Kerodon

Proof of Proposition 8.7.3.15. Let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibit $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Proposition 8.7.3.11 guarantees that the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is locally $\lambda $-small. It will therefore suffice to show that the set of isomorphism classes $\pi _0( \widehat{\operatorname{\mathcal{C}}}^{\simeq } )$ is $\lambda $-small (Proposition 4.7.8.7).

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

• We define $\widehat{\operatorname{\mathcal{C}}}_{0}$ to be the essential image of the functor $h$.

• If $\beta = \alpha +1$ is a successor ordinal, we define $\widehat{\operatorname{\mathcal{C}}}_{\beta }$ to be the smallest replete full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ which contains $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ and the colimit of every diagram $K \rightarrow \widehat{\operatorname{\mathcal{C}}}_{\alpha |}$ for $K \in \mathbb {K}$.

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

By assumption, every $K \in \mathbb {K}$ is $\kappa $-small. Consequently, every diagram $K \rightarrow \widehat{\operatorname{\mathcal{C}}}_{\kappa }$ factors through $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ for some $\alpha < \kappa $, and therefore has a colimit in $\widehat{\operatorname{\mathcal{C}}}_{\alpha +1} \subseteq \widehat{\operatorname{\mathcal{C}}}_{\kappa }$. It follows that $\widehat{\operatorname{\mathcal{C}}}_{\kappa }$ contains the essential image of $h$ and is closed under $K$-indexed colimits for each $K \in \mathbb {K}$, and therefore coincides with $\widehat{\operatorname{\mathcal{C}}}$. We are therefore reduced to showing that the set of isomorphism classes $\pi _0( \widehat{\operatorname{\mathcal{C}}}_{\kappa }^{\simeq } )$ is $\lambda $-small. Suppose otherwise. Then there is some smallest ordinal $\beta \leq \kappa $ such that $\pi _0( \widehat{\operatorname{\mathcal{C}}}_{\beta }^{\simeq } )$ is not $\lambda $-small. Our assumption that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small guarantees that $\beta > 0$. Since $\lambda $ has cofinality $> \kappa $, $\beta $ cannot be a limit ordinal. We must therefore have $\beta = \alpha +1$, for some ordinal $\alpha < \kappa $. In this case, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ is essentially $\mu $-small.

Note that every object of $\widehat{\operatorname{\mathcal{C}}}_{\alpha +1}$ either belongs to $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ or can be realized as the colimit of a diagram $F: K \rightarrow \widehat{\operatorname{\mathcal{C}}}_{\alpha }$, for some $K \in \mathbb {K}$. Note that we may assume that $\mathbb {K}$ is $\lambda $-small (Remark 8.7.3.16). Moreover, if $K \in \mathbb {K}$ is fixed, our assumption that $\lambda $ has exponential cofinality $\geq \kappa $ guarantees that the set of isomorphism classes $\pi _0( \operatorname{Fun}( K, \widehat{\operatorname{\mathcal{C}}}_{\alpha })^{\simeq } )$ is $\lambda $-small (Corollary 4.7.8.8). The regularity of $\lambda $ now guarantees that $\pi _0( \widehat{\operatorname{\mathcal{C}}}_{\alpha +1}^{\simeq } )$ is $\lambda $-small, contradicting our choice of $\beta $. $\square$