Let $f:X\to Y$ be a map of Deligne-Mumford stacks. Let's say that the map $f$ is a geometric equivalence if the induced map on small étale topoi is a geometric equivalence. Moreover, let's say that the map $f$ is a universal geometric equivalence if for every morphism $A\to Y$ with $A$ an affine scheme, the projection map $X\times_Y A \to A$ is a geometric equivalence (so without loss of generality, then, we can assume that $Y$ is affine). So then the question:

Is it true that given a universal geometric equivalence $f:X\to Y$, the map is an affine morphism?

It would in fact be enough to show that the map is schematic (using the classification of universal homeomorphisms between schemes together with the reduction to $Y$ affine), but I'm not sure if that's so clear either.

Bonus question: If it is true for Deligne-Mumford stacks, does it remain true for algebraic stacks?


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