Geometry of 2-arrows It is frequently said that one of the contributions of Grothendieck to geometry was to systematically think about the properties of morphisms, as opposed to the properties of spaces themselves (the latter could then be recovered as the properties of the structure morphism). In a sense, this means instead of thinking about objects, we think about arrows. 
Does this philosophy have an incarnation in the world of higher mathematics? Can we talk about the geometric properties of arrows between arrows, arrows between arrows between arrows, etc.? Maybe one answer would be "$n$-stacks" but I never really saw much serious geometry done with them. I think that SAG/DAG has nothing to do with this question but correct me if I am wrong.
I have heard that $(\infty, 2)$-categories are less understood than $(\infty, 1)$-categories, maybe this is relevant to this question.
 A: As I have understood it (though I’m neither an algebraic geometer nor a historian) the novelty of Grothendieck that you’re describing was considering morphisms as representing families of objects varying over a base, and taking such families of objects as a main focus of development (rather than just individual objects).
This idea is a pervasive and crucial ingredient in all the approaches to “higher mathematics” that I know of.  In Lurie’s development, for instance, it shows up as the various notions of “fibration” that appear (sometimes as the main objects of study, sometimes as auxiliary technical tools); in the frequent use of slice categories, codomain fibrations, and so on; and many other places.  (Though the question suggests that perhaps Lurie’s development isn’t the kind of direction you’re thinking of; if so, could you clarify which approaches you are more interested in?)
So there’s hardly a question of “can this concept be studied in the higher-categorical setting” — it’s hard to imagine developments of higher-categorical mathematics without this concept.
