MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


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?


share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.