TITLE: Analogue to covering space for higher homotopy groups?
QUESTION [47 upvotes]: The connection between the fundamental group and covering spaces is quite fundamental.  Is there any analogue for higher homotopy groups?  It doesn't make sense to me that one could make a branched cover over a set of codimension 3, since I guess, my intuition is all about 1-D loops, and not spheres.

REPLY [2 votes]: Since Peter Scholze in his answer used his condensed framework to consider higher covering spaces treating the topological and homotopic information separately, perhaps an alternative approach to doing this should also be mentioned, alluded to in comments there, namely, cohesive homotopy types. The constructions are very similar, see sections '5.2.5 Universal coverings and geometric Whitehead towers' and '6.3.12 Universal coverings and geometric Whitehead towers' of Urs Schreiber's 'Differential cohomology in a cohesive ∞-topos'.