$ is a full subcategory of $\widehat{\operatorname{\mathcal{D}}}$ and that $\iota $ is the inclusion functor. Proposition 9.7.1.6 guarantees the existence of a $\lambda $-cocontinuous functor $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{D}}}$ and an isomorphism $\overline{\alpha }: f \rightarrow \overline{F} \circ \overline{H}$. Since $\overline{F}$ is $\mathbb {K}$-cocontinuous, the inverse image $\overline{F}^{-1}( \operatorname{\mathcal{D}}) \subseteq \overline{\operatorname{\mathcal{C}}}$ is closed under $K$-indexed colimits for $K \in \mathbb {K}$ and therefore contains the full subcategory $\widehat{\operatorname{\mathcal{C}}} \subseteq \overline{\operatorname{\mathcal{C}}}$. It follows that $\overline{F}$ restricts to a $\mathbb {K}$-cocontinuous functor $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$, and that $\overline{\alpha }$ can be identified with an isomorphism $\alpha : f \rightarrow F \circ H$. The pair $(F, \alpha )$ then satisfies condition $(2)$ by construction.

We first show that the pair $(F,\alpha )$ satisfies $(1)$: that is, $\alpha $ exhibits $F$ as a left Kan extension of $f$ along $H$ (when regarded as a $\operatorname{\mathcal{D}}$-valued functor). In fact, we prove a stronger claim: the natural transformation $\overline{\alpha }$ exhibits $\overline{F}$ as a left Kan extension of $f$ along $\overline{H}$ (when regarded as a $\widehat{\operatorname{\mathcal{D}}}$-valued functor). Since $L: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) \rightarrow \overline{\operatorname{\mathcal{C}}}$ is a localization functor, the identity transformation $\operatorname{id}_{ \overline{F} \circ L}$ automatically exhibits $\overline{F}$ as a left Kan extension of $\overline{F} \circ L$ along $L$ (Proposition 7.3.1.18). Invoking the transitivity of Kan extensions (Proposition 7.3.8.24), we are reduced to showing that $\overline{\alpha }$ exhibits the functor $(\overline{F} \circ L): \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }) \rightarrow \widehat{\operatorname{\mathcal{D}}}$ as a left Kan extension of $f$ along the Yoneda embedding $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$. By virtue of Theorem 8.4.3.5, this is a reformulation of our assumption that $\overline{F}$ is $\lambda $-cocontinuous.

Now suppose we are given an arbitrary functor $F': \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\alpha ': f \rightarrow F' \circ H$. Invoking the universal property of Kan extensions, we may assume that $\alpha '$ is obtained from $\alpha $ by postcomposition with some natural transformation $\beta : F \rightarrow F'$ in the $\infty $-category $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$. Using Remark 7.3.1.12, we see that conditions $(1)$ and $(2)$ for the pair $(F', \alpha ')$ can be reformulated as follows:

$(1')$

The natural transformation $\beta $ is an isomorphism: that is, it induces an isomorphism $\beta _{X}: F(X) \rightarrow F'(X)$ for each $X \in \widehat{\operatorname{\mathcal{C}}}$.

$(2')$

The functor $F'$ is $\mathbb {K}$-cocontinuous and the map $\beta _{X}$ is an isomorphism when $X$ belongs to the essential image of $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$.

The implication $(1') \Rightarrow (2')$ follows immediately from the construction (since $F$ is $\mathbb {K}$-cocontinuous). To prove the reverse implication, we observe that condition $(2')$ guarantees that the collection of objects $X$ for which $\beta _{X}$ is an isomorphism contains the essential image of $H$ and is closed under $K$-indexed colimits for $K \in \mathbb {K}$, and therefore coincides with $\widehat{\operatorname{\mathcal{C}}}$. $\square$

