Kerodon

Remark 7.4.5.4. Let $\operatorname{\mathcal{C}}$ be a small simplicial set, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram, and let $\varinjlim (\mathscr {F} )$ denote its colimit (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$). Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is a Kan complex. Then $\varinjlim (\mathscr {F} )$ is also a Kan complex, which can be regarded as a limit of $\mathscr {F}$ in the subcategory $\operatorname{\mathcal{S}}\subset \operatorname{\mathcal{QC}}$. In particular, the inclusion functor $\operatorname{\mathcal{S}}\hookrightarrow \operatorname{\mathcal{QC}}$ preserves small colimits. This is a special case of Variant 7.1.4.25, since $\operatorname{\mathcal{S}}$ is a coreflective subcategory of $\operatorname{\mathcal{QC}}$ (Example 6.2.2.6). However, it can also be deduced directly from Corollary 7.4.5.2: the assumption that each $\mathscr {F}(C)$ is a Kan complex guarantees that the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration (Example 5.6.2.9). In particular, every edge of the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is $U$-cocartesian (Example 5.1.1.3), so that the localization $(\int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1}]$ is automatically a Kan complex (Proposition 6.3.1.20).

Remark 7.4.5.7. In the situation of Definition 7.4.5.6, suppose that $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is a covariant refraction diagram. Then, for every vertex $C \in \operatorname{\mathcal{C}}$, the restriction $\mathrm{Rf}|_{ \operatorname{\mathcal{E}}_{C} }: \operatorname{\mathcal{E}}_{C} \rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is given by covariant transport along the unique edge $e_{C}: C \rightarrow {\bf 1}$ of $\operatorname{\mathcal{C}}^{\triangleright }$, in the sense of Definition 5.2.2.4.

Remark 7.4.5.10. Suppose we are given a pullback diagram of simplicial sets

where $U$ and $\overline{U}$ are cocartesian fibrations. Let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ and let $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ be a covariant refraction diagram. For every $U$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$, the image $\mathrm{Rf}(e)$ is an isomorphism in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$. To prove this, we observe that there is a morphism $\Delta ^1 \times \Delta ^1 \rightarrow \overline{\operatorname{\mathcal{E}}}$ as indicated in the diagram

where the horizontal maps are $\overline{U}$-cocartesian. Applying Proposition 5.1.4.13, we deduce that $\mathrm{Rf}(e)$ is an $\overline{U}$-cocartesian edge of $\overline{\operatorname{\mathcal{E}}}$, and therefore an isomorphism in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ (Proposition 5.1.4.12).

Remark 7.4.5.12. In the statement of Theorem 7.4.5.11, the covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ and the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}^{< \lambda }$ are only well-defined up to isomorphism (as objects of the $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{E}}, \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} })$ and $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{QC}}^{< \lambda } )$, respectively). However, the hypothesis and conclusion of the theorem depend only on their isomorphism classes (see Exercise 6.3.1.11 and Corollary 7.1.3.14).

Remark 7.4.5.13. In the statement of Theorem 7.4.5.11, condition $(\ast )$ does not depend on the cardinal $\lambda $. Consequently, if condition $(\ast )$ is satisfied, then $\overline{\mathscr {F}}$ is a also colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \mu }$ for every cardinal $\mu \geq \lambda $.

Remark 7.4.5.17. Let $\lambda $ be an uncountable cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is essentially $\lambda $-small. Let $\kappa = \mathrm{cf}(\lambda )$ be the cofinality of $\lambda $. If $\operatorname{\mathcal{C}}$ is $\kappa $-small, then the simplicial set $\operatorname{\mathcal{E}}$ is essentially $\lambda $-small (Proposition 4.7.9.10), so any localization of $\operatorname{\mathcal{E}}$ is also essentially $\lambda $-small (Variant 6.3.2.6). It follows that any cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ satisfying the requirements of Proposition 7.4.5.16 is also essentially $\lambda $-small.