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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    $\begingroup$ 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: ncatlab.org/nlab/show/coverage $\endgroup$ – Zhen Lin Apr 19 '13 at 14:49
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    $\begingroup$ 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. $\endgroup$ – Colin McLarty Apr 21 '13 at 0:22

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 stability condition then a presheaf is a sheaf with respect to this family if and only if it is a sheaf with respect to the topology it generates. Hence to some extend the identity axiom is useless, as well is the locality axiom.

The reason why this two other axioms are important is that when you have all three axioms then you can prove that if two topologies defines the same notion of sheaf then they are equal: adding new covering will reduces the number of sheaf, and this is the case only if the three axioms are satisfied.

  • $\begingroup$ Indeed. And a reason why one needs stability under pullback is in order to be able to construct the sheafification functor. $\endgroup$ – Ingo Blechschmidt Apr 6 '17 at 18:30
  • $\begingroup$ You do not really need it to construct the sheafification functors (its existence can be obtained formally from the special adjoint functor theorem without any assumption other than smallness), but you definitely need it for proving that sheafification is left exact (and for the way it is usually constructed) $\endgroup$ – Simon Henry Dec 1 '17 at 13:34

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