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5.6.1 Covering Space Theory

Let $f: X \rightarrow S$ be a covering map of topological spaces. For every point $s \in S$, let $\operatorname{Tr}_{X/S}(s)$ denote the fiber $X_{s} = \{ s \} \times _{S} X$. The construction $s \mapsto \operatorname{Tr}_{X/S}(s)$ can be promoted to a functor from the fundamental groupoid $\pi _{\leq 1}(S)$ to the category of sets, which we will refer to as the monodromy representation of $f$ and denote by $\operatorname{Tr}_{X/S}$ (see Example 5.6.1.3 below). 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 5.6.1.1 (The Fundamental Theorem of Covering Space Theory). Let $S$ be a topological space which is semilocally simply connected. Then the construction $X \mapsto \operatorname{Tr}_{X/S}$ 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 5.6.1.1 can be broken into two parts:

$(a)$

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

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Covering maps of topological spaces $f: X \rightarrow S$} \} \ar [d] \\ \{ \textnormal{Covering maps of simplicial sets $\operatorname{\mathcal{E}}\rightarrow \operatorname{Sing}_{\bullet }(S)$} \} . } \]
$(b)$

For every Kan complex $\operatorname{\mathcal{C}}$, the formation of homotopy transport representations (Construction 5.2.3.2) induces an equivalence of categories

\[ \{ \textnormal{Covering maps $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} \rightarrow \operatorname{Fun}( \pi _{\leq 1}(\operatorname{\mathcal{C}}), \operatorname{Set}) \quad \quad \operatorname{\mathcal{E}}\mapsto \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}. \]

The proof of $(a)$ requires some point-set topology, and we defer the discussion to ยง. Our goal in this section is to give a proof of $(b)$ (see Corollary 5.6.1.10 below). We will deduce $(b)$ from a more general statement, which classifies left coverings of an arbitrary simplicial set $\operatorname{\mathcal{C}}$ (Corollary 5.6.1.6). Let us begin by introducing some notation.

Construction 5.6.1.2 (The Covariant Transport Representation). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets (Definition 4.2.3.8). Then $U$ is a left fibration (Remark 4.2.3.11), and therefore a cocartesian fibration (Proposition 5.1.4.14). For each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a discrete simplicial set (Remark 4.2.3.17). It follows that the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ of Construction 5.2.3.2 can be regarded as a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets (regarded as a full subcategory of the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). We will denote this functor by $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ and refer to it as the covariant transport representation of $U$.

Example 5.6.1.3 (The Monodromy Representation). Let $f: X \rightarrow S$ be a covering map of topological spaces. Then $\operatorname{Sing}_{\bullet }(\pi ): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a covering map of simplicial sets (Proposition 3.1.4.9). Applying Construction 5.6.1.2 to $\operatorname{Sing}_{\bullet }(\pi )$, we obtain a functor from the fundamental groupoid $\pi _{\leq 1}(S)$ to the category of sets, which we will denote by $\operatorname{Tr}_{X/S}: \pi _{\leq 1}(S) \rightarrow \operatorname{Set}$ and refer to as the monodromy representation of $f$. Concretely, it is given on objects by the formula $\operatorname{Tr}_{X/S}(s) = \{ s\} \times _{S} X$.

Example 5.6.1.4. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ be a morphism of simplicial sets, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be the simplicial set given by Definition 5.5.4.1, so that the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering map (see Example 5.5.4.8). Then the covariant transport representation $\operatorname{Tr}_{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} / \operatorname{\mathcal{C}}}$ is canonically isomorphic to the functor $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ induced by $\mathscr {F}$.

Proposition 5.6.1.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets, let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ be the covariant transport representation of Construction 5.6.1.2, and let us abuse notation by identifying $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ with a morphism of simplicial sets from $\operatorname{\mathcal{C}}$ to $\operatorname{N}_{\bullet }(\operatorname{Set})$. Then there is a canonical isomorphism $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$.

