What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?

Question

Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Actually I do not know many details about it. It says an analytic stack is a sheaf $\mathrm{AnRing}\rightarrow \mathrm{Ani}$ from the category of analytic rings to the category of anima (spaces) satisfying $!$-descent for $!$-hypercovers.

On the other hand, in their paper arXiv:1412.5166 Yu Yue and Porta have defined higher complex/non-archimedean analytic stacks. A higher complex analytic stack in their sense is a sheaf $\mathrm{Stn}_{\mathbb{C}}\rightarrow \mathrm{Ani}$ from the category of stein complex analytic spaces whose topology is defined by open immersions to animas. And for a non-achimedean field $k$, a higher non-achimedean analytic stack is a sheaf $\mathrm{Afd}_k\rightarrow \mathrm{Ani}$ from the category of $k$-affinoid spaces with quasi-etale topology to animas.

What's the relation between the two theories? Should the first theory unify the two notions in the latter? And how does the first theory unify analytic geometries, in which sense?

Answer 1

One way to think about categories of stacks (or more generally, (higher) topoi) is that one has some class of generating objects, and some class of allowed gluings; and then one builds the full category by taking the category generated freely by the generating objects subject to the gluing relations.

In the work of Yue Yu and Porta, the generating objects are (Stein) complex-analytic spaces or (affinoid) Berkovich analytic spaces, and the gluings are open covers or quasi-étale covers.

In our work, the generating objects are "affine analytic stacks" corresponding to "analytic rings", and the gluings !-covers.

There are functors from the categories considered by Yue Yu and Porta to our category. To construct this, one has to send the generating objects to our category, and show that allowed gluings go to allowed gluings. But there are natural analytic rings corresponding to (compact Stein) complex-analytic spaces, and affinoid Berkovich analytic spaces; and open covers or quasi-etale maps go to !-covers.

Those functors will not be fully faithful, but this is to be expected: In general, categories of stacks are very sensitive to the Grothendieck topologies chosen. (Questions such as comparisons of cohomologies in different sites are instances of this question of fully faithfulness.)

In any case, this shows that our notion of analytic stack "unifies" these other notions (along with many other ones). Something that is somewhat novel about our formalism is that classical algebraic geometry sits in there just as well as various flavors of analytic geometry, and they can play with each other. For one fun application (quite distant from the examples originally motivating us) of the versatility of our formalism, in particular of "algebraic geometry over smooth manifolds", see Dustin Clausen's recent talk at Esnault's birthday conference.