The hypercomplete derived-localization model on the actual general site

PDF p. 6: Proposition 2.6 proof

Assume the Axiom of Choice and the fixed universe and site-size bounds of the derived-sections suppliers. Fix the completed qcqs prime-field cdh site ${\mathcal{C}}_F$. Let $a$ be its abelian sheafification and $i$ its inclusion of abelian sheaves into presheaves. For a presheaf complex $C$ define $L_{\mathrm{c}\mathrm{d}\mathrm{h}}^{F}C=RiaC$: choose a K-injective sheaf complex $aC\to I$ and evaluate $I$ as a presheaf. This is localization at maps inducing isomorphisms on all associated cohomology sheaves; a complex is local precisely when its unit to this localization is an equivalence. By the exact linear input, hypercover-descent complexes are local in this same model. This defines the general-site version without modifying the completed narrower finite-type definition.

In comparisons using both reflectors, write $L_{\mathrm{c}\mathrm{d}\mathrm{h}}^{h,F}=L_{\mathrm{c}\mathrm{d}\mathrm{h}}^F=Ria$ for the hypercomplete reflector constructed in this item. Write $L_{\mathrm{c}\mathrm{d}\mathrm{h}}^{o,F}$ for ordinary covering-family/Čech sheafification of $D(\mathbb{Z})$-valued presheaves, as in the source comparisons. These notations do not identify the two reflectors on arbitrary unbounded inputs. The hypercover criterion used here concerns $L_{\mathrm{c}\mathrm{d}\mathrm{h}}^{h,F}$; the particular ordinary-locality consequences used by consumers are separate arguments.