Is there a name for a "rigid" sheaf?

Question

Is there a name for the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is connected and $U$ is nonempty?

In other words, this is a sheaf which satisfies the identity theorem of complex analysis (taking for $\mathcal F$ the sheaf of holomorphic functions).

Many sheaves in geometry satisfy this property, but I've never heard a name for it. Perhaps a "rigid" sheaf, in contrast to a "flasque" sheaf? (This rigidity property seems to be the one which gave rigid analytic geometry its name, but I am not completely sure about that, and in any case, I have never seen the words "rigid sheaf" written down.)

Also, if these sheaves do not have a name, I would be curious to know why that might be. Perhaps the property is not useful enough on its own to deserve a name?

Thank you.

Answer 1

The problem is that your definition is well behaved only if there is enough open subsets $V$ such that $V$ is connected (if there is no such open subset, then your condition is empty) hence the notion as you defined is well behaved only on a locally connected space.

Once you have fixed this issue (either by slightly changing your definition, or by restricting yourself to locally connected space) The name your are looking for is "decidable sheaf".

In general, an object $X$ of a topos is called decidable (or is said to have a decidable equality) if the diagonal embedding $X \rightarrow X \times X$ is complemented, that is if there exists a sub-object $\Delta^{c}$ of $X \times X$ such that $X \times X \simeq X \coprod \Delta^c$ (with the $X$ part corresponding to the diagonal inclusion).

In the case of the topos of sheaves over a topological space $X$ the sheaf $\mathcal{F}$ is decidable iff for each couple $f,g \in \mathcal{F}(U)$ the (open) set $W$ of points $x \in U$ such that $f$ and $g$ are equal in the stalks at $x$ is closed in $U$.

You can easily check that this imply the property you want and that on a locally connected space they are equivalent.

Also note that this is equivalent to the fact that the map from the etale space $Y$ of $\mathcal{F}$ to $X$ is relatively hausdorff in the sense that the diagonal embeddings $Y \rightarrow Y \times_X Y$ is a closed embeddings.

Also this version of "rigidity" is a local property, and stable by arbitrary pullback, whereas I don't think the original definition is.

Answer 2

Hausdorff sheaves have this property, and if I am not mistaken, this is even an equivalence if the base space is Hausdorff and locally connected.

Answer 3

If $\mathcal{F}$ happens to be the sheaf of solutions of some differential equation (for example, complex analytic functions satisfy the Cauchy-Riemann equations) then this property is called unique continuation (property or principle). If the map $\mathcal{F}(V) \to \mathcal{F}(U)$ also happens to be surjective (that is, a bijection) for $V$ and $U$ connected and sufficiently small, then the sheaf $\mathcal{F}$ is called locally constant.