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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{universal covering space} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{Definition}{In point-set topology}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{InCohesiveHomotopyTheory}{In cohesive homotopy theory}\dotfill \pageref*{InCohesiveHomotopyTheory} \linebreak \noindent\hyperlink{in_the_petit_toposes_over_the_space}{In the petit $\infty$-toposes over the space}\dotfill \pageref*{in_the_petit_toposes_over_the_space} \linebreak \noindent\hyperlink{Example}{Examples}\dotfill \pageref*{Example} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{Definition}{}\subsubsection*{{In point-set topology}}\label{Definition} The following is the classical discussion of universal covering spaces in [[point-set topology]]. \begin{prop} \label{SimplyConnectedCoveringUniqueness}\hypertarget{SimplyConnectedCoveringUniqueness}{} \textbf{([[essentially unique|essential uniqueness]] of simply connected covering spaces)} Let $X$ be a [[topological space]] which is \begin{enumerate}% \item [[path-connected topological space|path-connected]], \item [[locally path-connected topological space|locally path connected]]. \end{enumerate} Then if $E_i \overset{p_i}{\to} X$ are two [[covering spaces]] over $X$, $i \in \{1,2\}$, which are both [[path-connected topological space|path-connected]] and [[simply connected topological space|simply connected]], then they are [[isomorphism|isomorphic]] as covering spaces. \end{prop} \begin{proof} Since both $E_1$ and $E_2$ are simply connected, the assumption of the lifting theorem for covering spaces is satisfied (\href{covering+space#TheTheoremLifting}{this prop.}). This says that there are horizontal continuous function making the following diagrams commute: \begin{displaymath} \itexarray{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X } \phantom{AAAAAA} \itexarray{ E_2 && \overset{g}{\longrightarrow} && E_1 \\ & {}_{\mathllap{p_2}}\searrow && \swarrow_{\mathrlap{p_1}} \\ && X } \end{displaymath} \begin{displaymath} \itexarray{ E_i && \overset{id}{\longrightarrow} && E_i \\ & {}_{\mathllap{p_i}}\searrow && \swarrow_{\mathrlap{p_i}} \\ && X } \end{displaymath} and that these are \emph{unique} once we specify the image of a single point, which we may freely do (in the given fiber). So if we pick any point $x \in X$ and $\hat x_1 \in E_1$ with $p(\hat x) = x$ and $\hat x_2 \in E_2$ with $p(\hat x_2) = x$ and specify that $f(\hat x_1) = \hat x_2$ and $g(\hat x_2) = \hat x_1$ then uniqueness applied to the composites implies $f \circ g = id_{E_{2}}$ and $g \circ f = id_{E_1}$. \end{proof} \begin{defn} \label{CoveringUniversal}\hypertarget{CoveringUniversal}{} \textbf{(universal covering space)} Let $X$ be a [[topological space]] which is \begin{enumerate}% \item [[path-connected topological space|path-connected]], \item [[locally path-connected topological space|locally path connected]]. \end{enumerate} Then a [[path-connected topological space|path-connected]] and [[simply connected topological space|simply connected]] [[covering space]], is called [[generalized the|the]] \emph{universal covering space} of $X$. This is well-defined, if it exists, up to [[isomorphism]], by prop. \ref{SimplyConnectedCoveringUniqueness}. \end{defn} \begin{prop} \label{ReconstructCoveringForFreeAndTransitiveMonodromyRepresentation}\hypertarget{ReconstructCoveringForFreeAndTransitiveMonodromyRepresentation}{} \textbf{([[universal covering space]] reconstructed from [[free action|free]] and [[transitive action|transitive]] [[fundamental group]] [[representation]])} Let $X$ be a topological space which is \emph{[[well-connected space|well-connected]]} in that it is \begin{itemize}% \item [[path-connected space|path-connected]], \item [[locally path-connected space|locally path-connected]] \item and [[semi-locally simply-connected space|semi-locally simply-connected]] \end{itemize} Then a universal covering space of $X$ (def. \ref{CoveringUniversal}) exists. \end{prop} \begin{proof} By \href{covering+space#CoveringConnectivityViaMonodromy}{this prop.