Kerodon

Warning 7.4.3.9 (Fake Colimits). Let $\lambda $ be an uncountable cardinal and suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ indexed by a simplicial set $\operatorname{\mathcal{C}}$. It follows from Variant 7.4.1.4 that if $\mathscr {F}$ admits a limit in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$, then that limit is independent of $\lambda $: that is, it is automatically preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \mu }$ for every cardinal $\lambda \geq \kappa $. It follows from Corollary 7.4.3.8 that the same result holds for colimits, provided that the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small for $\kappa = \mathrm{cf}(\lambda )$. Beware that this cardinality assumption cannot be omitted: if the simplicial set $\operatorname{\mathcal{C}}$ is too large, then it is possible for a $X$ to be a colimit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$, which does not remain a colimit in $\operatorname{\mathcal{S}}^{< \mu }$ for $\mu \gg \lambda $ (see Exercise 7.1.2.12). In this case, we will generally reserve the notation $\varinjlim (\mathscr {F} )$ for the “true” colimit of $\mathscr {F}$ (computed in the $\infty $-category $\operatorname{\mathcal{S}}^{< \mu }$ for $\mu $ sufficiently large), characterized by the requirement that it is weakly homotopy equivalent to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$.