Proof of Theorem 9.7.1.4. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $Q = \{ \overline{q}_{j}: K_{j}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\} _{j \in J}$ be a collection of diagrams where each $K_ j$ belongs to $\mathbb {K}$. Fix regular cardinals $\kappa < \lambda $ such that each $K_ j$ is $\kappa $-small, $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, and $\lambda $ has exponential cofinality $\geq \kappa $. We will show that the functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ of Construction 9.7.1.10 exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ relative to $Q$. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $\mathbb {K}$-cocomplete, let $\operatorname{Fun}_{Q}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ be the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ such that each $f \circ \overline{q}_{j}$ is a colimit diagram, and define $\operatorname{Fun}_{Q}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ similarly. It follows from Lemma 9.7.1.11 that the restriction functor

admits a fully faithful left adjoint, given by left Kan extension along $H$. Moreover, its essential image is the full subcategory $\operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont} }_{Q}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by the $\mathbb {K}$-cocontinuous functors from $\widehat{\operatorname{\mathcal{C}}}$ to $\operatorname{\mathcal{D}}$ which carry each $\overline{q}_ j$ to a colimit diagram in $\operatorname{\mathcal{C}}$. In particular, the restriction functor induces an equivalence of $\infty $-categories $\operatorname{Fun}^{ \mathbb {K}-\mathrm{cocont} }_{Q}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{Q}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. $\square$

Proof. Combine Lemma 9.7.1.11 with (the proof of) Theorem 9.7.1.4. $\square$

Proof. Apply Corollary 9.7.1.12 in the special case $F = \operatorname{id}_{ \widehat{\operatorname{\mathcal{C}}} }$. $\square$

Proof. Since $H$ carries each $\overline{q}_ j$ to a colimit diagram in $\widehat{\operatorname{\mathcal{C}}}$, the implication $(1) \Rightarrow (2)$ follows from Variant 7.1.4.13. To prove the converse, it follows from the proof of Theorem 9.7.1.4 that we can enlarge the collection $\mathbb {K}$, and may therefore assume that it consists of all $\lambda $-small simplicial sets where $\lambda $ is a regular cardinal satisfying the hypotheses of Proposition 9.7.1.6. In this case, we can identify $\widehat{\operatorname{\mathcal{C}}}$ with the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ spanned by those functors $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ which carry each $\overline{q}_ j$ to a limit diagram in $\operatorname{\mathcal{S}}_{< \lambda }$. If condition $(2)$ is satisfied, then this subcategory contains all representable functors (Proposition 7.4.1.22), so that $H$ can be identified with the Yoneda embedding $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, which is fully faithful by virtue of Theorem 8.3.3.13. $\square$

Proof. Using the proof of Theorem 9.7.1.4, we may assume that $\widehat{\operatorname{\mathcal{C}}}$ is the $\infty $-category given by Construction 9.7.1.10: that is, the smallest full subcategory of $\overline{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ which contains the essential image of the functor $\overline{H} = L \circ h$ and is closed under $K$-indexed colimits for each $K \in \mathbb {K}$. We first show that $\widehat{\operatorname{\mathcal{C}}}$ is locally $\lambda $-small: that is, the morphism space $M = \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \mathscr {F}, \mathscr {G})$ is essentially $\lambda $-small for every pair of objects $\mathscr {F}, \mathscr {G} \in \widehat{\operatorname{\mathcal{C}}}$. Let us regard the object $\mathscr {G}$ as fixed. Since $\lambda $ has exponential cofinality $\geq \kappa $, the collection of objects $\mathscr {F}$ which satisfy this condition is closed under the formation of $\kappa $-small colimits (see Proposition 7.4.1.22 and Variant 7.4.1.15). We may therefore assume without loss of generality that $\mathscr {F} = L( h_ C )$ is the localization of the functor $h_{C}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ represented by an object $C \in \operatorname{\mathcal{C}}$. In this case, Proposition 8.3.1.3 supplies a homotopy equivalence

so that $M$ is $\lambda $-small as desired.