} the covering space is connected and simply connected precisely if its [[monodromy]] representation is free and transitive. By the [[fundamental theorem of covering spaces]] every [[permutation representation]] of the [[fundamental groupoid]] $\Pi_1(X)$ arises as the monodromy of some covering space. Hence it remains to see that a free and transitive representation of $\Pi_1(X)$ exists. Let $x \in X$ be any point, then $Hom_{\Pi_1(X)}(x,-)$ is such a representation. \end{proof} This is a functorial construction of a universal covering spaces \begin{displaymath} Top_*^{wc} \longrightarrow Cov_* \end{displaymath} where $Top_*^{wc}$ denotes the [[full subcategory]] of [[pointed topological spaces]] $Top_*$ on the well-connected spaces and $Cov$ is the subcategory of $Top_*^2$ of pointed maps of spaces with objects the covering space maps. Specifically, if $X$ is a space with basepoint $x_0$, we define $X^{(1)}$ to be the space whose points are homotopy classes of paths in $X$ starting at $x_0$, with the projection $X^{(1)}\to X$ projecting to the endpoint of a path. We can equip this set $X^{(1)}\to X$ with a topology coming from $X$ so that it becomes a universal covering space as above. As described at [[covering space]], under the correspondence between covering spaces and $\Pi_1(X)$-actions, the space $X^{(1)}$ corresponds to the ``regular representation'' of $\Pi_1(X)$. \begin{uremark} This [[regular representation]] can be seen to arise by taking a [[category of elements]] in the same way that the regular representation of a group is gotten by taking its action on itself: we can see that the \emph{universal covering groupoid} $\Pi(X)^{(1)}$ in the [[slice category]] $\mathbf{Grpd}/\Pi(X)$ (see the universal covering $\infty$-groupoid below) is just the category of elements of the action of $\Pi(X)$ on itself, and can be topologized in a natural way by lifting the topology on $X$ along the canonical projection $\Pi(X)^{(1)} \to \Pi(X)$; decategorifying this yields $X^{(1)}$. \end{uremark} \hypertarget{InCohesiveHomotopyTheory}{}\subsubsection*{{In cohesive homotopy theory}}\label{InCohesiveHomotopyTheory} We describe now how the universal cover construction may be understood from the [[nPOV]], using a [[fragment]] of [[cohesive homotopy theory]]. The basic idea is that the universal cover of a space $X$ is the [[homotopy fiber]] of the canonical morphism \begin{displaymath} X \longrightarrow \Pi_1(X) \end{displaymath} from $X$ to its [[fundamental groupoid]], which exists if both objects are regarded as ``[[topological ∞-groupoids]]''. We may think of this as a precise way of the intuitive idea of ``[[forcing]] $\Pi_1(X)$ to become trivial in a universal way''. \begin{example} \label{ExamplesOfCohesiveToposesOfTopologicalGroupoids}\hypertarget{ExamplesOfCohesiveToposesOfTopologicalGroupoids}{} \textbf{(cohesive higher toposes of topological groupoids)} Let $\mathcal{S}$ be some [[∞-cohesive site]] of [[spaces]] and consider \begin{displaymath} \mathbf{H} \coloneqq Sh_\infty(\mathcal{S}) \end{displaymath} the corresponding [[cohesive (∞,1)-topos]] over this site. This is the [[(∞,1)-category]] of ``[[topological ∞-groupoids]]'' modeled on $\mathcal{S}$. A canonical choice relating to the traditional discussion would be $\mathcal{S} = Top^\kappa_{lcont}$ a [[small category|small]] [[full subcategory]] of [[Top]] on [[locally contractible topological spaces]], in which case the objects of $\mathbf{H}$ might be called ``[[locally contractible topological ∞-groupoids]]''. A more restrictive choice would be $\mathcal{S} =$ [[CartSp]], in which case the objects of $\mathbf{H}$ might be called [[Euclidean-topological ∞-groupoids]]. In fact for the discussion of just universal [[covering spaces]] as opposed to the higher stages in the [[Whitehead tower]] it would be sufficient and more natural to take $\mathcal{S}$ the full subcategory of [[locally simply connected topological spaces]] and consider just the [[(2,1)-category]] $\mathbf{H}$ of [[stacks]] over this site. \end{example} But the following discussion is completely formal and applies globally to all such realizations. Namely all we need is that \begin{defn} \label{AssumptionOnH}\hypertarget{AssumptionOnH}{} Assume that \begin{enumerate}% \item $\mathbf{H}$ is a [[cohesive (∞,1)-topos|cohesive (n,1)-topos]] \begin{displaymath} \mathbf{H} \itexarray{ \overset{\Pi}{\longrightarrow} \\ \overset{\Delta}{\longleftarrow} \\ \overset{\Gamma}{\longrightarrow} \\ \overset{\nabla}{\longleftarrow} } \mathbf{B} \end{displaymath} over the given [[base (∞,1)-topos|base (n,1)-topos]] of [[n-groupoids]]. \item such that its [[shape modality]] preserves [[homotopy fiber products]] over [[discrete objects]] (objects in the essential image of $\Delta$). \end{enumerate} We write \begin{displaymath} ʃ \coloneqq \Delta \Pi \end{displaymath} for the induced [[shape modality]] and \begin{displaymath} X \overset{\eta_X}{\longrightarrow} ʃ X \end{displaymath} for its [[unit of an adjunction|unit]] morphism ona given object $X$. \end{defn} \begin{example} \label{}\hypertarget{}{} Assumption \ref{AssumptionOnH} is satisfied for the case of $(\infty,1)$-toposes over an [[∞-cohesive site]] and for $(n,1)$-toposes by an $n$-cohesive site as in example \ref{ExamplesOfCohesiveToposesOfTopologicalGroupoids}: by \href{cohesive+infinity-topos#PiPreservesPullbacksOverDiscretes}{this prop.}. \end{example} Now consider $X \in \mathbf{H}$ any object, for instance a [[topological space]] regarded as a [[0-truncated object|0-truncated]] [[topological infinity-groupoid]]. Assume, just for ease of discussion, that $X$ is \emph{geometrically connected} in that the [[0-truncated object|0-truncation]] of its [[shape]] is [[contractible]]: \begin{displaymath} \tau_0(ʃ X) \simeq \ast \,. \end{displaymath} Using the [[axiom of choice]] in the base topos $\mathbf{H}$ we choose a point \begin{displaymath} \ast \to \tau_1(ʃ X) \,. \end{displaymath} \begin{defn} \label{}\hypertarget{}{} The \emph{universal cover} of $X$ is the [[homotopy pullback]] in \begin{displaymath} \itexarray{ \hat X &\longrightarrow& \ast \\ \downarrow &(pb)& \downarrow \\ X &\underset{L_{\tau_1} \circ \eta_X}{\longrightarrow}& \tau_1(ʃ X) } \,. \end{displaymath} \end{defn} The [[universal property]] of $\hat X$ is immediate from the abstract setup: \begin{enumerate}% \item $\hat X$ is simply connected, in that $\tau_1 ʃ \hat X \simeq \ast$ This is because $ʃ X$ is discrete by construction, and hence so is $\tau_1(ʃ X)$. So by assumpotion \ref{AssumptionOnH} applying $ʃ$ to the above square yields another homotopy pullback of the form \begin{displaymath} \itexarray{ ʃ \hat X &\longrightarrow& \ast \\ \downarrow &(pb)& \downarrow \\ ʃ X &\underset{ʃ \eta_X}{\longrightarrow}& \tau_1(ʃ X) } \,. \end{displaymath} Now the [[long exact