Etale sheaves on algebraic spaces vs. Etale sheaves on affines

Question

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!

Answer 1

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).