topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
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fiber space, space attachment
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
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paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
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homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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(see also Chern-Weil theory, parameterized homotopy theory)
In topological homotopy theory, the fundamental theorem of covering spaces says that for a sufficiently well-behaved topological space $X$, the functor which sends a covering space of $X$ to the Set-action (permutation representation) of the fundamental groupoid of $X$ on the fibers of $E$ is an equivalence of categories.
This is a basic instance of the general principle of Galois theory.
It follows in particular that for connected $X$ then the automorphism group of the universal covering space of $X$ coincides with the fundamental group $\pi_1(X,x)$ itself (for any basepoint $x$). This often yields a convenient means to determine the fundamental group of $X$ in the first place.
This is closely related to the (∞,1)-Grothendieck construction; the equivalence $\infty Grpd_{/\esh X} \simeq \infty Grpd^{\esh X}$ restricts to an equivalence between the subcategories of bundles $Y \to \esh X$ with 0-truncated fibers and of set-valued functors on $\esh X$. Observe in particular that $Set^{\Pi_1(X)} \simeq Set^{\esh X}$.
(Here $\esh X$ denotes the shape of the topological space $X$, hence its fundamental $\infty$-groupoid).
(fundamental theorem of covering spaces)
Let $X$ be a locally path-connected and semi-locally simply-connected topological space. Then the operations on
extracting the monodromy $Fib_{E}$ of a covering space $E$ over $X$
constitute an equivalence of categories
between the category of covering spaces, and the category of permutation groupoid representations of the fundamental groupoid of $X$.
With the standard definitions of the two functors, both are in fact inverse isomorphisms of categories instead of just equivalences of categories (meaning that the required natural isomorphisms from the composites of the two functors to the identity functor are componentwise equalities), which establishes the claim right away. For definiteness, we make this explicit:
Given $\rho \in Set^{\Pi_1(X)}$ a permutation representation, we need to exhibit a natural isomorphism of permutation representations.
First consider what the right hand side is like: By this def. of $Rec$ and this def. of $Fib$ we have for every $x \in X$ an actual equality
To similarly understand the value of $Fib(Rec(\rho))$ on morphisms $[\gamma] \in \Pi_1(X)$, let $\gamma \colon [0,1] \to X$ be a representing path in $X$. As in the proof of the path lifting lemma for covering spaces (this lemma) we find a finite number of paths $\{\gamma_i\}_{i \in \{1,n\}}$ such that
regarded as morphisms $[\gamma_i]$ in $\Pi_1(X)$ they compose to $[\gamma]$:
each $\gamma_i$ factors through an open subset $U_i \subset X$ over which $Rec(\rho)$ trivializes.
Hence by functoriality of $Fib(Rec(\rho))$ it is sufficient to understand its value on these paths $\gamma_i$. But on these we have again by direct unwinding of the definitions that
This means that if we take
to be the above identification, then this is a natural transformation and hence in a particular a natural isomorphism, as required.
It remains to see that these morphisms $\eta_\rho$ are themselves natural in $\rho$, hence that for each morphism $\phi \colon \rho \to \rho'$ the diagram
commutes as a diagram in $Rep(\Pi_1(X), Set)$. Since these morphisms are themselves groupoid homotopies (natural isomorphisms) this is the case precisely if for all $x \in X$ the corresponding component diagram commutes. But by the above this is
and hence this means that the top and bottom horizontal morphism are in fact equal. Direct unwinding of the definitions shows that this is indeed the case.
Conversely, given $E \in Cov(X)$ a covering space, we need to exhibit a natural isomorphism of covering spaces of the form
Again by this def. of $Rec$ and this def. of $Fib$ the underlying set of $Rec(Fib(E))$ is actually equal to that of $E$, hence it is sufficient to check that this identity function on underlying sets is a homeomorphism of topological spaces.
By the assumption that $X$ is locally path-connected and semi-locally simply connected, it is sufficient to check for $U\subset X$ an open path-connected subset and $x \in X$ a point with the property that $\pi_1(U,x) \to \pi_1(X,x)$ lands is constant on the trivial element, that the open subsets of $E$ of the form $U \times \{\hat x\} \subset p^{-1}(U)$ form a basis for the topology of $Rec(Fib(E))$. But this is the case by definition of $Rec$.
It remains to see that $\epsilon_E$ is itself natural in $E$. But as for the converse direction, since the components of $\epsilon_E$ are in fact equalities, this follows by direct unwinding of the definitions.
This establishes an equivalence as required. In fact this is an adjoint equivalence.
In homotopy type theory, the fundamental theorem of covering spaces is really just a special case of the universal property of $1$-truncation.
Indeed, in HoTT a covering space over a type $X$ is usually defined as a dependent type $C \colon X \to \mathsf{Sets}$. We can get the more familiar bundle definition of a covering space by taking the dependent sum type $\sum_{x \colon X} C(x)$, which comes equipped with a canonical projection $\operatorname{proj}_1 \colon \sum_{x \colon X} C(x) \to X$.
Since $\mathsf{Sets}$ is a $1$-type, then by the universal property of 1-truncation we have an equivalence:
Since the 1-truncation $\Vert X \Vert_1$ is the fundamental groupoid of $X$, this really is the fundamental theorem of covering spaces.
Or rather, more faithful to the traditional concepts in topology, it is the shape modality $\esh$ in cohesive homotopy type theory which turns a geometric homotopy type into $X$ its fundamental $\infty$-groupoid $\esh X$ (see at shape via cohesive path ∞-groupoid) truncating to the actual fundamental groupoid $\Vert \esh X \Vert_1$.
Then the adjunction between the shape modality and the flat modality $\flat$ says that
or, equivalently, that
where $\flat Sets$ is the actual classifier for covering spaces in the generality of cohesive (e.g. topological) homotopy types. This reflects the fundamental theorem of covering spaces as traditionally understood in topology.
This is the topic of dcct, sec. 3.8.6, p. 358, see also Cherubini & Rijke 2020, Thm. 8.7.
Lecture notes on the classical theory:
Friedhelm Waldhausen, around p. 122 of Topologie (pdf)
Jesper Møller, The fundamental group and covering spaces (2011) (pdf)
Discussion in cohesive homotopy theory:
Discussion in cohesive homotopy type theory: