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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*{Euclidean-topological infinity-groupoid} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes} [[!include cohesive infinity-toposes - contents]] \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{ModelCategoryPresentation}{Model category presentation}\dotfill \pageref*{ModelCategoryPresentation} \linebreak \noindent\hyperlink{structures_in_the_cohesive_topos_}{Structures in the cohesive $(\infty,1)$-topos $ETop \infty Grpd$}\dotfill \pageref*{structures_in_the_cohesive_topos_} \linebreak \noindent\hyperlink{Groups}{Cohesive $\infty$-groups}\dotfill \pageref*{Groups} \linebreak \noindent\hyperlink{GeometricHomotopy}{Geometric homotopy and Galois theory}\dotfill \pageref*{GeometricHomotopy} \linebreak \noindent\hyperlink{PresentationOfPathGroupoid}{Paths and geometric Postnikov towers}\dotfill \pageref*{PresentationOfPathGroupoid} \linebreak \noindent\hyperlink{Cohomology}{Cohomology and principal $\infty$-bundles}\dotfill \pageref*{Cohomology} \linebreak \noindent\hyperlink{twisted_cohomology}{Twisted cohomology}\dotfill \pageref*{twisted_cohomology} \linebreak \noindent\hyperlink{WhiteheadTowers}{Universal coverings and geometric Whitehead towers}\dotfill \pageref*{WhiteheadTowers} \linebreak \noindent\hyperlink{PathInftinityGroupoids}{Path $\infty$-groupoid and geometric Postnikov towers}\dotfill \pageref*{PathInftinityGroupoids} \linebreak \noindent\hyperlink{homotopy_localization}{Homotopy localization}\dotfill \pageref*{homotopy_localization} \linebreak \noindent\hyperlink{idea_2}{Idea}\dotfill \pageref*{idea_2} \linebreak \noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{Euclidean-topological $\infty$-groupoid} is an [[∞-groupoid]] equipped with [[cohesive (∞,1)-topos|cohesion]] in the form of [[Euclidean topology]]. Examples of [[1-truncated]] type are [[topological groupoid]]s/[[topological stack]]s whose topologies are detectable by maps out of [[Euclidean topology|Euclidean topologies]], for instance [[internal groupoid]]s in [[topological manifold]]s. More generally, every [[simplicial topological space]] whose topology is degreewise detectable by Euclidean topologies canonically identifies with a Euclidean-topological $\infty$-groupoid. Various constructions with simplicial toppological spaces find their natural home in this [[(∞,1)-topos]]. For instance \begin{itemize}% \item [[geometric realization of simplicial topological spaces|geometric realization of simplicial topological manifolds]] is equivalently the image $\Pi(X)$ of the corresponding Euclidean-topological $\infty$-groupoid $X$ under the canonical [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]. \item topological [[simplicial principal bundle]]s over topological [[simplicial group]]s are the corresponding [[principal ∞-bundle]]s in $ETop\infty Grpd$ classified by its [[cohomology|internal cohomology]]. \end{itemize} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{udefn} Let [[CartSp]]${}_{top}$ be the [[site]] whose underlying [[category]] has as [[objects]] the [[Cartesian space]]s $\mathbb{R}^n$, $n \in \mathbb{N}$ equipped with the [[Euclidean topology]] and as [[morphism]]s the [[continuous maps]] between them; and whose [[coverage]] is given by [[good open cover]]s. \end{udefn} \begin{udefn} Define \begin{displaymath} ETop \infty Grpd := (\infty,1)Sh(CartSp_{top}) \end{displaymath} to be the [[(∞,1)-category of (∞,1)-sheaves]] on $CartSp_{top}$. \end{udefn} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{uprop} The [[(∞,1)-topos]] $ETop \infty Grpf$ is a [[cohesive (∞,1)-topos]]. \end{uprop} \begin{proof} The site [[CartSp]]${}_{top}$ an [[∞-cohesive site]]. See there for details. \end{proof} For completeness we record general properties of [[cohesive (∞,1)-topos]]es implied by this. \begin{uprop} $ETop\infty Grpd$ is \begin{itemize}% \item [[hypercomplete (∞,1)-topos|hypercomplete]]; \item of [[cohomological dimension]] 0; \item of [[homotopy dimension]] 0; \item of the [[shape of an (∞,1)-topos]] of the point. \end{itemize} \end{uprop} \begin{udef} We say that $ETop \infty Grpd$ defines \textbf{Euclidean-topological cohesion}. An object in $ETop \infty Grpd$ we call a \textbf{Euclidean-topological $\infty$-groupoid}. \end{udef} \begin{uprop} Write [[TopMfd]] for the category of [[topological manifold]]s. This becomes a [[large site]] with the [[open cover]] [[coverage]]. We have an [[equivalence of (∞,1)-categories]] \begin{displaymath} ETop\infty Grpd \simeq \hat Sh_{(\infty,1)}(TopMfd) \end{displaymath} with the [[hypercompletion]] of the [[(∞,1)-category of (∞,1)-sheaves]] on [[TopMfd]]. \end{uprop} \begin{proof} Since every [[topological manifold]] admits an [[open cover]] by [[open balls]] [[homeomorphic]] to a [[Cartesian space]] it follows that [[CartSp]]${}_{top}$ is a [[dense sub-site]] of $TopMfd$. Accordingly the [[categories of sheaves]] are [[equivalence of categories|equivalent]] \begin{displaymath} Sh(CartSp_{top}) \simeq Sh(TopMfd) \,. \end{displaymath} By the discussion at [[model structure on simplicial sheaves]] it follows that the [[hypercomplete (∞,1)-topos]]es over these sites are [[equivalence of (∞,1)-categories|equivalent]] \begin{displaymath} \hat