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Edit: In this question (Proof that the category of presheaves on a category $C$ is equivalent to the category of presheaves on its Karoubi envelope) it discusses that presheaves of sets on the Karoubi envelope of a small category is equivalent to presheaves on the original category, and gives a reference.

Edit: In this question (Proof that the category of presheaves on a category $C$ is equivalent to the category of presheaves on its Karoubi envelope) it discusses that presheaves of sets on the Karoubi envelope of a small category is equivalent to presheaves on the original category, and gives a reference.

(Start reading again here if you skipped): Consider the restriction functor $$i^*:Psh\left(\mathbf{Man}\right) \to Psh\left(\mathbf{Open} \right)$$ of presheaves of sets, which has a right adjoint $i_*$. (Explicitly, by the Yoneda lemma, one has that $i_*(F)(M)\cong Hom(i^*y(M),F)$ where $y$ is the Yoneda embedding.). Since $i^*$ also has a left adjoint $i_!,$ $i^*$ is left exact, and since $i$ is full and faithful, so is $i_*.$ So we have that the adjunction $(i^*,i_*)$ exhibits $Psh\left(\mathbf{Open} \right)$ as a left exact localization of $Psh\left(\mathbf{Man}\right)$. Hence, there is a unique Grothendieck topology $J$ on $\mathbf{Man}$ for which $Sh_J\left(\mathbf{Man}\right)\cong Psh\left(\mathbf{Open}\right)$ (more precisely, such that the localization induced by sheafification agrees with the one above). However, since we know that $i^*$ is an equivalence, it implies that this Grothendieck topology must be the trivial one. Notice that we also have an induced adjunction $(i^*,i_*)$ between $\infty$-presheaves. (Here there is no danger of the abuse of notation, since both functors are left exact, so preserves $n$-truncated objects for all $n$, so their restriction to presheaves of sets agree with the ones above). This adjunction is still a left exact localization, and since $Psh_\infty\left(\mathbf{Man}\right)$ is $1$-localic, it must again correspond to a unique Grothendieck topology. This left exact localization factors uniquely as a topological localization (one coming from a Grothendieck topology), followed by a cotopological one (one for which the only monos sent to equivalences are equivalences). Since the $\infty$-category of $\infty$-groupoids is hypercomplete and colimits are computed pointwise in presheaves, $Psh_\infty\left(\mathbf{Man}\right)$ is hypercomplete, so it follows that this localization must be topological. The covering sieves of this topology correspond exactly to those monos $f:S \to y(M)$ such that $i^*(f)$ is an equivalence. However, subobjects of representable objects in $Psh_\infty\left(\mathbf{Man}\right)$ are the same as subobjects in $Psh\left(\mathbf{Man}\right),$ so one sees the resulting class of covering sieves must be the same as for the $1$-categorical case, which we have argued only gave the maximal sieves (the trivial Grothendieck topology). Hence $$i^*:Psh_\infty\left(\mathbf{Man}\right) \to Psh_\infty\left(\mathbf{Open} \right)$$ is an equivalence, and by restriction to $n$-truncated objects, $$i^*:Psh_n\left(\mathbf{Man}\right) \to Psh_n\left(\mathbf{Open} \right)$$ is as well.

(Start reading again here if you skipped): Consider the restriction functor $$i^*:Psh\left(\mathbf{Man}\right) \to Psh\left(\mathbf{Open} \right)$$ of presheaves of sets, which has a right adjoint $i_*$. (Explicitly, by the Yoneda lemma, one has that $i_*(F)(M)\cong Hom(i^*y(M),F)$ where $y$ is the Yoneda embedding.). Since $i^*$ also has a left adjoint $i_!,$ $i^*$ is left exact, and since $i$ is full and faithful, so is $i_*.$ So we have that the adjunction $(i^*,i_*)$ exhibits $Psh\left(\mathbf{Open} \right)$ as a left exact localization of $Psh\left(\mathbf{Man}\right)$. Hence, there is a unique Grothendieck topology $J$ on $\mathbf{Man}$ for which $Sh_J\left(\mathbf{Man}\right)\cong Psh\left(\mathbf{Open}\right)$ (more precisely, such that the localization induced by sheafification agrees with the one above). However, since we know that $i^*$ is an equivalence, it implies that this Grothendieck topology must be the trivial one. Notice that we also have an induced adjunction $(i^*,i_*)$ between $\infty$-presheaves. (Here there is no danger of the abuse of notation, since both functors are left exact, so preserves $n$-truncated objects for all $n$, so their restriction to presheaves of sets agree with the ones above). This adjunction is still a left exact localization, and since $Psh_\infty\left(\mathbf{Man}\right)$ is $1$-localic, it must again correspond to a unique Grothendieck topology. This left exact localization factors uniquely as a topological localization (one coming from a Grothendieck topology), followed by a cotopological one (one for which the only monos sent to equivalences are equivalences). The covering sieves of the topology corresponding to the topological part of the localization correspond exactly to those monos $f:S \to y(M)$ such that $i^*(f)$ is an equivalence. However, subobjects of representable objects in $Psh_\infty\left(\mathbf{Man}\right)$ are the same as subobjects in $Psh\left(\mathbf{Man}\right),$ so one sees the resulting class of covering sieves must be the same as for the $1$-categorical case, which we have argued only gave the maximal sieves (the trivial Grothendieck topology). It follows that the localization must be cotopological. Since the $\infty$-category of $\infty$-groupoids is hypercomplete and colimits are computed pointwise in presheaves, $Psh_\infty\left(\mathbf{Man}\right)$ is hypercomplete, and hence the localization must be trivial. Hence $$i^*:Psh_\infty\left(\mathbf{Man}\right) \to Psh_\infty\left(\mathbf{Open} \right)$$ is an equivalence, and by restriction to $n$-truncated objects, $$i^*:Psh_n\left(\mathbf{Man}\right) \to Psh_n\left(\mathbf{Open} \right)$$ is as well.