We now define a transfinite sequence of full subcategories $\{ \widehat{\operatorname{\mathcal{C}}}( \alpha ) \subseteq \widehat{\operatorname{\mathcal{C}}} \} _{ \alpha \leq \kappa }$ as follows:

• Let $\widehat{\operatorname{\mathcal{C}}}(0)$ be the essential image of the functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$.

• If $\beta \leq \kappa $ is a nonzero limit ordinal, we define $\widehat{\operatorname{\mathcal{C}}}(\beta ) = \bigcup _{\alpha < \beta } \widehat{\operatorname{\mathcal{C}}}(\alpha )$.

• If $\beta = \alpha +1$ is a successor ordinal, we let $\widehat{\operatorname{\mathcal{C}}}(\beta )$ be the replete full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by $\widehat{\operatorname{\mathcal{C}}}(\alpha )$ together with the colimits of every diagram $q: K \rightarrow \widehat{\operatorname{\mathcal{C}}}(\alpha )$, where $K \in \mathbb {K}$.

Since each $K \in \mathbb {K}$ is $\kappa $-small, every diagram $q: K \rightarrow \widehat{\operatorname{\mathcal{C}}}(\kappa )$ factors through $\widehat{\operatorname{\mathcal{C}}}(\alpha )$ for some $\alpha < \kappa $, and therefore admits a colimit in $\widehat{\operatorname{\mathcal{C}}}(\alpha +1)$. It follows that $\widehat{\operatorname{\mathcal{C}}}(\kappa )$ is closed under $K$-indexed colimits for each $K \in \mathbb {K}$, and therefore coincides with $\widehat{\operatorname{\mathcal{C}}}$. Consequently, if $\widehat{\operatorname{\mathcal{C}}}$ is not essentially $\lambda $-small, then there is some ordinal $\beta \leq \kappa $ such that $\widehat{\operatorname{\mathcal{C}}}(\beta )$ is not essentially $\lambda $-small. Choose $\beta $ as small as possible. Since $\widehat{\operatorname{\mathcal{C}}}( \beta )$ is locally $\lambda $-small, it follows that the collection of isomorphism classes $\pi _0( \widehat{\operatorname{\mathcal{C}}}(\beta )^{\simeq } )$ is not $\lambda $-small (Proposition 4.9.7.7). For $\beta = 0$, this contradicts our assumption that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small. If $\beta $ is a nonzero limit ordinal, it contradicts our assumption that $\lambda $ is regular (and therefore of cofinality $> \kappa $). We may therefore assume without loss of generality that $\beta = \alpha +1$ is a successor ordinal, and that $\widehat{\operatorname{\mathcal{C}}}( \alpha )$ is essentially $\lambda $-small. In this case, every object of $\widehat{\operatorname{\mathcal{C}}}( \beta )$ either belongs to $\widehat{\operatorname{\mathcal{C}}}( \alpha )$ or can be realized as the colimit of a diagram $q: K \rightarrow \widehat{\operatorname{\mathcal{C}}}( \alpha )$, where $K \in \mathbb {K}$ is a $\kappa $-small simplicial set. Since $\kappa < \lambda $ and $\lambda $ has exponential cofinality $\geq \kappa $, the collection of isomorphism classes of $\kappa $-small simplicial sets $K$ is $\lambda $-small (Proposition 4.9.4.20). Moreover, if the simplicial set $K$ is fixed, then the collection of isomorphism classes of diagrams $K \rightarrow \widehat{\operatorname{\mathcal{C}}}(\alpha )$ is $\lambda $-small. It follows that the collection of isomorphism classes of objects of $\widehat{\operatorname{\mathcal{C}}}(\beta )$ is also $\lambda $-small, which is a contradiction. $\square$

Proof. By virtue of Remark 9.7.1.17, this is a special case of Theorem 9.7.1.4 (together with Corollaries 9.7.1.13 and 9.7.1.14). $\square$