Sh_{(\infty,1)}(CartSp_{top}) \simeq \hat Sh_{(\infty,1)}(TopMfd) \,. \end{displaymath} But by the \hyperlink{ETopInfGrpdAsCohesiveTopos}{above proposition} we have that before [[hypercompletion]] $Sh_{(\infty,1)}(CartSp_{top})$ is [[cohesive (∞,1)-topos|cohesive]]. This means that it is in particular a [[local (∞,1)-topos]]. By the discussion there, this means that it already coincides with its [[hypercompletion]], $Sh_{(\infty,1)}(CartSp_{top}) \simeq \hat Sh_{(\infty,1)}(CartSp_{top})$. \end{proof} \begin{udef} Write $Top_1$ for the 1-[[category]] of [[Hausdorff space|Hausdorff]] [[topological spaces]] and continuous maps. There is a canonical functor \begin{displaymath} j : Top_1 \to \tau_{\leq 0}ETop\infty Grpd \hookrightarrow ETop\infty Grpd \end{displaymath} given by sending a topological space $X$ to the 0-[[truncated]] [[(∞,1)-sheaf]] (= [[sheaf]]) on [[CartSp]]${}_{top}$ externally represented by $X$ under the embedding $CartSp_{top} \hookrightarrow Top$: \begin{displaymath} j(X) : (U \in CartSp_{top}) \mapsto Hom_{Top}(U,X) \in Set \hookrightarrow \infty Grpd \,. \end{displaymath} \end{udef} \begin{uprop} The functor $j$ exhibits [[TopMfd]] as a full [[sub-(∞,1)-category]] of $ETop\infty Grpd$ \begin{displaymath} TopMfd \hookrightarrow ETop\infty Grpd \,. \end{displaymath} \end{uprop} \begin{proof} With the \hyperlink{ToposOverTopMfd}{above proposition} this follows directly by the [[(∞,1)-Yoneda lemma]]. \end{proof} \hypertarget{ModelCategoryPresentation}{}\subsubsection*{{Model category presentation}}\label{ModelCategoryPresentation} We dicuss some aspects of the [[presentable (∞,1)-category|presentation]] of $ETop \infty Grpd$ by [[model category]] structures. \begin{uprop} Let $[CartSp_{top}^{op}, sSet]_{proj,loc}$ be the Cech-local projective [[model structure on simplicial presheaves]]. This is a [[presentable (∞,1)-category|presentation]] of $ETop \infty Grpd$ \begin{displaymath} ([CartSp_{top}^{op}, sSet]_{proj,loc})^\circ \simeq ETop \infty Grpd \,. \end{displaymath} Also the [[model structure on simplicial sheaves]] $sSh(CartSp_{{top})_{loc}$ is a presentation \begin{displaymath} (sSh(CartSp_{top})_{loc})^\circ \simeq ETop \infty Grpd \,. \end{displaymath} \end{uprop} \begin{proof} The first statement is a special case of the general discussion at [[model structure on simplicial presheaves]]. Similarly, by the general discussion at [[model structure on simplicial sheaves]] we have that this presents the [[hypercompletion]] of the [[(∞,1)-category of (∞,1)-sheaves]]. But by \hyperlink{PointLikePropertiesOfETopInfGrpd}{the above} $ETop\infty Grpd$ already is hypercomplete. \end{proof} Moreover: \begin{uprop} $ETop\infty Grpd$ is also the [[hypercompletion]] of the [[(∞,1)-topos]] presented by the local [[model structure on simplicial presheaves]] over all of [[Mfd]] (or over any [[small category|small]] [[dense sub-site]] such as for instance the full sub-category of manifolds bounded in size by some [[regular cardinal]]). \begin{displaymath} \hat{}([Mfd^{op}, sSet]_{proj,loc})^\circ \simeq ETop \infty Grpd \,. \end{displaymath} \end{uprop} \begin{proof} By the \hyperlink{ToposOverTopMfd}{above proposition}. \end{proof} While the model structures on simplicial presheaves over both sites present the same [[(∞,1)-category]], they lend themselves to different computations: the model structure over $CartSp_{top}$ has more fibrant objects and hence fewer cofibrant objects, while the model structure over $Mfd$ has more cofibrant objects and fewer fibrant objects. More specifically: \begin{uprop} Let $X \in [Mfd^{op}, sSet]$ be an object that is \emph{globally fibrant} , \emph{separated} and \emph{locally trivial}, meaning that \begin{enumerate}% \item $X(U)$ is an [[inhabited]] [[Kan complex]] for all $U \in Mfd$; \item for every [[covering]] $\{U_i \to U\}$ in [[Mfd]] the [[descent]] comparison morphism $X(U) \to [Mfd^{op}, sSet](C(\{U_i\}), X)$ is a [[full and faithful (∞,1)-functor]]; \item for [[contractible]] $U$ we have $\pi_0[Mfd^{op}, sSet](C(\{U_i\}), X) \simeq *$. \end{enumerate} Then the restriction of $X$ along $CartSp_{top} \hookrightarrow Mfd$ is a fibrant object in the local model structure $[CartSp_{top}^{op}, sSet]_{proj,loc}$. \end{uprop} \begin{proof} The fibrant objects in the local model structure are precisely those that are Kan complexes over every object and for which the descent morphism is an equivalence for all covers. The first condition is given by the first assumption. The second and third assumptions imply the second condition over contractible manifolds, such as the [[Cartesian space]]s. \end{proof} \begin{uexample} Let $G$ be a [[topological group]], regarded as the presheaf over [[Mfd]] that it [[representable functor|represents]]. Write $\bar W G$ (see the notation at [[simplicial group]]) for the simplicial presheaf on $Mfd$ given by the [[nerve]] of the [[topological groupoid]] $(G \stackrel{\to}{\to} *)$. (This is a presentation of the [[delooping]] of the [[0-truncated]] [[∞-group]] $G \in ETop\infty Grpd$, see the discussion \hyperlink{Groups}{below}. ) The fibrant [[resolution]] of $\bar W G$ in $[Mfd^{op}, sSet]_{proj,loc}$ is (the rectification of) its [[stackification]]: the [[stack]] $G Bund$ of topological $G$-[[principal bundle]]s. But the canonical morphism \begin{displaymath} \bar W G \to G Bund \end{displaymath} is a [[full and faithful functor]] (over each object $U \in Mfd$): it includes the single object of $\bar W G$ as the trivial $G$-[[principal bundle]]. The automorphism of the single object in $\bar W G$ over $U$ are $G$-valued [[continuous function]]s on $U$, which are precisely the automorphisms of the trivial $G$-bundle. Therefore this inclusion is full and faithful, the presheaf $\bar W G$ is a [[separated prestack]]. Moreover, it is locally trivial: every [[Cech cohomology|Cech cocycle]] for a $G$-bundle over a [[Cartesian space]] is equivalent to the trivial one. Equivalently, also $\pi_0 G Bund(\mathbb{R}^n) \simeq *$. Therefore $\bar W G$, when restricted $CartSp_{top}$, does become a fibrant object in $[CartSp_{top}^{op}, sSet]_{proj,loc}$. On the other hand, let $X \in Mfd$ be any non-contractible manifold. Since in the projective [[model structure on simplicial presheaves]] every representable is cofibrant, this is a cofibrant object in $[Mfd^{op}, sSet]_{proj,loc}$. However, it fails to be cofibrant in $[CartSp_{top}^{op}, sSet]_{proj,loc}$. Instead, there a cofibrant replacement is given by the [[Cech nerve]] $C(\{U_i\})$ of any [[good open cover]] $\{U_i \to X\}$. This yields two different ways to compute the first [[nonabelian cohomology]] \begin{displaymath} H^1_{ETop}(X,G) := \pi_0 ETop\infty Grpd (X, \mathbf{B}G) \end{displaymath} in $ETop\infty Grpd$ on $X$ with coefficients in $G$, as \begin{enumerate}% \item $\cdots \simeq \pi_0 [Mfd^{op}, sSet](X, G Bund) \simeq \pi_0 G Bund(X)$; \item $\cdots \simeq \pi_0 [CartSp_{top}^{op}, sSet](C(\{U_i\}), \bar W G) \simeq H^1_{Ch}(X,G)$. \end{enumerate} In the first case we need to construct the fibrant replacement $G Bund$. This amounts to computing $G$-cocycles = $G$-bundles over \emph{all} manifolds and then evaluate on the given one, $X$, by the [[2-Yoneda lemma]]. In the second case however we cofibrantly replace $X$ by a good open cover, and then find the [[Cech cohomology|Cech cocycles]] with coefficients in $G$ on that. For ordinary $G$-bundles the difference between the two computations may be irrelevant in practice, because ordinary $G$-bundles are very well understood. However for more general coefficient objects, for instance general topological [[simplicial group]]s $G$, the first approach requires to find the full [[∞-stackification]] to the [[∞-stack]] of all [[principal ∞-bundle]]s, while the second approach requires only to compute specific coycles over one specific base object. In practice the latter is often all that one needs. \end{uexample} \hypertarget{structures_in_the_cohesive_topos_}{}\subsection*{{Structures in the cohesive $(\infty,1)$-topos $ETop \infty Grpd$}}\label{structures_in_the_cohesive_topos_} We discuss what some of the general abstract look like in the model $ETop \infty Grpd$. As usual, write \begin{displaymath} (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : ETop \infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd \end{displaymath} for the defining quadruple of [[adjoint (∞,1)-functor]]s that refine the [[global section]] [[(∞,1)-geometric morphism]] to [[∞Grpd]]. \hypertarget{Groups}{}\subsubsection*{{Cohesive $\infty$-groups}}\label{Groups} By the general properties of [[cohesive (∞,1)-topos]]es with an [[∞-cohesive site]] of definition, every [[∞-group]] object is presented by a presheaf of [[simplicial group]]s. For $ETop\infty Grpd$ among these are the [[simplicial topological group]]s. See there for more details. \hypertarget{GeometricHomotopy}{}\subsubsection*{{Geometric homotopy and Galois theory}}\label{GeometricHomotopy} We discuss the realization of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] in $ETop \inft Grpd$. \begin{prop} \label{FundGroupoidOfParacompact}\hypertarget{FundGroupoidOfParacompact}{} Let $X$ be a [[paracompact topological space]] such that that $X$ admits a [[good open cover]] by [[open ball]]s (for instance a [[paracompact manifold]]). Then $\Pi(X) := \Pi(i(X)) \in \infty Grpd$ is equivalent to the standard [[fundamental ∞-groupoid]] of a [[topological space]] that is presented by the [[singular simplicial complex]] $Sing X$ \begin{displaymath} \Pi(X) \simeq Sing X \,. \end{displaymath} Equivalently, under [[geometric realization]] $\mathbb{L}|-| : \infty Grpd \to Top$ we have that there is a [[weak homotopy equivalence]] \begin{displaymath} X \simeq |\Pi(X)| \,. \end{displaymath} \end{prop} \begin{proof} By the discussion at [[∞-cohesive site]] we have an equivalence $\Pi(-) \simeq \mathbb{L} \lim_\to$ to the [[derived functor]] of the [[sSet]]-[[colimit]] functor $\lim_\to : [CartSp^{op}, sSet]_{proj,loc} \to sSet_{Quillen}$. To compute this derived functor, let $\{U_i \to X\}$ be a [[good open cover]] by [[open balls]], hence [[homeomorphic]]ally by [[Cartesian space]]. By goodness of the cover the [[Cech nerve]] $C(\coprod_i U_i \to X) \in [CartSp^{op}, sSet]$ is degreewise a [[coproduct]] of [[representable functor|representable]]s, hence a [[split hypercover]]. By the discussion at [[model structure on simplicial presheaves]] we have that in this case the canonical morphism \begin{displaymath} C(\coprod_i U_i \to