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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*{geometric homotopy groups in an (infinity,1)-topos} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory} [[!include (infinity,1)-topos - contents]] \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] This is a sub-entry of [[homotopy groups in an (∞,1)-topos]]. It discusses the general notions of \textbf{[[étale homotopy]]} in the context of [[locally ∞-connected (∞,1)-toposes]]. For the other notion of homotopy groups in an $(\infty,1)$-topos see [[categorical homotopy groups in an (∞,1)-topos]]. \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{GeomIdea}{Idea}\dotfill \pageref*{GeomIdea} \linebreak \noindent\hyperlink{GeomDef}{Definition}\dotfill \pageref*{GeomDef} \linebreak \noindent\hyperlink{LocalContraction}{In terms of local contractions}\dotfill \pageref*{LocalContraction} \linebreak \noindent\hyperlink{LocalContractionRef}{References}\dotfill \pageref*{LocalContractionRef} \linebreak \noindent\hyperlink{Monodromy}{In terms of monodromy and Galois theory}\dotfill \pageref*{Monodromy} \linebreak \noindent\hyperlink{InTermsOfMonodromyReferences}{References}\dotfill \pageref*{InTermsOfMonodromyReferences} \linebreak \noindent\hyperlink{Paths}{In terms of concrete paths}\dotfill \pageref*{Paths} \linebreak \noindent\hyperlink{references_3}{References}\dotfill \pageref*{references_3} \linebreak \noindent\hyperlink{GeomExamples}{Examples}\dotfill \pageref*{GeomExamples} \linebreak \noindent\hyperlink{Pi0Ofsheafontopspace}{Geometric $\Pi_0$ of a sheaf on a locally connected topological space}\dotfill \pageref*{Pi0Ofsheafontopspace} \linebreak \noindent\hyperlink{Pi0InLocConTop}{Geometric $\Pi_0$ of a general object in a locally connected topos}\dotfill \pageref*{Pi0InLocConTop} \linebreak \noindent\hyperlink{geometric__of_objects_in_a_1topos}{Geometric $\pi_1$ of objects in a 1-topos}\dotfill \pageref*{geometric__of_objects_in_a_1topos} \linebreak \noindent\hyperlink{geometric__of_a_topological_space}{Geometric $\Pi_2$ of a topological space}\dotfill \pageref*{geometric__of_a_topological_space} \linebreak \noindent\hyperlink{ooStackOnTopSpace}{Geometric $\Pi_\infty$ of a topological space}\dotfill \pageref*{ooStackOnTopSpace} \linebreak \noindent\hyperlink{GeomPiOfTermObj}{Geometric $\Pi_\infty$ of the terminal object in a locally $\infty$-connected $(\infty,1)$-topos}\dotfill \pageref*{GeomPiOfTermObj} \linebreak \noindent\hyperlink{GeneralExamples}{Examples}\dotfill \pageref*{GeneralExamples} \linebreak \hypertarget{GeomIdea}{}\subsection*{{Idea}}\label{GeomIdea} An ordinary [[topos]] $E$ is a [[locally connected topos]] if the [[global section]]s [[geometric morphism]] $(LConst \dashv \Gamma) : E \stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set$ is in fact an [[essential geometric morphism]] in that $LConst$ has also a [[left adjoint]] $(\Pi_0 \dashv LConst)$: \begin{displaymath} (\Pi_0 \dashv LConst \dashv \Gamma) : E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \,. \end{displaymath} This left adjoint $\Pi_0$ sends each object $X$ of $A$ to its [[set]] $\Pi_0$ of connected components. In other words this left adjoint produces the degree 0-part of the homotopy groups of objects of $E$. This has an obvious generalization of [[(∞,1)-topos]]es. \hypertarget{GeomDef}{}\subsection*{{Definition}}\label{GeomDef} The obvious generalization of the notion of $\Pi_0$ for a [[locally connected topos]] is to say that for $n \in \mathbb{N}$ an [[(n,1)-topos]] $\mathbf{H}$ is a [[locally n-connected (∞,1)-topos|locally n-connected (n,1)-topos]] if again the terminal geometric morphism is an essential geometric