sequence of homotopy groups]] (in $\mathbf{B}$) applied to this [[homotopy fiber sequence]] is of the form \begin{displaymath} 0 \to \pi_1(ʃ \hat X) \longrightarrow \pi_1(ʃ X) \overset{=}{\longrightarrow} \pi_1(\tau_1 ʃ X) \to \cdots \end{displaymath} which implies that $\pi_1( ʃ \hat X )$ and hence $\tau_1 ʃ \hat X$ is indeed trivial. \item Let $E \longrightarrow X$ be any other object of $\mathbf{H}_{/X}$ such that $\tau_1(ʃ E) \simeq \ast$, then there is a morphism \begin{displaymath} \itexarray{ E && \longrightarrow && \hat X \\ & \searrow && \swarrow \\ && X } \end{displaymath} This is because the naturality of the shape unit and of the truncation unit gives a homotopy commuting square of the form \begin{displaymath} \itexarray{ E &\overset{\eta_E}{\longrightarrow}& ʃ E &\overset{}{\longrightarrow}& \tau_1(ʃ E ) \simeq \ast \\ \downarrow && && \downarrow \\ X &\underset{\eta_X}{\longrightarrow}& ʃ X &\longrightarrow& \tau_1 ʃ X } \end{displaymath} and thus a [[cone]] over the diagram which defines $\hat X$ via its [[universal property]]. \end{enumerate} This shows that $\hat X$ is the universal cover on abstract grounds. We may also check explicitly that $\hat X$ is given as the space of homotopy classes of [[paths]] in $X$ from the given basepoint. To that end we use that, at least over the site [[CartSp]], the [[shape modality|shape]] of $X$ is represented by the topological [[path ∞-groupoid]]. See at \emph{[[shape via cohesive path ∞-groupoid]]}. \begin{quote}% The following is old material that deserves to be harmonized a bit more with the above stuff. \end{quote} \begin{uprop} Let $X$ be a suitably well behaved pointed space. The universal cover $X^{(1)}$ of $X$ is (equivalent to) the [[homotopy fiber]] of $X \to \Pi(X)$ in the [[(∞,1)-category]] $\mathbf{H} = Sh_{(\infty,1)}(Top_{cg})$ of [[topological ∞-groupoid]]s. In other words, the [[principal ∞-bundle]] classified by the [[cocycle]] $X \to \Pi_1(X)$ is the universal cover $X^{(1)}$: we have a [[homotopy pullback]] square \begin{displaymath} \itexarray{ X^{(1)} &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \Pi(X) } \,. \end{displaymath} \end{uprop} [[Urs Schreiber]]: may need polishing. \begin{proof} We place ourselves in the context of [[topological ∞-groupoid]]s and regard both the space $X$ as well as its [[schreiber:homotopy ∞-groupoid]] $\Pi(X)$ and its truncation to the [[fundamental groupoid]] $\Pi_1(X)$ as objects in there. The canonical morphism $X \to \Pi(X)$ hence $X \to \Pi_1(X)$ given by the inclusion of constant paths may be regarded as a [[cocycle]] for a $\Pi(X)$-[[principal ∞-bundle]], respectively for a $\Pi_1(X)$-principal bundle. Let $\pi_0(X)$ be the set of connected components of $X$, regarded as a topological $\infty$-groupoid, and choose any [[section]] $\pi_0(X) \to \Pi(X)$ of the projection $\Pi(X) \to \pi_0(X)$. Then the $\Pi(X)$-principal $\infty$-bundle classified by the [[cocycle]] $X \to \Pi(X)$ is its [[homotopy fiber]], i.e. the [[homotopy pullback]] \begin{displaymath} \itexarray{ UCov(X) &\to& \pi_0(X) \\ \downarrow && \downarrow \\ X &\to& \Pi(X) } \,. \end{displaymath} We think of this topological $\infty$-groupoid $UCov(X)$ as the \textbf{universal covering $\infty$-groupoid} of $X$. To break this down, we check that its [[decategorification]] gives the ordinary universal covering space: for this we compute the [[homotopy pullback]] \begin{displaymath} \itexarray{ UCov_1(X) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{x}} \\ X &\to& \Pi_1(X) } \,, \end{displaymath} where we assume $X$ to be connected with chosen baspoint $x$ just to shorten the exposition a little. By the