X) \to X \end{displaymath} is a cofibrant [[resolution]] of $X$ in $[CartSp^{op}, sSet]_{proj,loc}$. Accordingly we have \begin{displaymath} \Pi(X) \simeq (\mathbb{L} \lim_\to) (X) \simeq \lim_\to C(\coprod_i U_i \to X) \,. \end{displaymath} Using the [[equivalence of categories]] $[CartSp^{op}, sSet] \simeq [\Delta^{op}, [CartSp^{op}, Set]]$ and that [[colimit]]s in [[presheaf categories]] are computed objectwise and finally using that the colimit of a [[representable functor]] is the point (an incarnation of the [[Yoneda lemma]]) we have that $\Pi(X)$ is presented by the [[Kan complex]] that is obtained by contracting in the [[Cech nerve]] $C(\coprod_i U_i)$ each open subset to a point. The classical [[nerve theorem]] asserts that this implies the claim. \end{proof} \begin{remark} \label{}\hypertarget{}{} We may regard [[Top]] itself as a [[cohesive (∞,1)-topos]]. $(\Pi_{Top}\dashv Disc_{Top} \dashv \Gamma_{Top} \dashv coDisc_{Top}) Top \stackrel{\simeq}{\to} \infty Grpd$. This is discussed at [[discrete ∞-groupoid]]. Using this the above proposition may be stated as saying that for $X$ a [[paracompact topological space]] that admits a [[good open cover]] we have \begin{displaymath} \Pi_{ETop\infty Grpd}(X) \simeq \Pi_{Top}(X) \,. \end{displaymath} \end{remark} \begin{prop} \label{FundGroupoidOfSimplicialParacompact}\hypertarget{FundGroupoidOfSimplicialParacompact}{} Let $X_\bullet$ be a [[good simplicial topological space]] that is degreewise [[paracompact topological space|paracompact]] and degreewise admits a [[good open cover]], regarded naturally as an object $X_\bullet \in Top^{\Delta^{op}} \to ETop \infty Grpd$. We have that the intrinsic $\Pi(X_\bullet) \in \infty Grpd$ coincides under [[geometric realization]] $\mathbb{L}|-| : \infty Grpd \stackrel{\simeq}{\to} Top$ with the ordinary [[geometric realization of simplicial topological spaces]] $|X_\bullet|_{Top^{\Delta^{op}}}$ \begin{displaymath} |\Pi(X_\bullet)| \simeq |X_\bullet|_{Top^{\Delta^{op}}} \,. \end{displaymath} \end{prop} \begin{proof} Write $Q$ for Dugger's cofibrant replacement functor on $[CartSp^{op}, sSet]_{proj,loc}$ (discussed at [[model structure on simplicial presheaves]]). On a simplicially constant simplicial presheaf $X$ it is given by \begin{displaymath} Q X := \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{U_0 \to \cdots \to U_n \to X} U_0 \right) \,, \end{displaymath} where the [[coproduct]] in the integrand of the [[coend]] is over all sequences of morphisms from [[representable functor|representable]]s $U_i$ to $X$ as indicated. On a general [[simplicial presheaf]] $X_\bullet$ it is given by \begin{displaymath} Q X_\bullet := \int^{[k] \in \Delta} \Delta[k] \cdot Q X_k \,, \end{displaymath} which is the simplicial presheaf that over any $\mathbb{R}^n \in CartSp$ takes as value the [[diagonal]] of the [[bisimplicial set]] whose $(n,r)$-entry is $\coprod_{U_0 \to \cdots \to U_n \to X_k} CartSp_{top}(\mathbb{R}^n,U_0)$. Since [[coend]]s are special [[colimit]]s, the colimit functor itself commutes with them and we find \begin{displaymath} \begin{aligned} \Pi(X_\bullet) & \simeq (\mathbb{L} \lim_\to) X_\bullet \\ & \simeq \lim_\to Q X_\bullet \\ & \simeq \int^{[n] \in \Delta} \Delta[k] \cdot \lim_\to (Q X_k) \,. \end{aligned} \end{displaymath} By the discussion at [[Reedy model structure]] this [[coend]] is a [[homotopy colimit]] over the simplicial [[diagram]] $\lim_\to Q X_\bullet : \Delta \to sSet_{Quillen}$ \begin{displaymath} \cdots \simeq hocolim_\Delta \lim_\to Q X_\bullet \,. \end{displaymath} By the \hyperlink{FundGroupoidOfParacompact}{above proposition} we have for each $k \in \mathbb{N}$ weak equivalences $\lim_\to Q X_k \simeq (\mathbb{L} \lim_\to) X_k \simeq Sing X_k$, so that \begin{displaymath} \begin{aligned} \cdots &\simeq hocolim_\Delta Sing X_k \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot Sing X_k \\ & \simeq diag Sing(X_\bullet)_\bullet \end{aligned} \,. \end{displaymath} By the discussion at [[geometric realization of simplicial topological spaces]], this maps to the [[homotopy colimit]] of the simplicial topological space $X_\bullet$, which is just its geometric realizaiton if it is proper. \end{proof} \hypertarget{PresentationOfPathGroupoid}{}\subsubsection*{{Paths and geometric Postnikov towers}}\label{PresentationOfPathGroupoid} We discuss the notion of \emph{\href{http://ncatlab.org/nlab/show/cohesive+%28infinity,1%29-topos+--+structures#Paths}{geometric path ∞-groupoids}} realized in $ETop\infty Grpd$. In the \hyperlink{GeometricRealization}{above} constructions of $\Pi(X)$ the actual paths are not explicit. We discuss here presentations of $\mathbf{\Pi}(X)$ in terms of actual paths. By prop. \ref{FundGroupoidOfParacompact} we have \begin{prop} \label{DiscretePathGroupoid}\hypertarget{DiscretePathGroupoid}{} Let $X$ be a a [[paracompact topological space]], regarded as an object of $ETop\infty Grpd$. Then $\mathbf{\Pi}(X)$ is presented by the constant [[simplicial presheaf]] \begin{displaymath} Disc Sing(X) \,:\, (U,[k]) \mapsto Hom_{Top}(\Delta^k, X) \,. \end{displaymath} \end{prop} Possibly more natural would seem to look at the topological Kan complex that remembers the topology on the spaces of paths: \begin{defn} \label{}\hypertarget{}{} For $X$ a paracompact topological space, define the simplicial presheaf \begin{displaymath} \mathbf{Sing} X : (U,[k]) \mapsto Hom_{Top}(U \times \Delta^k, X) \,. \end{displaymath} \end{defn} \begin{prop} \label{}\hypertarget{}{} Also $\mathbf{Sing} X$ is a presentation of $\mathbf{\Pi}(X)$ \end{prop} \begin{proof} For each fixed $U \in CartSp$ the inclusion of simplicial sets \begin{displaymath} Sing X \to \mathbf{Sing}(X)(U) \end{displaymath} is a [[weak homotopy equivalence]], since $U \in CartSp$ is [[contractible space|contractible]]. Therefore the inclusion of simplicial presheaves \begin{displaymath} Disc Sing X \to \mathbf{Sing} X \end{displaymath} is a weak equivalence in $[CartSp^{op}, sSet]_{proj}$. This implies the claim with prop. \ref{DiscretePathGroupoid}. \end{proof} \begin{remark} \label{}\hypertarget{}{} Typically one is interested in mapping out of $\mathbf{\Pi}(X)$. While it is clear that $Disc Sing X$ is cofibrant in $[CartSp^{op}, sSet]_{proj,loc}$, it is harder to determine the necessary [[resolution]]s of $\mathbf{Sing}X$. \end{remark} \hypertarget{Cohomology}{}\subsubsection*{{Cohomology and principal $\infty$-bundles}}\label{Cohomology} We dicuss aspects of the intrinsic [[cohomology]] of $E Top \infty Grpd$ and of the [[principal ∞-bundle]]s that it classifies. \begin{udef} Let $A \in$ [[∞Grpd]] be any [[discrete ∞-groupoid]]. Write $|A| \in$ [[Top]] for its [[geometric realization]]. For $X$ any [[topological space]], the [[nonabelian cohomology]] of $X$ with coefficients in $A$ is the set of [[homotopy]] classes of maps $X \to |A|$ \begin{displaymath} H_{Top}(X,A) := \pi_0 Top(X,|A|) \,. \end{displaymath} We say $Top(X,|A|)$ itself is the [[cocycle]] [[∞-groupoid]] for $A$-valued [[nonabelian cohomology]] on $X$. Similarly, for $X, \mathbf{A} \in ETop \infty Grpd$ two e-topological $\infty$-groupoids, write \begin{displaymath} H_{ETop}(X,\mathbf{A}) := \pi_0 ETop\infty Grpd(X,\mathbf{A}) \end{displaymath} for the intrinsic [[cohomology]] of $ETop \infty Grpd$ on $X$ with coefficients in $\mathbf{A}$. \end{udef} \begin{uprop} Let $A \in$ [[∞Grpd]], write $Disc A \in ETop \infty Grpd$ for the corresponding [[discrete ∞-groupoid|discrete topological ∞-groupoid]]. Let $X \in Top_1 \stackrel{i}{\hookrightarrow} ETop \infty Grpd$ be a [[paracompact topological space]] regarded as a 0-[[truncated]] Euclidean-topological $\infty$-groupoid. We have an [[isomorphism]] of cohomology sets \begin{displaymath} H_{Top}(X,A) \simeq H_{ETop}(X,Disc A) \end{displaymath} and in fact an [[equivalence in an (∞,1)-category|equivalence]] of [[cocycle]] [[∞-groupoid]]s \begin{displaymath} Top(X,|A|) \simeq ETop\infty Grpd(X, Disc A) \,. \end{displaymath} \end{uprop} \begin{proof} By the $(\Pi \dashv Disc)$-[[adjunction]] of the [[locally ∞-connected (∞,1)-topos]] $ETop \infty Grpd$ we have \begin{displaymath} ETop\infty Grpd(X, Disc A) \simeq \infty Grpd(\Pi(X), A) \underoverset{\simeq}{|-|}{\to} Top(|\Pi X|, |A|) \,. \end{displaymath} From this the claim follows by the \hyperlink{FundGroupoidOfParacompact}{above proposition}. \end{proof} \begin{uprop} Let $G$ be a [[well-pointed simplicial topological group]] degreewise in [[TopMfd]]. Then the $(\infty,1)$-functor $\Pi : \mathrm{ETop}\infty\mathrm{Grpd} \to \infty \mathrm{Grpd}$ preserves [[homotopy fiber]]s of all morphisms of the form $X \to \mathbf{B}G$ that are presented in $[\mathrm{CartSp}_{\mathrm{top}}^{\mathrm{op}}, \mathrm{sSet}]_{proj}$ by morphism of the form $X \to \bar W G$ with $X$ fibrant. \end{uprop} \begin{proof} Notice that since [[(∞,1)-sheafification]] preserves finite [[(∞,1)-limit]]s we may indeed discuss the homotopy fiber in the global model structure on simplicial presheaves. Write $Q X \stackrel{\simeq}{\to} X$ for the global cofibrant resolution given by $Q X : [n] \mapsto \coprod_{\{U_{i_0} \to \cdots \to U_{i_n} \to X_n\}} U_{i_0}$, where the $U_{i_k}$ range over $\mathrm{CartSp}_{\mathrm{top}}$ . (Discussed at . ) This has degeneracies splitting off as direct summands, and hence is a [[good simplicial topological space]] that is degreewise in [[TopMfd]]. Consider then the pasting of two [[pullback]] diagrams of simplicial presheaves \begin{displaymath} \itexarray{ P' &\stackrel{\simeq}{\to}& P &\to& W G \\ \downarrow && \downarrow && \downarrow \\ Q X &\stackrel{\simeq}{\to}& X &\to & \bar W G } \,. \end{displaymath} By the discussion at [[geometric realization of simplicial topological spaces]] we have that the rightmost vertical morphism is a fibration in $[CartSp_{top}^{op}, sSet]_{proj}$. Since fibrations are stable under pullback, the middle vertical morphism is also a fibration (as is the leftmost one). Since the global model structure is a right [[proper model category]] it follows then that also the top left horizontal morphism is a weak Since the square on the right is a pullback of fibrant objects with one morphism being a fibration, $P$ is a presentation of the [[homotopy fiber]] of $X \to \bar W G$. Hence so is $P'$, which is moreover the pullback of a diagram of good simplicial spaces. By prop. \ref{FundGroupoidOfSimplicialParacompact} we have that on the outer diagram $\Pi$ is presented by [[geometric realization of simplicial topological spaces]] $|-|$. By the discussion of realization