Conventions: In the below,
- unless otherwise stated, terms regarding derived algebraic geometry will follow the conventions of Yaylali.
- an algebraic stack will be a stack $\mathscr{S}$ over a base scheme $S$ such that the diagonal $\Delta : \mathscr{S}\to\mathscr{S}\times_S\mathscr{S}$ is representable by algebraic spaces and such that there exists a smooth surjective morphism $U\to\mathscr{S}$ with $U$ a scheme (in particular, there is no finiteness hypothesis on the diagonal).
- a derived algebraic stack is a derived geometric stack in the sense of definition 3.11 in loc. cit., and a representable morphism of derived algebraic stacks is a geometric morphism in the sense of definition 3.11 of loc. cit.
- a derived algebraic space will be a $0$-DM derived stack in the sense of definition 3.11 of loc. cit. whose diagonal is a monomorphism.
If $f : X\to Y$ is an etale morphism of ordinary algebraic spaces, $f$ has many desirable properties. One such property is that its diagonal $\Delta_f : X\to X\times_Y X$ is a Zariski open immersion (and in particular, a monomorphism).
My question is about generalizing this result to (derived) algebraic stacks. Namely:
If $f : X\to\mathscr{S}$ is an etale morphism from a (derived) algebraic space to a (derived) algebraic stack, is $\Delta_f : X\to X\times^{\mathbf{R}}_{\mathscr{S}} X$ an open immersion?
I know that classically, this holds more generally for unramified morphisms. However, I haven't seen much made of unramified morphisms in derived algebraic geometry (and I also only care about the etale situation), so I'll restrict to the etale case.
Moreover, the problem above reduces to the question of whether $\Delta_f$ is a monomorphism: it is automatically etale, and an etale monomorphism in derived algebraic geometry is a Zariski open immersion.
I am also interested in the same question when $X$ is replaced by a (derived) algebraic stack $\mathscr{S}',$ and the morphism $f : \mathscr{S}'\to\mathscr{S}$ is representable and etale.
Thoughts: For a morphism of ordinary algebraic spaces, it's easy to show that the diagonal is a monomorphism; in fact, this is true in any $1$-category, since the diagonal composed with the projection to either factor is the identity (hence a monomorphism). Of course, for higher categories (even just $2$-categories), this is no longer quite so simple. I do know that it would suffice to prove that $X\times^{\mathbf{R}}_{\mathscr{S}}X\xrightarrow{\pi} X$ is $0$-truncated or that $X\times^{\mathbf{R}}_{X\times^{\mathbf{R}}_{\mathscr{S}} X}X\xrightarrow{\pi} X$ is $(-1)$-truncated by Kerodon, 9.2.3.16 (where $\pi$ is either projection). This would follow if etale morphisms were $0$-truncated, but I don't know if I should expect this to be true -- it seems like this might be too much to hope for.
I also do know that an unramified morphism of algebraic stacks is the same as a morphism which is locally of finite type with etale diagonal, so even in the classical situation of a representable morphism between two general algebraic stacks, this would need to crucially make use of the representability hypothesis or need extra assumptions to work.
