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Let me expand on my comments. Assuming a morphism is flat and locally of finite presentation, it is surjective if and only if it is a universally effective epimorphism. A morphism $f : X \to Y$ of s …

A Grothendieck topology by definition consists of sieves – what Johnstone calls a sifted coverage – whereas a Grothendieck pretopology in any non-trivial case will contain a non-sieve. (Recall that $\ …

I would just say that the inclusion preserves covering families (in the naïve sense). You don't need Grothendieck pretopologies to make sense of this – just plain coverages (in the sense of Johnstone; …

There is no hope for this in any subcanonical topology coarser than the fppf topology, or more generally, any subcanonical topology in which morphisms $\operatorname{Spec} C \to \operatorname{Spec} K$ …

To answer your question directly, WISC does not imply the existence of subobject classifiers. Notice that when there are only trivial covers, WISC is trivially satisfied, so it suffices to find a cate …