morphism in that the [[constant stack|constant n-stack]] functor $LConst$ has a [[left adjoint]] $\Pi_n$ \begin{displaymath} (\Pi_n \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi_n}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} n Grpd \,. \end{displaymath} Here we may take $n = \infty$ and say that an [[(∞,1)-topos]] is [[locally n-connected (∞,1)-topos|locally contractible]] if we have an [[essential geometric morphism]] \begin{displaymath} (\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} \infty Grpd \end{displaymath} to [[? Grpd]], with $\Pi$ the left [[adjoint (∞,1)-functor]] to the [[constant ∞-stack]] [[(∞,1)-functor]] $LConst$. For $X \in \mathbf{H}$ any object, the [[∞-groupoid]] $\Pi(X)$ deserves to be called the \textbf{[[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]} of $X$ Its ordinary [[homotopy group]]s are the homotopy groups of $X$. While an obvious slight generalization or refinement of what is considered in previous literature, it seems that the simple picture of a left [[adjoint (∞,1)-functor]] to the [[constant ∞-stack]] functor has not been made explicit in the existing literature (though possibly in the thesis by [[Richard Williamson]]). However, up to some straightforward translations of concepts and notation, it turns out that essentially all aspects of this picture are present and well known -- if somewhat implicitly -- in existing literature. A detailed commented account of what is in the literature is in the following subsection and in particular in the section \hyperlink{GeomExamples}{Examples} below. There are essentially three different methods concretely constructing the abstractly defined [[schreiber:homotopy ∞-groupoid]]-functor $\Pi(-)$. \begin{enumerate}% \item by constructing the left adjoint $\Pi(-)$ as the functor that takes an object to its \textbf{local contraction} -- this is described in the section \emph{\hyperlink{LocalContraction}{In terms of local contractions};} \item by using \textbf{monodromy}/[[Galois theory]] of [[locally constant ∞-stack]]s to reproduce $\Pi()$ by [[Tannaka duality]] -- this is described in the section \href{Monodromy}{In terms of monodromy and Galois theory}; \item by constructing $\Pi(-)$ explicitly as a path $\infty$-groupoid in terms of paths modeled on an [[interval object]] in $\mathbf{H}$ -- this is described in the section \emph{\href{Paths}{In terms of concrete paths}} . \end{enumerate} \hypertarget{LocalContraction}{}\subsection*{{In terms of local contractions}}\label{LocalContraction} If the [[locally contractible (∞,1)-topos]] $\mathbf{H}$ has a [[site]] $C$ with $\mathbf{H} \simeq Sh_{(\infty,1)}(C)$ such that the objects of the site are \textbf{geometrically contractible} in that \emph{constant} [[(∞,1)-presheaf|(∞,1)-presheaves]] already satisfy [[descent]] over objects in $C$, then the [[left adjoint]] $\Pi : \mathbf{H} \to \infty Grpd$ to $LConst$ may be constructed explicitly as follows. Following the discussion at [[models for ∞-stack (∞,1)-toposes]] there is a [[model structure on simplicial presheaves]] $sPSh(C)_{proj}^{loc}$ wich [[presentable (∞,1)-category|presents]] $\mathbf{H}$. \textbf{Proposition} The [[adjoint (∞,1)-functor|(∞,1)-adjunction]] $(\Pi \dashv LConst) : Sh_{(\infty,1)}(C) \stackrel{\leftarrow}{\to} \infty Grpd$ is presented by an [[SSet]]-[[enriched category theory|enriched]] [[Quillen adjunction]] \begin{displaymath} (\Pi \dashv LConst) : sPSh(C)_{proj}^{loc} \stackrel{\overset{\Pi}{\to}}{\overset{LConst}{\leftarrow}} sSet_{Quillen} \,, \end{displaymath} where \begin{itemize}% \item for $S \in sSet$ the presheaf $LConst_S$ sends all $U \mapsto S$, for all $U$; \item the functor $\Pi$ acts by $Pi(X) = \int^{U \in C} X = \lim_\to X$. \end{itemize} The total left [[derived functor]] of $\Pi$ first takes an object $X$ to a [[simplicial presheaf]] that is degreewise a [[coproduct]] of [[representable functor|representables]] and then \emph{contracts} all these representables to the [[terminal object]], regarding the resulting constant simplicial presheaf as a simplicial set: \begin{displaymath} \mathbb{L} \Pi : X \mapsto Q X = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} U_i \right) \mapsto \Pi(Q X) = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} * \right) \,. \end{displaymath} \textbf{Proof} This is discussed at [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]] and [[cohesive (∞,1)-topos]]. \hypertarget{LocalContractionRef}{}\subsubsection*{{References}}\label{LocalContractionRef} Essentially the construction of $\mathbb{L} \Pi$ as above is an old construction in terms of -- somewhat implicitly -- the structure of a [[category of fibrant objects]] on [[simplicial object]]s in a [[topos]]: the discussion on page 18 of \begin{itemize}% \item [[Ieke Moerdijk]], \emph{Classifying Spaces and Classifying Topoi} Lecture Notes in Mathematics 1616, Springer (1995) . \end{itemize} which goes back to \begin{itemize}% \item Artin, Mazur, \emph{Etale Homotopy} Springer Lecture Notes in Mathematics 100, Berlin (1969) \end{itemize} goes as follows: Let $E = Sh(C)$ be a [[locally connected topos]] \begin{displaymath} (\Pi_0 \dashv LConst) : Sh(C) \stackrel{\leftarrow}{\to} Set \end{displaymath} that here we think of as a [[petit topos|petit]] over-topos over a given object $X$ in some ambient [[gros topos]]. Accordingly we write $X = *$ for the [[terminal object]] in $Sh(C)$. Assume that $E$ has [[point of a topos|enough point]]. Then [[stalk]]wise [[Kan fibration|Kan-fibrant]] [[simplicial object]]s in $E$, i.e. [[stalk]]-wise Kan-fibrant simplicial sheaves on $C$ form a [[category of fibrant objects]]. In particular a fibrant simplicial object $Y \in [\Delta^{op}, Sh(C)]$ equipped with an acyclic fibration $Y \to X$ to the [[terminal object]] $X = *$ is a [[hypercover]] of $X$. The definition of the [[∞-groupoid]] $\Pi(X)$ as defined in the above references (notice that only its homotopy groups are written down explicitly there, but it's immediate to equivalently write it as we do now) is \begin{displaymath} \Pi(X) = \lim_\to \Pi_0( Y_\bullet) \,, \end{displaymath} where \begin{itemize}% \item the [[colimit]] is taken over the category of acyclic fibrations/hypercovers $Y \to X$; \item the connected components functor $\Pi_0 : Sh(C) \to Set$ is applied degreewise to the simplicial sheaf $Y = (Y_\bullet)$ to produce a [[simplicial set]]. \end{itemize} In Artin-Mazur it is discussed that this prescription does produce the right homotopy groups for $X$ a [[topological space]] if one assumes that this space is [[locally contractible space]]. If we therefore interpret this as saying that for the above prescription to yield the correct result we generally ought to assume that $Sh_{(\infty,1)}(C)$ is a [[locally contractible (∞,1)-topos]], then this prescription can be seen to model implicitly the left Quillen functor $\Pi(-)$ that we described above: In terms of the full [[model category]] structure on $sPSh(C)_{proj}^{loc}$ among all these hypercovers is one that is the cofibrant object \begin{displaymath} Y = Q X = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} U_i \right) \end{displaymath} mentioned above, consisting degreewise of coproducts of representables with $\Pi_0(U_i) = *$. For instance if $X$ admits a [[good open cover]], we can take $Y$ to be the [[Cech nerve]] of that good cover. (For more on this see [[∞-Lie