# Epimorphisms and quotients in Sch versus $\mathrm{Sh}(\mathrm{Ring}^{\mathrm{op}},\mathrm{Zar})$

$$\DeclareMathOperator\Ring{Ring}\DeclareMathOperator\Aff{Aff}\DeclareMathOperator\op{op}\DeclareMathOperator\Sch{Sch}\DeclareMathOperator\Zar{Zar}$$The category of schemes sits, fully faithfully, in various categories of sheaves on $$\Ring^{\op} =\Aff$$ (for the Zariski topology, etale topology, etc.) Are there epimorphisms in $$\Sch$$ which are not epimorphisms in the larger category of sheaves on $$\Ring^{\op}$$ for a given topology? Are there useful criterions which tell me that a epimorphism in $$\Sch$$ is also a sheaf-epimorphism of the associated functor of points?

More specifically, are there categorical quotients of a scheme $$X$$ by an internal equivalence relation $$R$$ on $$X$$ such that the quotient map $$p:X\to X/R$$ is a quotient in $$\Sch$$ but not in for example the big Zariski topos?

Note that the categorical quotient $$p:X\to X/R$$ in $$\Zar$$ always exist (because it is a topos). If $$X/R$$ happens to be a scheme, then $$p$$ is also a quotient in the category $$\Sch$$. But if $$X/R$$ is not a scheme, then there might still be a different map $$X\to Y$$ which satisfies the weaker universal property in $$\Sch$$ but not in $$\Zar$$. Does this happen and do such quotients show up in practice?

• Certainly this happens in practice. Take a finite group acting freely on a quasi-projective variety. The quotient exists as a scheme and is the same as the étale sheaf quotient, but not the Zariski one. Nov 4, 2022 at 16:47