(Start reading again here if you skipped): Consider the restriction functor $$i^*:Psh\left(\mathbf{Man}\right) \to Psh\left(\mathbf{Open} \right)$$ of presheaves of sets, which has a right adjoint $i_*$. (Explicitly, by the Yoneda lemma, one has that $i_*(F)(M)\cong Hom(i^*y(M),F)$ where $y$ is the Yoneda embedding.). Since $i^*$ also has a left adjoint $i_!,$ $i^*$ is left exact, and since $i$ is full and faithful, so is $i_*.$ So we have that the adjunction $(i^*,i_*)$ exhibits $Psh\left(\mathbf{Open} \right)$ as a left exact localization of $Psh\left(\mathbf{Man}\right)$. Hence, there is a unique Grothendieck topology $J$ on $\mathbf{Man}$ for which $Sh_J\left(\mathbf{Man}\right)\cong Psh\left(\mathbf{Open}\right)$ (more precisely, such that the localization induced by sheafification agrees with the one above). However, since we know that $i^*$ is an equivalence, it implies that this Grothendieck topology must be the trivial one. Notice that we also have an induced adjunction $(i^*,i_*)$ between $\infty$-presheaves. (Here there is no danger of the abuse of notation, since both functors are left exact, so preserves $n$-truncated objects for all $n$, so their restriction to presheaves of sets agree with the ones above). This adjunction is still a left exact localization, and since $Psh_\infty\left(\mathbf{Man}\right)$ is $1$-localic, it must again correspond to a unique Grothendieck topology. The covering sieves of this topology correspond exactly to those monos $f:S \to y(M)$ such that $i^*(f)$ is an equivalence. However, subobjects of representable objects in $Psh_\infty\left(\mathbf{Man}\right)$ are the same as subobjects in $Psh\left(\mathbf{Man}\right),$ so one sees the resulting class of covering sieves must be the same as for the $1$-categorical case, which we have argued only gave the maximal sieves (the trivial Grothendieck topology). Hence $$i^*:Psh_\infty\left(\mathbf{Man}\right) \to Psh_\infty\left(\mathbf{Open} \right)$$ is an equivalence, and by restriction to $n$-truncated objects, $$i^*:Psh_n\left(\mathbf{Man}\right) \to Psh_n\left(\mathbf{Open} \right)$$ is as well.

(Start reading again here if you skipped): Consider the restriction functor $$i^*:Psh\left(\mathbf{Man}\right) \to Psh\left(\mathbf{Open} \right)$$ of presheaves of sets, which has a right adjoint $i_*$. (Explicitly, by the Yoneda lemma, one has that $i_*(F)(M)\cong Hom(i^*y(M),F)$ where $y$ is the Yoneda embedding.). Since $i^*$ also has a left adjoint $i_!,$ $i^*$ is left exact, and since $i$ is full and faithful, so is $i_*.$ So we have that the adjunction $(i^*,i_*)$ exhibits $Psh\left(\mathbf{Open} \right)$ as a left exact localization of $Psh\left(\mathbf{Man}\right)$. Hence, there is a unique Grothendieck topology $J$ on $\mathbf{Man}$ for which $Sh_J\left(\mathbf{Man}\right)\cong Psh\left(\mathbf{Open}\right)$ (more precisely, such that the localization induced by sheafification agrees with the one above). However, since we know that $i^*$ is an equivalence, it implies that this Grothendieck topology must be the trivial one. Notice that we also have an induced adjunction $(i^*,i_*)$ between $\infty$-presheaves. (Here there is no danger of the abuse of notation, since both functors are left exact, so preserves $n$-truncated objects for all $n$, so their restriction to presheaves of sets agree with the ones above). This adjunction is still a left exact localization, and since $Psh_\infty\left(\mathbf{Man}\right)$ is $1$-localic, it must again correspond to a unique Grothendieck topology. This left exact localization factors uniquely as a topological localization (one coming from a Grothendieck topology), followed by a cotopological one (one for which the only monos sent to equivalences are equivalences). Since the $\infty$-category of $\infty$-groupoids is hypercomplete and colimits are computed pointwise in presheaves, $Psh_\infty\left(\mathbf{Man}\right)$ is hypercomplete, so it follows that this localization must be topological. The covering sieves of this topology correspond exactly to those monos $f:S \to y(M)$ such that $i^*(f)$ is an equivalence. However, subobjects of representable objects in $Psh_\infty\left(\mathbf{Man}\right)$ are the same as subobjects in $Psh\left(\mathbf{Man}\right),$ so one sees the resulting class of covering sieves must be the same as for the $1$-categorical case, which we have argued only gave the maximal sieves (the trivial Grothendieck topology). Hence $$i^*:Psh_\infty\left(\mathbf{Man}\right) \to Psh_\infty\left(\mathbf{Open} \right)$$ is an equivalence, and by restriction to $n$-truncated objects, $$i^*:Psh_n\left(\mathbf{Man}\right) \to Psh_n\left(\mathbf{Open} \right)$$ is as well.