Let's fix a field $k$. First, consider $Aff\_k$ to be the category of affine finite type $k$ schemes. On this category, one can define the etale topology and thus consider the site $Aff\_k^{et}$, then define the category of sheaves:


$Sh(Aff^{et}\_k)$


Similarly, it seems to me that one could also take the category $ \text{AlgSp}\_k$ of finite type algebraic spaces and define the etale topology and get a site $\text{AlgSp}\_k^{et}$, and then also consider the category of sheaves:


$Sh(\text{AlgSp}\_k^{et})$


**My question**: can I expect an equivalence of topoi between $Sh(Aff^{et}\_k)$ and $Sh(\text{AlgSp}\_k^{et})$?


Secretly, when I write "sheaves", I mean sheaves of spaces in the $\infty$-categorical sense, which are not necessarily hyper-complete. But I really don't have so much intuition for this, so even if we take sheaves of sets I'd be interested in the answer. Thanks!



Etale sheaves on algebraic spaces vs. Etale sheaves on affines
Yes, the two $\infty$-topoi are equivalent. Let $u: \mathrm{Aff} \to \mathrm{AlgSp}$ be the inclusion. Then $u$ preserves étale covering families and pullbacks, hence commutes with the formation of the Čech nerve of such a covering. This implies that $u^\*$ preserves sheaves. The functor $u$ is also cocontinuous: if $X\in\mathrm{Aff}$ and $R\subset \mathrm{AlgSp}\_{/X}$ is a covering sieve, then $u^\*(R)\subset \mathrm{Aff}\_{/X}$ is still a covering sieve. This implies that the right Kan extension functor $u\_\*$ preserves sheaves also. We therefore have an adjunction $(u^\*,u\_\*)$ between the $\infty$-categories of sheaves, where $u\_\*$ is fully faithful. By looking at the triangle identities, it remains to show that $u^\*$ is conservative. To see this we can look at the two inclusions $\mathrm{Aff} \subset \mathrm{SepSch} \subset \mathrm{AlgSp}$. Each inclusion $\mathcal{C}\subset\mathcal{D}$ has the property that every object of $\mathcal D$ admits a covering by objects of $\mathcal C$ such that all the fiber products occurring in the Čech nerve are still in $\mathcal C$ (because the diagonal of a separated scheme is affine and the diagonal of an algebraic space is representable by separated schemes); this immediately implies that restriction from $\mathcal D$ to $\mathcal C$ detects equivalences between sheaves.


**Remark.** We also have an equivalence between the $\infty$-topoi of Nisnevich sheaves. The cocontinuity of $u$ and the conservativity of $u^\*$ in that case are less obvious; they follow from a result of Gruson and Raynaud (see Prop. 3.7.5.3 in Lurie's Spectral Algebraic Geometry).


