In SGA 1, Expose 5, there is a proposition (3.5), which states that for any two objects $X$ and $X'$ in $\textbf{Et}/S$, all $S$ morphisms between them factor into a surjective etale morphism $X\to X''$ and a canonical immersion $X''\to X'$. If I am understanding this proof correctly, this is by the fact that to be in $\textbf{Et}/S$, any morphism from an object in the category to $S$ is finite etale, and therefore both open and closed, and so any morphism between objects in the category is both open and closed, so our image will be open and closed, and thus a finite union of connected components of $X'$.

There is a corollary following this proposition, rephrasing it in terms of an effective epimorphism followed by a monomorphism. What are the advantages of this? The corollary follows by the fact that the first map is faithfully flat, and thus a strict epimorphism, and a similar result for the monomorphism. Perhaps this is just my lack of familiarity with category theory, but would every strict epimorphism be a faithfully flat morphism in this category? Would the reason to introduce these terms just be to define Galois categories later, and should we always just think about strict epimorphisms/monomorphisms in the sense of Proposition 3.5?

(Also, is there a typo in the statement, switching the roles of $X'$ and $X''$?)


I can offer two answers, but I am not sure either is what you are looking for.

First, any category has the so called ``canonical topology'', which is the finest topology in which all representable pre-sheaves are sheaves. The covers in this topology can be described (essentially) as precisely the strict epimorphisms (I think this is in SGA4, but it is also easy). The statement you mention says that the flat topology is sub-canonical, and I believe it is at least rather close to the canonical one (but I have nothing to back it up). For etale maps, this just says that $X\rightarrow X''$ from the proposition is a cover in a sub-canonical topology.

From another point of view, one often thinks of a finite etale scheme over $S$ as a "finite set" over $S$ (as opposed to a finite scheme over $S$, that could, for instance, have nilpotents). The corollary then provides evidence that this category indeed behaves in that manner: like for sets, any map is a surjection onto a subset. The fact that strict epimorphisms are reasonable analogues of surjections is discussed (for instance) in a book of Makkai and Reyes, ``First order categorical logic'' (for example, section 3.3), which also discusses some other notions from SGA4 from this point of view.

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