Higher covering spaces in an $\infty$-topos

Abstract

We develop a systematic theory of $(n-1)$-truncated maps, called $n$-covering maps, in a fixed $\infty$-topos $\mathscr{E}$, guided by the analogy with classical covering spaces. We prove an equivalence of $n$-categories between $n$-coverings over a pointed connected object $(X,x)$ and $\infty$-actions of the fundamental $n$-group $\Pi_n(X,x)$ on $(n-1)$-truncated objects, which restricts to a classification of pointed connected $n$-coverings in terms of sub-$n$-groups of $\Pi_n(X,x)$. We study the $n$-group of deck transformations $\mathscr{D}\mathrm{eck}(p)$, identifying it with $\Pi_n(X,x)$-equivariant autoequivalences of the fiber $F$. For normal $n$-coverings, it is further described as a quotient of $\Pi_n(X,x)$, yielding a classification of such coverings in terms of normal subgroups of $\pi_n(X,x)$. For an arbitrary $n$-covering, the deck $n$-group arises as a quotient of a suitable normalizer. Our approach relies on a careful study of $n$-groups and their $\infty$-actions, on the use of univalent universes, and on an internal Yoneda embedding. When $n=1$ and $\mathscr{E}$ is the $\infty$-category of homotopy types, our results recover the classical theory of covering spaces. We further illustrate the theory in sheaf and \'etale $\infty$-topoi, where the external deck group recovers cohomology of the base, and in cohesive $\infty$-topoi, where it recovers the $1$-covering theory of manifolds.