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Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to I\mathcal Y$.

My questions:

  1. Is $\tilde f$ representable if $f$ is representable?
  2. Is $\tilde f$ $P$ if $f$ is $P$? (We assume $f$ and $\tilde f$ to be representable.)
  3. Can we have a (relatively weak) condition (on $f$ or stacks) such that (2) holds? (See the second note below.)

Here $P$ is a property stable under pullbacks. (e.g. etale, surjective, injective, open, etc. In particular I am interested in the case $P$ = etale.)

Notes:

  • I am thinking about the stacks on the category of schemes over $\mathbb C$ or the category of smooth manifolds.
  • I found a relevant description at the stack project. According to this, if $f$ is a monomorphism, then the questions (1) and (2) are both true. But the monomorphism condition is rather strong. In fact, if both $\mathcal X$ and $\mathcal Y$ are representable, then the above question (2) is always true, whichever $f$ is representable or not. ($\because I \mathcal X=\mathcal X$.) Therefore I feel that there is, at least, a relaxed condition for ((1) and) (2). (I also think that there is a relevant paper in literature. But I could not find any.)
  • I wrote the question in terms of stacks, but any answers in terms (proper etale) Lie groupoids are welcome.
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