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Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of fibrations. Suppose $$\left(f_\alpha:C_\alpha \to C\right)_\alpha$$ is a cover in $C$, then if $f$ denotes the map $$f:\coprod\limits_\alpha C_\alpha \to C$$ we have that $C = \operatorname{colim} \check{C}(f)$, i.e. $C$ is the colimit of the Čech nerve of this cover in $C$. Because all the covering families consist of fibrations, all the pullbacks involved in constructing this Čech nerve are homotopy pullbacks, so its image of $\check{C}(f)$ in the simplicial localization $\mathcal{C}\left[W^{-1}\right]_\infty$ (i.e. the associated $\left(\infty,1\right)$-category) is the Čech nerve of the image of $f$ in $\mathcal{C}\left[W^{-1}\right]_\infty$ (provided we take the untruncated simplicial diagram for $\check{C}(f)$).

Question: Under what conditions on $J$ and $\left(\mathcal{C},W,F\right)$ will I have that $C = \operatorname{hocolim} \check{C}(f)$ (the homotopy colimit of the entire simplicial diagram) in the simplicial localization $C\left[W^{-1}\right]_\infty$? In other words, under what conditions will the effective epimorphism $$f:\coprod\limits_\alpha C_\alpha \to C$$ still be an effective epimorphism in the associated $\left(\infty,1\right)$-category?

Remark: I imagine that a necessary condition is that weak equivalences can be detected locally, in the sense that if $g:D \to C$ is such that for some cover $$\left(f_\alpha:C_\alpha \to C\right)_\alpha,$$ each induced map $$f_\alpha^*D \to C_\alpha$$ is a weak equivalence, then so is $f$. But is this enough?

Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of fibrations. Suppose $$\left(f_\alpha:C_\alpha \to C\right)_\alpha$$ is a cover in $C$, then if $f$ denotes the map $$f:\coprod\limits_\alpha C_\alpha \to C$$ we have that $C = \operatorname{colim} \check{C}(f)$, i.e. $C$ is the colimit of the Čech nerve of this cover in $C$. Because all the covering families consist of fibrations, all the pullbacks involved in constructing this Čech nerve are homotopy pullbacks, so its image of $\check{C}(f)$ in the simplicial localization $\mathcal{C}\left[W^{-1}\right]_\infty$ (i.e. the associated $\left(\infty,1\right)$-category) is the Čech nerve of the image of $f$ in $\mathcal{C}\left[W^{-1}\right]_\infty$ (provided we take the untruncated simplicial diagram for $\check{C}(f)$).

Question: Under what conditions on $J$ and $\left(\mathcal{C},W,F\right)$ will I have that $C = \operatorname{hocolim} \check{C}(f)$ (the homotopy colimit of the entire simplicial diagram) in the simplicial localization $C\left[W^{-1}\right]_\infty$? In other words, under what conditions will the effective epimorphism $$f:\coprod\limits_\alpha C_\alpha \to C$$ still be an effective epimorphism in the associated $\left(\infty,1\right)$-category?

Remark: I imagine that a necessary condition is that weak equivalences can be detected locally, in the sense that if $g:D \to C$ is such that for some cover $$\left(f_\alpha:C_\alpha \to C\right)_\alpha,$$ each induced map $$f_\alpha^*D \to C_\alpha$$ is a weak equivalence, then so is $f$. But is this enough?

Update: I can prove that in the situation in the Remark, that the Yoneda embedding into sheaves is at least faithful and conservative.

Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of fibrations. Suppose $$\left(f_\alpha:C_\alpha \to C\right)_\alpha$$ is a cover in $C$, then if $f$ denotes the map $$f:\coprod\limits_\alpha C_\alpha \to C$$ we have that $C = \operatorname{colim} \check{C}(f)$, i.e. $C$ is the colimit of the Čech nerve of this cover in $C$. Because all the covering families consist of fibrations, all the pullbacks involved in constructing this Čech nerve are homotopy pullbacks, so its image of $\check{C}(f)$ in the simplicial localization $\mathcal{C}\left[W^{-1}\right]_\infty$ (i.e. the associated $\left(\infty,1\right)$-category) is the Čech nerve of the image of $f$ in $\mathcal{C}\left[W^{-1}\right]_\infty$ (provided we take the untruncated simplicial diagram for $\check{C}(f)$).

Question: Under what conditions on $J$ and $\left(\mathcal{C},W,F\right)$ will I have that $C = \operatorname{hocolim} \check{C}(f)$ (the homotopy colimit of the entire simplicial diagram) in the simplicial localization $C\left[W^{-1}\right]_\infty$? In other words, under what conditions will the effective epimorphism $$f:\coprod\limits_\alpha C_\alpha \to C$$ still be an effective epimorphism in the associated $\left(\infty,1\right)$-category?

Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of fibrations. Suppose $$\left(f_\alpha:C_\alpha \to C\right)_\alpha$$ is a cover in $C$, then if $f$ denotes the map $$f:\coprod\limits_\alpha C_\alpha \to C$$ we have that $C = \operatorname{colim} \check{C}(f)$, i.e. $C$ is the colimit of the Čech nerve of this cover in $C$. Because all the covering families consist of fibrations, all the pullbacks involved in constructing this Čech nerve are homotopy pullbacks, so its image of $\check{C}(f)$ in the simplicial localization $\mathcal{C}\left[W^{-1}\right]_\infty$ (i.e. the associated $\left(\infty,1\right)$-category) is the Čech nerve of the image of $f$ in $\mathcal{C}\left[W^{-1}\right]_\infty$ (provided we take the untruncated simplicial diagram for $\check{C}(f)$).

Question: Under what conditions on $J$ and $\left(\mathcal{C},W,F\right)$ will I have that $C = \operatorname{hocolim} \check{C}(f)$ (the homotopy colimit of the entire simplicial diagram) in the simplicial localization $C\left[W^{-1}\right]_\infty$? In other words, under what conditions will the effective epimorphism $$f:\coprod\limits_\alpha C_\alpha \to C$$ still be an effective epimorphism in the associated $\left(\infty,1\right)$-category?

Remark: I imagine that a necessary condition is that weak equivalences can be detected locally, in the sense that if $g:D \to C$ is such that for some cover $$\left(f_\alpha:C_\alpha \to C\right)_\alpha,$$ each induced map $$f_\alpha^*D \to C_\alpha$$ is a weak equivalence, then so is $f$. But is this enough?

Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of fibrations. Suppose $$\left(f_\alpha:C_\alpha \to C\right)_\alpha$$ is a cover in $C$, then if $f$ denotes the map $$f:\coprod\limits_\alpha C_\alpha \to C$$ we have that $C = \operatorname{colim} \check{C}(f)$, i.e. $C$ is the colimit of the $\check{\mbox{C}}$ech nerve of this cover in $C$. Because all the covering families consist of fibrations, all the pullbacks involved in constructing this $\check{\mbox{C}}$ech nerve are homotopy pullbacks, so its image of $\check{C}(f)$ in the simplicial localization $\mathcal{C}\left[W^{-1}\right]_\infty$ (i.e. the associated $\left(\infty,1\right)$-category) is the $\check{\mbox{C}}$ech nerve of the image of $f$ in $\mathcal{C}\left[W^{-1}\right]_\infty$ (provided we take the untruncated simplicial diagram for $\check{C}(f)$).

Question: Under what conditions on $J$ and $\left(\mathcal{C},W,F\right)$ will I have that $C = \operatorname{hocolim} \check{C}(f)$ (the homotopy colimit of the entire simplicial diagram) in the simplicial localization $C\left[W^{-1}\right]_\infty$? In other words, under what conditions will the effective epimorphism $$f:\coprod\limits_\alpha C_\alpha \to C$$ still be an effective epimorphism in the associated $\left(\infty,1\right)$-category?

Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of fibrations. Suppose $$\left(f_\alpha:C_\alpha \to C\right)_\alpha$$ is a cover in $C$, then if $f$ denotes the map $$f:\coprod\limits_\alpha C_\alpha \to C$$ we have that $C = \operatorname{colim} \check{C}(f)$, i.e. $C$ is the colimit of the Čech nerve of this cover in $C$. Because all the covering families consist of fibrations, all the pullbacks involved in constructing this Čech nerve are homotopy pullbacks, so its image of $\check{C}(f)$ in the simplicial localization $\mathcal{C}\left[W^{-1}\right]_\infty$ (i.e. the associated $\left(\infty,1\right)$-category) is the Čech nerve of the image of $f$ in $\mathcal{C}\left[W^{-1}\right]_\infty$ (provided we take the untruncated simplicial diagram for $\check{C}(f)$).

Question: Under what conditions on $J$ and $\left(\mathcal{C},W,F\right)$ will I have that $C = \operatorname{hocolim} \check{C}(f)$ (the homotopy colimit of the entire simplicial diagram) in the simplicial localization $C\left[W^{-1}\right]_\infty$? In other words, under what conditions will the effective epimorphism $$f:\coprod\limits_\alpha C_\alpha \to C$$ still be an effective epimorphism in the associated $\left(\infty,1\right)$-category?