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Let $X$ be locally path connected and semi-locally simply connected topological space.

The fundamental theorem of covering spaces says the local system functor functor taking a covering space to the path-lifting action of its fundamental groupoid on the fibers is a category equivalence $$L:\mathsf{Cov}_{/X}\overset{\simeq}\longrightarrow [\pi_{\leq 1}(X),\mathsf{Set}]$$ where $\pi_{\leq 1}X$ is the fundamental groupoid of the topological space $X$. (The qusai-inverse is constructed by using the topological assumptions on $X$ to construct from each such action a total space whose projection onto $X$ is a covering map.)

Fix a basepoint $x\in X$. How can I use the fundamental theorem to formally express the subgroup lattice of $\pi_1(X,x)$ in terms of (possibly pointed) coverings of $X$?

Arrow
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Write $G = \pi_1(X, x)$. The object representing the fiber functor sending a covering to its fiber over $x$ is, on the one hand, the universal cover, and on the other hand, $G$ regarded as a $G$-set. Now consider the lattice of quotient objects of this object (equivalently, the lattice of intermediate covers between $X$ and the universal cover); I claim it's the opposite of the lattice of subgroups of $G$.

A slightly different thing you can do, which doesn't require picking a basepoint, is to restrict attention to the full subcategory of connected objects. These are the connected covers on the cover side and the transitive $G$-sets on the $G$-set side, so isomorphism classes of objects recover (conjugacy classes of) subgroups, but the morphisms are more interesting. To get subgroups on the nose you can look at pointed connected objects, although saying “pointed” does require a baseplint.

Qiaochu Yuan
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