laws of [[homotopy pullback]]s in general and [[homotopy fiber]]s in particular, we may compute this as the ordinary [[pullback]] of a weakly equivalent diagram, where the point $*$ is resolved to the [[generalized universal bundle|universal]] $\Pi_1(X)$-principal bundle \begin{displaymath} \mathbf{E}_x \Pi_1(X) = T_x \Pi_1(X) \,. \end{displaymath} \begin{quote}% (More in detail, what we do behind the scenes is this: we regard the diagram as a diagram in the \emph{global} [[model structure on simplicial presheaves]] on [[Top]]. In there all our topological groupoids are fibrant, hence all we have to ensure is that one of the morphisms of the diagram becomes a fibration, which is what the passage to $\mathbf{E}_x \Pi_1(X)$ achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in $\infty$-prestacks. Then by left exactness of $\infty$-stackification, the image of that in $\infty$-stacks is still a homotopy pullback. ) \end{quote} The topological groupoid $\mathbf{E}_x \Pi_1(X)$ has as objects homotopy classes rel endpoints of paths in $X$ starting at $x$ and as morphisms $\kappa : \gamma \to \gamma'$ it has commuting triangles \begin{displaymath} \itexarray{ && x \\ &{}^{\mathllap{\gamma}}\swarrow && \searrow^{\mathrlap{\gamma'}} \\ y &&\stackrel{\kappa}{\to}&& y' } \end{displaymath} in $\Pi_1(X)$. The topology on this can be deduced from thinking of this as the [[pullback]] \begin{displaymath} \itexarray{ \mathbf{E}_x \Pi_1(X) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Pi_1(X)^I &\stackrel{d_0}{\to}& \Pi_1(X) } \end{displaymath} in simplicial presheaves on [[Top]]. Unwinding what this means we find that the open sets in $Mor(\mathbf{E}_x \Pi_1(X))$ are those where the endpoint evaluation produces an open set in $X$. Now it is immediate to read off the homotopy pullback as the ordinary pullback \begin{displaymath} \itexarray{ UCov_1(X) &\to& \mathbf{E}_x \Pi_1(X) \\ \downarrow && \downarrow \\ X &\to& \Pi_1(X) \,. } \end{displaymath} Since $X$ is categorically discrete, this simply produces the space of objects of $\mathbf{E}_x \Pi_1(X)$ over the points of $X$, which is just the space of all paths in $X$ starting at $x$ with the projection $UCov_1(X) \to X$ being endpoint evaluation. This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in \begin{itemize}% \item \emph{\href{http://www.mathreference.com/at-cov,build.html}{Covering spaces, Building a universal cover}} , \href{http://www.mathreference.com/}{Math Reference Project} \end{itemize} \end{proof} \hypertarget{in_the_petit_toposes_over_the_space}{}\subsubsection*{{In the petit $\infty$-toposes over the space}}\label{in_the_petit_toposes_over_the_space} On the other hand, we can view a space $X$ as the little $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ of $(\infty,1)$-sheaves on $X$. If $X$ is locally connected and locally simply connected in the ``coverings'' sense, then $Sh(X)$ is locally 1-connected. In fact, for the construction of the universal cover we require only the (2,1)-topos $Sh_{(2,1)}(X)$ of sheaves (stacks) of [[groupoids]] on $X$, so we will work in that context because it is simpler. The construction can be adapted, however, to produce a ``universal cover'' of any locally 1-connected $(\infty,1)$-topos. Let $E$ be any (2,1)-topos which is locally 1-connected. This means that in the unique [[global sections]] [[geometric morphism]] $(E^*,E_*)\colon E\to Gpd$, the functor $E^*$ has a left adjoint $E_!\colon E \to Gpd$, which is automatically $Gpd$-[[indexed functor|indexed]]. The [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|fundamental groupoid]] of $E$ is defined to be $\Pi_1(E)\coloneqq E_!