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5.2.7 Covering Space Theory

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:

$(a)$

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 monodromy representations determines an equivalence of categories

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

The proof of $(a)$ requires some point-set topology; we defer a discussion to §. 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

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

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

$( \mathscr {F} \in \operatorname{Fun}(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) ) \mapsto \operatorname{\mathcal{C}}\times _{\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}) } \operatorname{N}_{\bullet }(\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathscr {F}) \in \operatorname{LCov}(\operatorname{\mathcal{C}})$

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:

$(1)$

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).

$(2)$

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).

$(3)$

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).

$(4)$

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

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

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

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

• Variant 5.2.6.2 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 5.2.7.6. 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

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