# Other interesting notions when we change topology on $\text{Sch}/S$

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 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 algebraic space over $$S$$ as a generalization of a scheme over $$S$$ in this sense (please correct me if I am mistaken). An algebraic 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, 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, 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 "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)?

• Anatoly Preygel at tolypreygel.com/notes/note_stacks.pdf, and many others at other places (please feel free to add links if you can recall exact references), says that "Algebriac spaces possess better formal properties than schemes, but are still geometric enough to allow us to import many definitions and properties from scheme theory". I wish the same (or better) properties hold for the geometric objects over a scheme $S$ in above question. – Praphulla Koushik Sep 7 '19 at 15:01