Relative cocompletion as presheaves | Higher Topos Theory notes

Remark 6.180.

Assume for simplicity that $$C$$ is small. If $$K'$$ consists of all small categories and $$K$$ of none, $\Pp^{\all}(C) = \PSh(C)$ is simply the presheaf category. For arbitrary $$K$$ (but still $K'=\mathrm{all}$ ), we may identify $\Pp^{\all}_K(C)$ with the localization of $\PSh(C)$ at the class of maps

\{\alpha_X\colon \colim_{i \in I} y(X_i) \to y(\colim_{i \in I} X_i) \mid I \in K, X \colon I \to C\}:

for every cocomplete category $$D$$ , a colimit-preserving functor $\PSh(C)\to D$ is determined by its restriction to $$C$$ , and it factors through the localization if and only if it inverts each $\alpha_X$ , which is exactly the condition that the restriction preserve $$K$$ -indexed colimits.

It follows from Yoneda that we may identify $\Pp^{\all}_{K}(C)$ with the full subcategory of $\PSh(C)$ consisting of those presheaves $F\colon C\catop \to \An$ which send $$K$$ -indexed colimits in $$C$$ to limits in $\An$ .