Let $S$ be a topological space. Every covering map $f: X \rightarrow S$ determines a functor from the fundamental groupoid $\pi _{\leq 1}(S)$ to the category of sets, given by the monodromy representation of Example 5.2.0.5. Under some mild assumptions on the topological space $S$, the converse is also true: every functor $\pi _{\leq 1}(S) \rightarrow \operatorname{Set}$ can be obtained as the monodromy representation of an essentially unique covering map $f: X \rightarrow S$. More precisely, we have the following:
Theorem 5.2.7.1 (The Fundamental Theorem of Covering Space Theory). Let $S$ be a topological space which is semilocally simply connected. Then the construction $X \mapsto \operatorname{hTr}_{X/S}$ determines an equivalence of categories
\[ \{ \textnormal{Covering maps $f: X \rightarrow S$} \} \rightarrow \operatorname{Fun}( \pi _{\leq 1}(S), \operatorname{Set}). \]
The proof of Theorem 5.2.7.1 can be broken into two parts:
If $S$ is a topological space which is semilocally simply connected, then the construction $X \mapsto \operatorname{Sing}_{\bullet }(X)$ induces an equivalence of categories
For every Kan complex $\operatorname{\mathcal{C}}$, the formation of monodromy representations determines an equivalence of categories
The proof of $(a)$ requires some point-set topology; we defer a discussion to ยง
. Our goal in this section is to give a proof of $(b)$ (see Corollary 5.2.7.6). We will deduce $(b)$ from a more general statement, which classifies left coverings of an arbitrary simplicial set $\operatorname{\mathcal{C}}$ (Corollary 5.2.7.3).
Proposition 5.2.7.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets, and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ be the homotopy transport representation of Proposition 5.2.0.3. Then there is a canonical isomorphism of simplicial sets
\[ \operatorname{\mathcal{E}}\simeq \operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )} \operatorname{N}_{\bullet }( \int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}). \]
Proof. Every vertex $X \in \operatorname{\mathcal{E}}$ can be regarded as an element of the set $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( U(X) )$, and the construction $(X \in \operatorname{\mathcal{E}}) \mapsto ( \operatorname{\mathcal{E}}_{U(X)}, X)$ determines a functor $\widetilde{\operatorname{hTr}}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \operatorname{Set}_{\ast }$. Let us identify $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ with a morphism of simplicial sets from $\operatorname{\mathcal{C}}$ to $\operatorname{N}_{\bullet }(\operatorname{Set})$ and $\widetilde{\operatorname{hTr}}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ with a morphism of simplicial sets from $\operatorname{\mathcal{E}}$ to $\operatorname{N}_{\bullet }(\operatorname{Set}_{\ast } )$, so that we have a commutative diagram of simplicial sets
which we can identify with a morphism of simplicial sets $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}) } \operatorname{N}_{\bullet }( \int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}})$. Since $U$ and the projection map $\int _{\operatorname{\mathcal{C}}} \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are both left covering maps (Remark 5.2.6.9), it follows that $V$ is a left covering map (Remark 4.2.3.14). By construction, $V$ is bijective at the level of vertices, and is therefore an isomorphism of simplicial sets (Proposition 4.2.3.19). $\square$
Corollary 5.2.7.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and let $\operatorname{LCov}_{\operatorname{\mathcal{C}}}$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ spanned by the left covering maps $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Then the formation of homotopy transport representations supplies an equivalence of categories
\[ \operatorname{LCov}_{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) \quad \quad (U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}. \]
Proof. Proposition 5.2.7.2 shows that the functor
is a left homotopy inverse to the functor $\operatorname{\mathcal{E}}\mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. By virtue of Example 5.2.0.6 and Remark 5.2.5.6, it is also a right homotopy inverse. $\square$
Corollary 5.2.7.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
There exists a pullback diagram of simplicial sets
where $V: \operatorname{\mathcal{D}}\rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a left covering functor (in the sense of Definition 4.2.3.1).
For every category $\operatorname{\mathcal{C}}'$ and every morphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}'$ and the projection $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$ is left covering functor (in the sense of Definition 4.2.3.1).
For every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}'$ and the projection $\operatorname{\mathcal{E}}' \rightarrow [n]$ is a left covering functor (in the sense of Definition 4.2.3.1).
The morphism $U$ is a left covering map of simplicial sets (in the sense of Definition 4.2.3.8).
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{N}_{\bullet }( \operatorname{\mathcal{D}}) \ar [d]^{ \operatorname{N}_{\bullet }(V) } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ), } \]
Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 4.2.3.6, the implication $(2) \Rightarrow (3)$ is trivial, and the implication $(3) \Rightarrow (4)$ follows by combining Remark 4.2.3.15 with Proposition 4.2.3.16. The implication $(4) \Rightarrow (1)$ follows from Proposition 5.2.7.2. $\square$
Corollary 5.2.7.5. Let $\operatorname{\mathcal{C}}$ be a category. Then:
Construction 5.2.6.1 determines a fully faithful functor
whose essential image consists of the left covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.
Variant 5.2.6.2 determines a fully faithful functor
whose essential image consists of the right covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}, \]
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int ^{\operatorname{\mathcal{C}}} \mathscr {F}, \]
Corollary 5.2.7.6. Let $\operatorname{\mathcal{C}}$ be a Kan complex. Then the construction $(U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ induces an equivalence of categories
\[ \{ \textnormal{Covering maps $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} \rightarrow \operatorname{Fun}( \pi _{\leq 1}(\operatorname{\mathcal{C}}), \operatorname{Set}). \]