When can a scheme be recovered from its descent groupoid?

Question

Suppose that $ Y $ is a scheme and $ f\colon X\to Y $ a covering of $ Y $ in some Grothendieck topology on the category of schemes (i.e. if $\{ U_i\to Y\}$ is a covering in the topological sense, then $ X = \coprod U_i$). Then we may consider the descent groupoid $\Gamma_0 = X$ and $ \Gamma_1 = X \times_{Y} X $ with source and target maps $ s,t\colon\Gamma_1 \rightrightarrows \Gamma_0$ given by the two projections (so for coverings in the topological sense, $ \Gamma_1 = \coprod U_i\cap U_j$).

It is natural to expect from a covering that $ Y $ can be recovered from $\Gamma_0$ by gluing along the intersections, i.e. $ Y =\mathrm{Coeq}( \Gamma_1 \rightrightarrows \Gamma_0 ) $. Equivalently, we would like the pullback square $$ \require{AMScd} \begin{CD} X\times_{Y} X @>{t}>> X\\ @V{s}VV @VV{f}V \\ X @>>{f}> Y \end{CD} $$ to be also a pushout square.

Do there exist sufficient conditions on a Grothendieck topology on schemes under which $Y$ is the pushout $$\displaystyle Y = \Gamma_0\coprod_{\Gamma_1} \Gamma_0 = X\coprod_{X\times_Y X} X$$ as above? Is this true for all the standard topologies (Zariski, etalé, fpqc, fppf)?

Answer 1

In any topos, if $Y \rightarrow X$ is an epimorphism then:

$$Y \times_X Y \rightrightarrows Y \rightarrow X $$

is indeed a colimit diagram.

If you have a site $S$ and a cover $Y \rightarrow X$ then it is an epimorphisms in the topos of sheaves, and if the fiber product $Y \times_X Y$ exists, then it also the fiber product in the topos and hence:

$$Y \times_X Y \rightrightarrows Y \rightarrow X $$

is a colimit in the topos of sheaves.

If the topology is subcanonical (i.e. the representables are sheaves) then the site is a full subcategory of the topos of sheaves so this concludes the proof. This works for all the topology you mentioned.

If the topology is not subcanonical, then you only know that one get a colimit after applying the sheafification functors to the representable functors, which might not say much on the site itself.