(*)$, where $*$ is the [[terminal object]] of $E$. As discussed [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|here]] in the $(\infty,1)$-case, the construction of $\Pi_1(E)$ is a left adjoint to the inclusion of groupoids into locally 1-connected (2,1)-toposes (which sends $G\mapsto Gpd/G \simeq Gpd^G$). Thus we have a geometric morphism $E\to \Pi_1(E)$ (where we regard $\Pi_1(E)$ as the (2,1)-topos $Gpd^{\Pi_1(E)}$). Suppose, for simplicity, that $E$ is connected. Then $\Pi_1(E)$ is also connected, and so we have an essentially unique functor $*\to \Pi_1(E)$. We define the \textbf{universal cover} of $E$ to be the pullback (2,1)-topos: \begin{displaymath} \itexarray{ E^{(1)} & \to & E \\ \downarrow & & \downarrow \\ * & \to & \Pi_1(E)} \end{displaymath} (where of course $*$ denotes the terminal (2,1)-topos $Gpd$). Now observe that $*\to \Pi_1(E)$ is a [[local homeomorphism of toposes]], since we have $Gpd \simeq Gpd/\Pi_1(E)/(*\to \Pi_1(E))$. Since local homeomorphisms of toposes are stable under pullback, $E^{(1)}\to E$ is also a local homeomorphism, i.e. there exists an object $\widetilde{E}\in E$ and an equivalence $E^{(1)} \simeq E/\widetilde{E}$ over $E$. Moreover, it is not hard to see that $\widetilde{E}$ can be identified with the pullback \begin{displaymath} \itexarray{ \widetilde{E} & \to & * \\ \downarrow & & \downarrow \mathrlap{\eta} \\ * & \to & E^*(\Pi_1(E)) \mathrlap{= E^*(E_!(*))}} \end{displaymath} in $E$. Note that the bottom map $*\to E^*(E_!(*))$ is $E^*$ applied to the unique map $*\to E_!(*)$, while the right-hand map is the unit of the adjunction $E_!\dashv E^*$. In order to see that this is a sensible definition, we first observe that $E^{(1)}$ is itself locally 1-connected (since it is etale over $E$). Moreover, it is actually 1-connected, which is equivalent to saying that $E_!(\widetilde{E}) = *$. This is because the ``Frobenius reciprocity'' condition for the adjunction $E_!\dashv E^*$ (which is equivalent to saying that $E_!$ is $Gpd$-indexed) applied to the defining pullback of $\widetilde{E}$ implies that we also have a pullback \begin{displaymath} \itexarray{ E_!(\widetilde{E}) & \to & E_!(*) \\ \downarrow & & \downarrow \mathrlap{id} \\ * & \to & E_!(*)} \end{displaymath} which clearly implies that $E_!(\widetilde{E}) = *$. Thus, $E^{(1)}$ is a connected and simply connected space with a local homeomorphism to $E$, but is it a covering space? In other words, is it locally trivial? Since we have supposed that $E$ is locally 1-connected, as a (2,1)-category it can be generated by 1-connected objects, i.e. objects $U$ such that $E_!(U)\simeq *$. In particular, we have a 1-connected object $U$ and a [[regular 1-epic]] $U\to *$. We claim that if $U$ is any 1-connected object of $E$, then $\widetilde{E}$ is trivialized (or split) over $U$, in that $U\times \widetilde{E}$ is equivalent, over $U$, to $U\times E^*S$ for some $S\in Gpd$. For pulling back the defining pullback to $U$, we obtain \begin{displaymath} \itexarray{ U\times \widetilde{E} & \to & U \\ \downarrow & & \downarrow \mathrlap{U\times \eta} \\ U & \to & U\times E^*(\Pi_1(E)).} \end{displaymath} But $U\times E^*(\Pi_1(E)) \cong (E/U)^* (\Pi_1(E))$, so to give a map $U \to (E/U)^* (\Pi_1(E))$ over $U$ is the same as to give a map $(E/U)_!(*) \to \Pi_1(E)$ in $Gpd$. But $(E/U)_!