of [[simplicial principal bundle]]s there, we have a pullback in $\mathrm{Top}_{\mathrm{Quillen}}$ \begin{displaymath} \itexarray{ {|P|} &\to& {|W G|} \\ \downarrow && \downarrow \\ {|Q X|} & \to & {|\bar W G|} } \end{displaymath} which exhibits $|P|$ as the [[homotopy fiber]] of $|Q X| \to |\bar W G|$. But this is a model for $|\Pi(X \to \bar W G)|$. \end{proof} \hypertarget{twisted_cohomology}{}\subsubsection*{{Twisted cohomology}}\label{twisted_cohomology} See \emph{[[twisted bundle]]} . \hypertarget{WhiteheadTowers}{}\subsubsection*{{Universal coverings and geometric Whitehead towers}}\label{WhiteheadTowers} We discuss in $ETop\infty Grpd$. \begin{uprop} Let $X$ be a [[pointed object|pointed] [[paracompact topological space]] that admits a [[good open cover]]. Then its ordinary [[Whitehead tower]] $* \to \cdots X^{(2)} \to X^{(1)} \to X^{(0)} = X$ in [[Top]] coincides with the image under the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|intrinsic fundamental ∞-groupoid functor]] $|\Pi(-)|$ of its [[Whitehead tower in an (∞,1)-topos|geometric Whitehead tower]] $X^{\mathbf{(\infty)}} \to \cdots X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} = X$ in $ETop \infty Grpd$: \begin{displaymath} \begin{aligned} |\Pi(-)| & : (X^{\mathbf{(\infty)}} \to \cdots X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} = X) \in ETop\infty Grpd \\ & \mapsto (* \to \cdots X^{(2)} \to X^{(1)} \to X^{(0)} = X) \in Top \end{aligned} \,. \end{displaymath} \end{uprop} \begin{proof} By the general discussion at [[Whitehead tower in an (∞,1)-topos]] the geometric Whitehead tower is characterized for each $n$ by the [[fiber sequence]] \begin{displaymath} X^{\mathbf{(n)}} \to X^{\mathbf{(n-1)}} \to \mathbf{B}^n \mathbf{\pi}_n(X) \to \mathbf{\Pi}_n(X) \to \mathbf{\Pi}_{(n-1)}(X) \,. \end{displaymath} By the above \hyperlink{FundGroupoidOfParacompact}{proposition on the fundamental ∞-groupoid} we have that $\mathbf{\Pi}_n(X) \simeq Disc Sing X$. Since $Disc$ is [[right adjoint]] and hence preserves [[homotopy fiber]]s this implies that $\mathbf{B} \mathbf{\pi}_n(X) \simeq \mathbf{B}^n Disc \pi_n(X)$, where $\pi_n(X)$ is the ordinary $n$th [[homotopy group]] of the pointed topological space $X$. Then by the above \hyperlink{GeometricRealizationOfHomotopyFibers}{proposition on geometric realization of homotopy fibers} we have that under $|\Pi(-)|$ the space $X^{\mathbf{(n)}}$ maps to the homotopy fiber of $|\Pi(X^{\mathbf{(n-1)}})| \to B^n |Disc \pi_n(X)| = B^n \pi_n(X)$. By induction over $n$ this implies the claim. \end{proof} \hypertarget{PathInftinityGroupoids}{}\subsubsection*{{Path $\infty$-groupoid and geometric Postnikov towers}}\label{PathInftinityGroupoids} Let $C$ be an [[∞-connected site]]. We give an explicit presentation of the \emph{} $X \to \mathbf{\Pi}(X)$ in the [[locally ∞-connected (∞,1)-topos]] over $C$ such that the component maps are cofibrations. \begin{uremark} The projective [[model structure on simplicial presheaves]] $[C^{op}, sSet]_{proj}$ has a set of generating cofibrations \begin{displaymath} I = \{ U \cdot \partial \Delta[n] \hookrightarrow U \cdot \Delta[n] | U \in C, n \in \mathbb{N}) \} \,. \end{displaymath} \end{uremark} See [[model structure on functors]] for details. \begin{udef} Write \begin{displaymath} \mathbf{Sing} : C \to [C^{op}, sSet] \end{displaymath} for the functor given by applying the [[small object argument]] to this set $I$ to obtain a functorial factorization of the [[terminal object|terminal morphisms]] $U \to *$ into a cofibration followed by an acyclic fibration \begin{displaymath} U \hookrightarrow \mathbf{Sing} U \stackrel{\simeq}{\to} * \,. \end{displaymath} Let \begin{displaymath} \mathbf{Sing} : [C^{op}, sSet] \to [C^{op}, sSet] \end{displaymath} be the [[Yoneda extension]] (left [[Kan extension]] through the [[Yoneda embedding]]) of this functor to all of $[C^{op}, sSet]$. \end{udef} \begin{uremark} For $U \in C$ the [[simplicial presheaf]] $\mathbf{Sing}U$ is a [[resolution]] of the ([[nerve]] of the) [[fundamental groupoid]] $\Pi_1(U)$: the non-degenerate components of $\mathbf{Sing}U$ at the first stage of the [[small object argument]] are such that a map out of them into a simplicial presheaf $A$ are given by commuting diagrams \begin{displaymath} \itexarray{ U_0 \coprod U_0 &\stackrel{(s,t)}{\to}& U \\ \downarrow && \downarrow \\ U_0 \times \Delta[1] &\to& A } \,. \end{displaymath} This is a $U$-parameterized family of objects of $A$ together with a $U_0$-parameterized family of morphisms of $A$ associated to the pairs of points $(s,t) \in U$, hence to the ``straight paths'' from $s$ to $t$. At the next stage for every triangle of such straight path a 2-morphism is thrown in, and so on. So $\mathbf{Sing}U$ indeed is an $\infty$-groupoid of paths in $U$. \end{uremark} \begin{uprop} The functor $\mathbf{Sing}$ is the [[left adjoint]] of a [[Quillen adjunction]] \begin{displaymath} (\mathbf{Sing} \dashv R) : [C^{op}, sSet]_{proj, loc} \to [C^{op}, sSet]_{proj, loc} \,. \end{displaymath} Its left [[derived functor]] is equivalent to the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] \begin{displaymath} \mathbb{L}\mathbf{Sing}(-) \simeq \Pi(-) \end{displaymath} and the \emph{constant path inclusion} $Id \to \Pi$ is