Hi,

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?

Thanks!

topos) is a coverage. See here: ncatlab.org/nlab/show/coverage – Zhen Lin Apr 19 '13 at 14:49