(*)\simeq *$, since $U$ is 1-connected, and $\Pi_1(E)$ is connected, so there is only one such morphism. Therefore, the two maps $U\to U\times E^*(\Pi_1(E))$ in the pullback above are in fact the same, and in particular both are the pullback to $E/U$ of the map $*\to \Pi_1(E)$. Thus, $U\times \widetilde{E}$ is equivalent to $(E/U)^*(S) \cong U\times E^*(S)$, where $S= \Omega(\Pi_1(E))$ is the loop object of $\Pi_1(E)$, i.e. what we might call the fundamental \emph{group} of the connected (2,1)-topos $E$. Therefore, since $\widetilde{E}$ is trivialized over any 1-connected object, and $E$ is generated by 1-connected objects, $\widetilde{E}$ is locally trivial. Moreover, since $*$ is a [[discrete object]] of $E$, so is $\widetilde{E}$. Thus, if we specialize all this to the case $E=Sh_{(2,1)}(X)$ of (2,1)-sheaves on a topological space, then we conclude that $\widetilde{E}$ is an honest 1-sheaf on $X$ which, when regarded as a local homeomorphism over $X$, is locally trivial (hence a covering space), connected, and 1-connected---i.e. a universal cover of $X$. \hypertarget{Example}{}\subsection*{{Examples}}\label{Example} \begin{example} \label{UniversalCoveringOfCircleRealLine}\hypertarget{UniversalCoveringOfCircleRealLine}{} \textbf{([[real line]] is universal covering of [[circle]])} Let \begin{enumerate}% \item $\mathbb{R}^1$ be the [[real line]] with its [[Euclidean space|Euclidean]] [[metric topology]]; \item $S^1 \coloneqq \left\{ x\in \mathbb{R}^2 \;\vert\; {\Vert x\Vert} = 1 \right\} \subset \mathbb{R}^2$ be the [[circle]] with its [[subspace topology]] induced from the [[Euclidean plane]]. \end{enumerate} Consider the [[continuous function]] \begin{displaymath} \itexarray{ \mathbb{R}^1 &\overset{p}{\longrightarrow}& S^1 \subset \mathbb{R}^2 \\ t &\mapsto& (\cos(2\pi t), \sin(2\pi t)) } \,. \end{displaymath} This exhibits the universal covering space (def. \ref{CoveringUniversal}) of the circle. \end{example} \begin{proof} Let $p \in S^1$ be any point. It is clear that we have a [[homeomorphism]] of the form \begin{displaymath} \itexarray{ S^1 \setminus p &\overset{\simeq}{\longrightarrow}& (0, 1) } \,. \end{displaymath} and hence a [[homeomorphism]] of the form \begin{displaymath} \itexarray{ S^1 \times Disc(\mathbb{Z}) &\simeq& (0,1) \times Disc(\mathbb{Z}) &\overset{\simeq}{\longrightarrow}& p^{-1}(S^1 \setminus \{p\}) \\ && (t,n) &\mapsto& (cos(2\pi n t), \sin(2\pi n t)) } \,. \end{displaymath} Now for $p_1 \neq p_2$ two distinct point in $S^1$, their [[complements]] constitute an [[open cover]] \begin{displaymath} \left\{ S^1 \setminus p_i \subset S^1 \right\}_{i \in \{1,2\}} \end{displaymath} and so this exhibits $p \colon \mathbb{R}^1 \to S^1$ as being covering spaces. Now \begin{enumerate}% \item $S^1$ is [[path-connected topological space|path-connected]] and [[locally path-connected topological space|locally path connected]] (\href{LocallyPathConnectedCircle}{this example}); \item $\mathbb{R}^1$ is [[simply connected topological space|simply connected]] (\href{fundamental+group#EuclideanSpaceFundamentalGroup}{this example}). \end{enumerate} Therefore $p$ exhibits $\mathbb{R}^1$ as a universal covering space of $S^1$, by def. \ref{CoveringUniversal}. \end{proof} \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[fundamental theorem of covering spaces]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} See the references at \emph{[[covering space]]}. [[!redirects universal cover]] [[!redirects universal covers]] [[!redirects universal covering spaces]] [[!redirects universal cover of a topos]] [[!redirects universal cover of a (2,1)-topos]] [[!redirects universal cover of an (∞,1)-topos]] [[!redirects universal cover of an (infinity,1)-topos]] [[!redirects universal covering topos]] [[!redirects universal covering (2,1)-topos]] [[!redirects universal covering (∞,1)-topos]] [[!redirects universal covering (infinity,1)-topos]] [[!redirects universal covering groupoid]] \end{document}