presented by the canonical [[natural transformation]] $Id \to \mathbf{Sing}$. \end{uprop} \begin{proof} On an arbitrary simplicial presheaf $X$ the functor $\mathbf{Sing}$ is given by the [[coend]] \begin{displaymath} \mathbf{Sing} : X \mapsto \int^{U \in C} X(U) \cdot \mathbf{Sing}U \,. \end{displaymath} By construction this preserves all colimits. Hence by the [[adjoint functor theorem]] (using that domain and codomain are [[presheaf categories]]) we have that $\mathbf{Sing}$ is a [[left adjoint]]. Explicitly, the [[right adjoint]] is given by \begin{displaymath} R X : U \mapsto [C^{op}, sSet](\mathbf{Sing}U, X) \,. \end{displaymath} We check that $\mathbf{Sing}$ is also a [[Quillen adjunction|left Quillen functor]] first for the [[model structure on simplicial presheaves|global projective model structure]]. For that, notice that the above expression is the evaluation of the left [[Quillen bifunctor]] (see the examples-section there for details) \begin{displaymath} \int^C (-) \cdot (-) : [C^{op}, sSet]_{proj} \times [C, [C^{op}, sSet]_{proj}]_{inj} \to [C^{op}, sSet]_{proj} \,. \end{displaymath} Since every representable $U$ is cofibrant in $[C^{op}, sSet]_{proj}$ and since $U \to \mathbf{Sing}U$ is a cofibration by the [[small object argument]], we have that $\mathbf{Sing}U$ is cofibrant in $[C^{op}, sSet]_{proj}$ for all $U$. This means that also $\mathbf{Sing}(-)$ is cofibrant in $[C, [C^{op}, sSet]_{pro}]_{inj}$. Since $\int^C (-) \cdot (-)$ is a left Quillen bifunctor it follows that $\int^C (-)\cdot \mathbf{Sing}$ is a left Quillen functor. Hence it preserves cofibrations and acyclic cofibrations. This establishes that $\mathbf{Sing}$ is a left simplicial Quillen functor on $[C^{op}, sSet]_{proj}$. Since this is a [[left proper model category]] we have by the discussion at [[simplicial Quillen adjunction]] that for showing that this does descend to the local model structure it is sufficient to check that the right adjoint preserves local fibrant objects. Which, in turn, is implied if $\mathbf{Sing}$ send covering [[Cech nerve]]s to weak equivalences. Let therefore $C(\coprod_i U_i \to U)$ be the Cech nerve of a covering family in the [[site]] $C$. We may write this as the [[coend]] \begin{displaymath} C(\coprod_i U_i) = \int^{[k] \in \Delta} \Delta[k] \cdot \left( \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n} \right) \,, \end{displaymath} where by assumption on the [[∞-connected site]] $C$ all the $U_{i_0, \cdots, i_n}$ are representable. By precomposing the projection $C(\coprod_i U_i) \to X$ with the objectwise [[Bousfield-Kan map]] that replaces the simplices with the [[fat simplex]] $\mathbf{\Delta} : \Delta \to sSet$, we get the morphisms \begin{displaymath} C(\coprod_i U_i) = \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n} \right) \stackrel{\simeq}{\to} C(\coprod_i U_i) \to U \,. \end{displaymath} Here the first map is an objectwise weak equivalence by Bousfield-Kan (see the examples at [[Reedy model structure]] for details). Hence by 2-out-of-3 we may equivalently check that $\mathbf{Sing}$ sends these morphisms to weak equivalences in $[C^{op}, sSet]_{proj}$. Since $\mathbf{Sing}$ commutes with all colimits and hence coends the result of applying it to this morphism is \begin{displaymath} \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} \mathbf{Sing} U_{i_0, \cdots, i_n} \right) \to \mathbf{Sing}U \,. \end{displaymath} Since the [[fat simplex]] is cofibrant in $[\Delta, sSet_{Quillen}]_{proj}$ and since the above is an evaluation of the left [[Quillen bifunctor]] \begin{displaymath} \int^\Delta (-) \cdot (-) : [\Delta, sSet_{Quillen}]_{proj} \times [\Delta^{op}, [C^{op}, sSet]_{proj}]_{inj} \to [C^{op}, sSet]_{proj} \end{displaymath} the functor $\int^\Delta \mathbf{\Delta} \cdot (-)$ is left Quillen and hence preserves weak equivalences between cofibrant objects (by the [[factorization lemma]]), such as the morphisms $\mathbf{Sing}U \stackrel{\simeq}{\to} *$. Therefore we have a commuting diagram \begin{displaymath} \itexarray{ \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} \mathbf{Sing} U_{i_0, \cdots, i_n} \right) &\stackrel{\simeq}{\to}& \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \left( \coprod_{i_0, \cdots, i_n} * \right) \\ \downarrow && \downarrow^{\simeq} \\ \mathbf{Sing}U &\stackrel{\simeq}{\to}& * } \,, \end{displaymath} with weak equivalences in $[C^{op}, sSet]_{proj}$ as indicated: the top morphism is a weak equivalence by the argument just given, the bottom one by the [[small object argument]]-construction of $\mathbf{Sing}$ and the right vertical morphism is a weak equivalence by the assumption on an [[∞-connected site]]. It follows by 2-out-of-3 that also the left vertical morphism is a weak equivalence. This establishes the fact that $\mathbf{Sing}$ is left Quillen on the local model structure on simplicial presheaves. By the discussion at [[simplicial Quillen adjunction]] this implies that its left [[derived functor]] is a [[left adjoint|left]] [[adjoint (∞,1)-functor]]. Hence it preserves [[(∞,1)-colimit]]s and so is determined on representatives. There $\mathbf{Sing} U \simeq *$ does coindice with $\Pi(U) \simeq *$, hence both [[(∞,1)-functor]]s are equivalent. \end{proof} \begin{ucorollary} For all