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The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a covering family.

Why is this axiom needed? Obviously a functor $F : C^{opp}\to Sets$ will satisfy the sheaf condition with respect to this family, so nothing seems to be gained by adding this covering family...

Am I missing something?


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You can make do with much less than a Grothendieck pretopology: all you need to have a good notion of sheaf (i.e. one that generates a topos) is a coverage. See here: – Zhen Lin Apr 19 '13 at 14:49
I'll just add that Grothendieck probably thought of this as analogous to saying a topology on a set $S$ always has $S$ itself as an open subset. – Colin McLarty Apr 21 '13 at 0:22

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