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Suppose that $\mathscr{C}$ is a (possibly higher) category and $J$ is a Grothendieck topology which is not subcanonical. Denote the composite

$$\mathscr{C} \hookrightarrow \mathbf{Set}^{\mathscr{C}^{op}} \stackrel{a}{\longrightarrow} \mathbf{Sh}\left(\mathscr{C}\right)$$

by $\theta$, where $a$ is the sheafification functor. Let $W$ denote the class of morphisms in $\mathscr{C}$ which become isomorphisms after applying $\theta.$ Is the essential image of $\theta$ equivalent to the category $\mathscr{C}\left[W^{-1}\right]$ obtained from $\mathscr{C}$ by formally inverting the morphisms $W$?

Question: There are certainly examples where this is true, which I provide below. Is this always the case? If not, what further conditions are needed for it to be true?

Here are two examples where this is true:

Example 1: Let $\mathscr{C}=\mathbf{Haus}$ be the category of Hausdorff topological spaces. Let $J$ be the Grothendieck topology whose covering families for a space $X$ consist of subspace inclusions $$\left(V_\alpha \hookrightarrow X\right)_{\alpha \in A}$$ such that for each compact subspace $K$ of $X$ there exists a finite subset $A(K) \subset A$ such that $$\left(V_\alpha\cap K \hookrightarrow K\right)_{\alpha \in A(K)}$$ can be refined by an open cover of $K$. The essential image of $\theta$ in this case is equivalent to the category of compactly generated Hausdorff spaces, and $W$ consists of all those maps $$X \to Y$$ such that for each compact Hausdorff space $C$, the induced map $$Hom(C,Y) \to Hom(C,X)$$ is an isomorphism. Formally inverting such morphisms yields a category equivalent to compactly generated Hausdorff spaces.

Example 2: Let $\mathscr{C}$ be the $\left(2,1\right)$-category of Lie groupoids and smooth functors and natural transormations. Let $i$ denote the full and faithful inclusion $$i:\mathit{Mfd} \hookrightarrow \mathit{LieGpd}.$$ The adjoint pair $i^* \dashv i_*$ exhibits $\mathbf{Fun}\left(\mathit{Mfd}^{op},\mathbf{Gpd}\right)$ as a left-exact localization of $\mathbf{Fun}\left(\mathit{LieGpd}^{op},\mathbf{Gpd}\right).$ Moreover, stacks of groupoids $$\mathbf{St}\left(\mathit{Mfd}\right)$$ is a left-exact localization of $\mathbf{Fun}\left(\mathit{Mfd}^{op},\mathbf{Gpd}\right),$ so by composition is also a left-exact localization of $\mathbf{Fun}\left(\mathit{LieGpd}^{op},\mathbf{Gpd}\right).$ Therefore, there exists a unique Grothendieck topology $J$ on $\mathit{LieGpd}$ such that $$\mathbf{St}\left(\mathit{LieGpd},J\right) \simeq \mathbf{St}\left(\mathit{Mfd}\right).$$ This Grothendieck topology is not subcanonical because the assignment $$M \mapsto Hom\left(M,\mathcal{G}_1\right) \rightrightarrows Hom\left(M,\mathcal{G}_0\right)$$ does not satisfy descent unless the Lie groupoid $\mathcal{G}$ is really a manifold. $\theta$ in this case is the functor assigning a Lie groupoid $\mathcal{G}$ its stack of principal bundles. The essential image is the $\left(2,1\right)$-category of differentiable stacks, and $W$ consists of Morita equivalences. It is well known that the bicategory of fractions of Lie groupoids with respect to Morita equivalences is equivalent to differentiable stacks.

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    $\begingroup$ If the underlying category of the site is a groupoid then this always fails. (For example, take a non-trivial group considered as a groupoid; then the Grothendieck topology generated by the empty sieve is not subcanonical.) $\endgroup$ – Zhen Lin Jul 17 '14 at 14:03
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I don't know whether this includes your two examples, but I believe one situation where this is true is if the topology $J$ is supercanonical, i.e. includes the canonical topology. For example, if $C$ is itself a topos, then supercanonical topologies on $C$ are the same as subtoposes of $C$, which can be obtained from $C$ by inverting the local isomorphisms.

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