Consider the category of sheaves (of sets) on the affine étale site. It's a well known fact that a morphism of schemes is a Zariski-open immersion if and only if it is an étale monomorphism, so we extend this idea to all sheaves as follows:

We say that a *sheaf* $S$ *satisfies the condition* if given any two étale monomorphisms $A\to S$ and $B\to S$, then $A\times_S B$ is representable by $Spec(0)$ (the image of the 0 ring in the opposite category of commutative rings) if and only if either $A$ or $B$ is representable by $Spec(0)$.

Motivation: It would be nice if we could define the notion of irreducibility only in terms of functors of points. The condition that we are trying to simulate is the intersection of two open subsets of an irreducible topological space being empty if and only if one of the open subsets is empty. The problem is that the fiber-product of schemes does not necessarily coincide with the fiber product of the underlying topological space.

## Questions:

Are there any cases of schemes where this condition and irreducibility are not equivalent?

If this definition does work for schemes, does it work for algebraic spaces (perhaps with some tweaking)?

Edit: Recall that a morphism of sheaves $F\to G$ is an étale monomorphism if it is a monomorphism and the pullback (fiber product) by any morphism from an affine scheme $X\to G$ is an algebraic space with an atlas of affine schemes given by $\{U_i\to F \times_G X\}$ such that the composition $U_i\to F \times_G X\to X$ with the projection is an étale morphism of affine schemes (maps corresponding to étale maps of rings).

monomorphism, of course! (Also, in (2) I meant that such a map of sheaves isdefinedto be etale if it is formally smooth and locally of finite presentation.) $\endgroup$etaleand locally of finite presentaiton. $\endgroup$6more comments