The Stacks project

Definition 7.38.1. Let $\mathcal{C}$ be a site.

  1. A family of points $\{ p_ i\} _{i\in I}$ is called conservative if every map of sheaves $\phi : \mathcal{F} \to \mathcal{G}$ which is an isomorphism on all the fibres $\mathcal{F}_{p_ i} \to \mathcal{G}_{p_ i}$ is an isomorphism.

  2. We say that $\mathcal{C}$ has enough points if there exists a conservative family of points.

Comments (4)

Comment #2175 by Kestutis Cesnavicius on

"if for every map" ---> "if every map"

Comment #3840 by Rene on

Maybe it should be clarified if 'family of points' means 'set of points'

Comment #3933 by on

Well, actually a family of things means a collection of things indexed by some set. So you could have doubles. This can occasionally be useful. So let's leave it as is.

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View Definition 7.38.1 as pdf