I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds and algebraic varieties (sheaves, schemes, stacks etc).

This concrete question is about some algebro-geometric abstractions. It could be seen as an extension of this other one. I am wondering specifically about how to visualize $n$-stacks. For visualize I understand trying to describe their underlying geometrical space in a graphical way, in the spirit of Mumford´s illustrations of schemes and Eisenbud-Harris' book about schemes.

To my understanding a $n$-stack is roughly a sheaf taking values in an $n$-category. This would mean that if by generalizing form varieties to schemes we allow closed and nonclosed points (enriching the relations of inclusions and exclusions in our space) by using stacks and higher stacks we would enrich this by allowing points to be equipped with nontrivial automorphisms. So the idea would be replace points by points that are equal to other points up to some "higher isomorphisms" or equivalences. Hence, to my mind is difficult to see a geometrical difference between, say, a $4$-stack and a $5$-stack. To my eye the geometrical intuition misses among the algebraically flavoured composition rules of the $n$-morphisms.

Could anybody give me some kind of visual insight? For example it could be nice if someone could explain to me the geometrical differences between a stack (i.e. $(2,1)$-sheaf) and a $2$-stack (i.e. $(3,1)$-sheaf)).

PD: Techniques and ideas for visualization of other abstract algebro-geometric objects (derived manifolds, derived schemes etc) are welcome!

EDIT: The second answer here has some good insights for (1-)stacks!

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    $\begingroup$ I would start with just plain old common stacks in groupoids. For these, a good geometric replacements are internal groupoids in schemes. Most of the time they suffice to "see" all the relevant information carried by the stacks made from them. What is missing though is the intuition for nonisomorphic but equivalent such gadgets. $\endgroup$ – მამუკა ჯიბლაძე Mar 11 '19 at 14:47

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