ScalingStacks

Proof. Given a Waldhausen category ${\mathcal{C}}$, let ${\mathcal{P}}({\mathcal{C}})$ denote here the pointed simplicial presheaves on ${\mathcal{C}}$ with the projective model structure (i.e., weak equivalences and fibrations are determined pointwise). We can successively localize ${\mathcal{P}}({\mathcal{C}})$ to produce a category of presheaves which are pointwise Kan complexes, preserve weak equivalences, and take homotopy cocartesian squares in ${\mathcal{C}}$ to homotopy pullback squares in ${\mathcal{P}}({\mathcal{C}})$; denote this category by ${\mathcal{P}}_{\ex}({\mathcal{C}})$ [19, 4.10]. Let ${\mathcal{M}}({\mathcal{C}})$ denote the full subcategory of the localized category consisting of the objects which are cofibrant and weakly equivalent to representable presheaves; this can be regarded as a Waldhausen category, inheriting structure from the model structure on ${\mathcal{P}}_{\ex}({\mathcal{C}})$. The Yoneda embedding induces a DK-equivalence ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{M}}({\mathcal{C}})$ [19, 4.11], and a weakly exact functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}^{\prime}$ induces a left Quillen functor ${\mathcal{P}}_{\ex}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{P}}_{\ex}({\mathcal{C}}^{\prime})$ by left Kan extension, and hence an exact functor ${\mathcal{M}}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{M}}({\mathcal{C}}^{\prime})$ by restriction. ∎