Let $\text{Sch}$ be the category of schemes. Let $S$ be an object of $\text{Sch}$. Consider the category $\text{Sch}/S$.


Some interesting  topologies on $\text{Sch}/S$ are Zariski, fpqc, étale, fppf...

Given an object $X$  of $\text{Sch}/S$, consider the functor $h_X:(\text{Sch}/S)^{op}\rightarrow \text{Set}$ associated to $X$. It is easy to see that this is a sheaf if one gives Zariski topology to $\text{Sch}/S$. Then, Angelo Vistoli's [notes][1] on descent theory says that (Theorem $2.55$, page $36$), $h_X$ is a sheaf ("not easy at all") if I consider the fpqc or etale or fppf topology on $\text{Sch}/S$.

Given a scheme $X$ over $S$, one can see this  as a functor $h_X:(\text{Sch}/S)^{op}\rightarrow \text{Set}$ is a sheaf (with Zariski or fpqc or etale or fppf). I see/think algebriac space over $S$ as a generalization of a scheme over $S$ in this sense (please correct me if I am mistaken). An algebriac space over $S$ is a functor $\mathcal{F}:(\text{Sch}/S)^{op}\rightarrow \text{Set}$ that is a sheaf (and some other extra conditions). 

I did not mention the topology on $(\text{Sch}/S)^{op}$ because different references use different topologies. For [Algebraic Spaces and Stacks][2],  an algebraic space is a functor  $\mathcal{F}:(\text{Sch}/S)^{op}\rightarrow \text{Set}$ that is a **sheaf with respect to etale topology** (and some other extra conditions). For [Stacks Project][3], an algebraic space is a functor  $\mathcal{F}:(\text{Sch}/S)^{op}\rightarrow \text{Set}$ that is a **sheaf with respect to fppf topology** (and some other extra conditions). There may be other equivalent notions of algebraic spaces using some other topology on $\text{Sch}/S$ (please point me to references if there are any). Stacks project [says][4] "Instead of working with sheaves over the big fppf site $(\text{Sch}/S)_{fppf}$ we could work with sheaves over the big étale site $(\text{Sch}/S)_{etale}$."   


**Question** : Are there other (interesting) geometric objects over a scheme $S$ introduced using this approach, that is $\mathcal{F}:(\text{Sch}/S)^{op}\rightarrow \text{Set}$ is a sheaf with respect to some other topology (along with some other conditions)?


  [1]: http://homepage.sns.it/vistoli/descent.pdf
  [2]: https://bookstore.ams.org/coll-62/
  [3]: https://stacks.math.columbia.edu/tag/025Y
  [4]: https://stacks.math.columbia.edu/tag/076L