cofibrant $X \in [C^{op}, sSet]_{proj,loc}$, the $\mathbf{\Pi}_{dR} X$ is presented by the ordinary [[pushout]] \begin{displaymath} \itexarray{ X &\to& * \\ \downarrow && \downarrow \\ \mathbf{Sing}X &\to& \mathbf{\Pi}_{dR} X } \end{displaymath} in $[C^{op}, sSet]$. \end{ucorollary} \begin{proof} By definition we have that $\mathbf{\Pi}_{dR}$ is the [[(∞,1)-pushout]] $\mathbf{\Pi}(X) \coprod_X *$ in $Sh_{(\infty,1)}(C)$. By the above proposition we have a cofibrant presentation of the pushout diagram as indicated (all three objects cofibrant, at least one of the two morphisms a cofibration). By the general discussion at [[homotopy colimit]] the ordinary pushout of that diagram does compute the [[(∞,1)-colimit]]. \end{proof} \hypertarget{homotopy_localization}{}\subsection*{{Homotopy localization}}\label{homotopy_localization} We discuss that the [[homotopy localization]] of topological $\infty$-groupoids reproduces [[Top]] $\simeq$ [[∞Grpd]], following (\hyperlink{Dugger}{Dugger}). \hypertarget{idea_2}{}\subsubsection*{{Idea}}\label{idea_2} A central result about the [[(∞,1)-topos]] $Sh_{(\infty,1)}(Top)$ of [[∞-stack]]s on [[Top]] is that the [[homotopy localization]] is equivalent to [[Top]] itself \begin{displaymath} Sh_{(\infty,1)}(Top)^I \simeq Top \,. \end{displaymath} A discussion of this is in (the nice but not quite finished) (\hyperlink{Dugger}{Dugger}). In fact, this is true even for [[Lie ∞-groupoid]]s, i.e. [[∞-stack]]s on [[Diff]]: the homotopy invariant ones model plain [[topological space]]s. This provides a useful resolution of [[topological space]]s that often helps to disentangle the two different roles played by a topological space: on the one hand as a model for an [[∞-groupoid]], in the other as a [[locale]]. \hypertarget{details}{}\subsubsection*{{Details}}\label{details} Let $SPSh(Diff)^{loc}$ be the local [[model structure on simplicial presheaves]] obtained by left [[Bousfield localization of model categories|Bousfield localization]] at the [[Cech nerve]]s of [[Cech cover]]s with respect to the standard [[Grothendieck topology]] on [[Diff]]. This is a [[models for ∞-stack (∞,1)-toposes|model for ∞-stacks]] on [[Diff]]. Let $SPSh(Diff)^{loc}_I$ be furthermore the left [[Bousfield localization of model categories|Bousfield localization]] at the set of projection morphisms out of [[product]]s of the form $X \times \mathbb{R} \to X$ for all $X \in Diff$. The $\infty$-stacks that are [[local object]]s with respect to these morphisms are the \emph{homotopy invariant} $\infty$-stacks, so this localization models the [[(∞,1)-topos]] of homotopy invariant $\infty$-stacks on $Diff$. There is a [[adjunction]] \begin{displaymath} L : SSet \stackrel{\leftarrow}{\to} SPSh(Diff) : R \end{displaymath} where $L$ sends a simplicial set to the simplicial presheaf constant on that simplicial set, and where evaluates a simplicial presheaf on the manifold that is the [[point]]. \begin{utheorem} This adjunction $(L \dashv R)$ is a [[Quillen equivalence]] with respect to the standard [[model structure on simplicial sets]] on the left and the above model structure $SPSh(Diff)_{loc}^I$ on the right. \end{utheorem} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[cohesive (∞,1)-topos]] \begin{itemize}% \item [[discrete ∞-groupoid]] \item [[topological infinity-groupoid]] \begin{itemize}% \item \textbf{Euclidean-topological ∞-groupoid} \item [[locally contractible topological infinity-groupoid]] \end{itemize} \item [[smooth ∞-groupoid]] \item [[synthetic differential ∞-groupoid]] \item [[super ∞-groupoid]] \item [[smooth super ∞-groupoid]] \item [[synthetic differential super ∞-groupoid]] \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Section 3.2 in \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} Some discussion of the $(\infty,1)$-category of $(\infty,1)$-sheaves on the category of manifolds and its restriction to open balls and a discussion of its homotopy localization is in: \begin{itemize}% \item [[Dan Dugger]], \emph{Sheaves and homotopy theory} (\href{http://www.uoregon.edu/~ddugger/cech.html}{web}, \href{http://www.uoregon.edu/~ddugger/cech.dvi}{dvi}, \href{http://ncatlab.org/nlab/files/cech.pdf}{pdf}) \end{itemize} Discussion of [[geometric realization of simplicial topological spaces|geometric realization of simplicial topological]] principal bundles and of their [[classifying spaces]] is in \begin{itemize}% \item [[David Roberts]], [[Danny Stevenson]], \emph{Simplicial principal bundle in parameterized spaces} (\href{http://arxiv.org/abs/1203.2460}{arXiv:1203.2460}) \end{itemize} \begin{itemize}% \item [[Danny Stevenson]], \emph{Classifying theory for simplicial parametrized groups} (\href{http://arxiv.org/abs/1203.2461}{arXiv:1203.2461}) \end{itemize} [[!redirects Euclidean-topological infinity-groupoid]] [[!redirects Euclidean-topological infinity-groupoids]] [[!redirects Euclidean-topological infnity-groupoid]] [[!redirects Euclidean-topological ∞-groupoid]] [[!redirects Euclidean-topological ∞-groupoids]] [[!redirects Euclidean topological infinity-groupoid]] [[!redirects Euclidean topological infinity-groupoids]] [[!redirects Euclidean topological ∞-groupoid]] [[!redirects Euclidean topological ∞-groupoids]] [[!redirects ETop∞Grpd]] \end{document}