Proof. Every vertex $X \in \operatorname{\mathcal{E}}$ can be regarded as an element of the set $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( U(X) )$, and the construction $(X \in \operatorname{\mathcal{E}}) \mapsto ( \operatorname{\mathcal{E}}_{U(X)}, X)$ determines a functor $\widetilde{\operatorname{Tr}}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \operatorname{Set}_{\ast }$. Let us identify $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ with a morphism of simplicial sets from $\operatorname{\mathcal{C}}$ to $\operatorname{N}_{\bullet }(\operatorname{Set})$ and $\widetilde{\operatorname{Tr}}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ 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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\operatorname{Tr}}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } & \operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } & \operatorname{N}_{\bullet }( \operatorname{Set}) } \]

which we can identify with a morphism $V: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$. We will complete the proof by showing that $V$ is an isomorphism. Since $U$ and the projection map $\int _{\operatorname{\mathcal{C}}} \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ are both left covering maps (see Example 5.5.4.8), it follows that $V$ is a left covering map. 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 5.6.1.6. 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 Construction 5.6.1.2 supplies an equivalence of categories

\[ \operatorname{LCov}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) \quad \quad (U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}, \]

with homotopy inverse given by the construction

\[ ( \mathscr {F} \in \operatorname{Fun}(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) ) \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F} \in \operatorname{LCov}(\operatorname{\mathcal{C}}). \]

Corollary 5.6.1.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

There exists a morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Set})$ and an isomorphism $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which carries $U$ to the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.

$(2)$

There exists a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{N}_{\bullet }( \operatorname{\mathcal{D}}) \ar [d]^{ \operatorname{N}_{\bullet }(V) } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ), } \]

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).

$(3)$

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).

$(4)$

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).

$(5)$

The morphism $U$ is a left covering map of simplicial sets (in the sense of Definition 4.2.3.8).

Proof. Every morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Set})$ determines a functor of categories $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$, and Example 5.5.4.8 supplies a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d] & \operatorname{N}_{\bullet }( \int _{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\mathscr {F}} ) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ). } \]

The implication $(1) \Rightarrow (2)$ follows from Remark 5.5.1.9, the implication $(2) \Rightarrow (3)$ follows from Remark 4.2.3.6, the implication $(3) \Rightarrow (4)$ is trivial, and the implication $(4) \Rightarrow (5)$ follows by combining Remark 4.2.3.15 with Proposition 4.2.3.16. The implication $(5) \Rightarrow (1)$ follows from Proposition 5.6.1.5. $\square$

Corollary 5.6.1.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then $U$ is a left covering map (in the sense of Definition 4.2.3.8) if and only if the following pair of conditions is satisfied:

$(1)$

The induced map $\mathrm{h} \mathit{U}: \mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a left covering functor (in the sense of Definition 4.2.3.1).

$(2)$

The diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{E}}} ) \ar [d]^{\operatorname{N}_{\bullet }( \mathrm{h} \mathit{U} )} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}) } \]

is a pullback square.

Proof. The sufficiency of conditions $(1)$ and $(2)$ follows from Proposition 4.2.3.16 and Remark 4.2.3.15. To prove the converse, assume that $U$ is a left covering map. By virtue of Corollary 5.6.1.7, we may assume that $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for some morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$. Let us abuse notation by identifying $\mathscr {F}$ with a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets. Using Proposition 5.5.5.2, we can identify $\mathrm{h} \mathit{\operatorname{\mathcal{E}}}$ with the category of elements $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathscr {F}$ of Construction 5.5.1.1. Condition $(1)$ now follows from Remark 5.5.1.9, and condition $(2)$ by combining Example 5.5.4.8 with Remark 5.5.4.18. $\square$

Corollary 5.6.1.9. Let $\operatorname{\mathcal{C}}$ be a category. Then:

  • Construction 5.5.1.1 determines a fully faithful functor

    \[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}, \]

    whose essential image consists of the left covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.

  • Variant 5.5.1.2 determines a fully faithful functor

    \[ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{/\operatorname{\mathcal{C}}} \quad \quad \mathscr {F} \mapsto \int ^{\operatorname{\mathcal{C}}} \mathscr {F}, \]

    whose essential image consists of the right covering functors $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.

Corollary 5.6.1.10. 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

\[ \{ \textnormal{Covering maps $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} \rightarrow \operatorname{Fun}( \pi _{\leq 1}(\operatorname{\mathcal{C}}), \operatorname{Set}). \]