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The only important axiom in order to define a notion of sheaf is the stability under pullback. There is a proposition in SGA4 saying that if you have a family of sieves only satisfying the pullback st …

This is essentially correct, and there is no need for the topology to be subcanonical. But let me clarify: Whether the topology is subcaninical or not, we have the following: given any family of maps …

Let $ \chi : X \rightarrow \Omega$ be the characteristic function of $U$. By definition of a subobject classifier, the characteristic function of the pullback of $U$ by $U_i \rightarrow X$ is just th …

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 $Y \rightarrow X$ the …

There is (as always) a finer topology $T$ making this into a sheaf, the question is whether this topology is different from the usual one or not. The cover for $T$ of an open subset $V \subset \mathb …

This will not be the case in general. A family is a cover in the canonical topology if all its pullbacks are jointly regular epimorphism. So this will for example be the case if coproducts are univers …