groupoid]].) Due to the lifting property of cofibrant objects, any colimit over all hypercovers can be computed by evaluating just at that hypercover. There the Artin-Mazur-Moerdijk-prescription yields \begin{displaymath} \Pi(Q X) = \Pi_0((Q X)_\bullet) = \int^{[n] \in \Delta} \Delta[n] \cdot \Pi_0\left( \coprod_{i_n} U_{i_n} \right) = \int^{[n] \in \Delta} \Delta[n] \cdot \Pi_0\left( \coprod_{i_n} * \right) \,. \end{displaymath} This is indeed the action of the left Quillen functor from above. It is the [[nerve theorem]] that asserts that for $Y$ the [[Cech nerve]] of a [[good open cover]], this simplicial set is homotopy equivalent to the original [[paracompact space]]. A closely related, implicitly slightly more general statement is in on p. 25 of \begin{itemize}% \item [[Daniel Dugger]], \emph{[[DuggerUniv.pdf:file]]} \end{itemize} which describes this construction for the case $\mathbf{H} = Sh_{(\infty,1)}(Diff)$ (the [[gros topos]] of $\infty$-stacks on [[Diff]]). With even more general sites allowed, but working only at the level of [[homotopy category|homotopy categories]] the left adjoint $\Pi$ and its construction is described in \href{http://math.berkeley.edu/~teleman/math/simpson.pdf#page=5}{Proposition 2.18} of \begin{itemize}% \item [[Carlos Simpson]], [[Constantin Teleman]], \emph{de Rham's theorem for $\infty$-stacks} (\href{http://math.berkeley.edu/~teleman/math/simpson.pdf}{pdf}) \end{itemize} See also the discussion at [[locally contractible (∞,1)-topos]]. \hypertarget{Monodromy}{}\subsection*{{In terms of monodromy and Galois theory}}\label{Monodromy} Given an [[(∞,1)-topos]] $\mathbf{H} = Sh_{(\infty,1)}(C)$ we define the [[∞-groupoid]] of [[locally constant ∞-stack]]s on an object $X \in \mathbf{H}$ to be \begin{displaymath} \infty CovBund(X) := \mathbf{H}(X, LConst_{Core(\infty Grpd)}) \,, \end{displaymath} where $LConst_{Core(\infty Grpd)}$ is the [[constant ∞-stack]] on the [[core]] [[∞-groupoid]] of [[? Grpd]]. If $\mathbf{H}$ is a [[locally contractible (∞,1)-topos]] in that $LConst$ has the left [[adjoint (∞,1)-functor]] $\Pi(-)$, then by definition of adjunction we have the equivalence \begin{displaymath} \infty CovBund(X) \simeq Func(\Pi(X), \infty Grpd) \end{displaymath} with [[locally constant ∞-stack]]s/$\infty$-[[covering space]]s on the one hand and [[(∞,1)-functor]]s from $\Pi(X)$ to [[∞Grpd]] on the other. Concrete realizations of this equivalence are discussed in the \href{GeomExamples}{Examples}-section below. Here we describe how one may \emph{[[reconstruction theorem|reconstruct]]} in terms [[Tannaka duality]] $\Pi(X)$ from just knowing $\infty CovBund(X)$ in terms of the [[automorphism]] [[∞-group]] of a fiber functor \begin{displaymath} F_x : \infty CovBund(X) \to \infty Grpd \end{displaymath} from $\infty$-[[covering space|coverin bundle]]s/[[locally constant ∞-stack]]s over $X$ to [[∞-groupoid]]. -- these automorphism are called the \textbf{monodromy} of $X$. We want to show that these automorphism [[∞-group]]s are the [[loop space object]]s of $\Pi(X)$, hence the geometric homotopy $\infty$-groups. \begin{displaymath} Aut (F_x) = \Omega^{geom}_x X =: \Omega_x \Pi(X) \,. \end{displaymath} This is the reconstruction of the geometric homotopy [[∞-group]]s of an [[∞-stack]] $X$ from its \textbf{monodromy} or \textbf{[[Galois theory]]}. \textbf{Proof} The underlying mechanism is just $(\infty,1)$-[[Tannaka duality]], i.e. essentially the [[(∞,1)-Yoneda lemma]] applied a few times in a row: suppose we knew $\Pi(X)$, so that by adjunction we have \begin{displaymath} CovBund(X) \simeq \infty Func(\Pi(X), \infty Grpd) \,. \end{displaymath} Then for each point $x \in \Pi(X)$ given by a morphism $i : {*} \to \Pi(X)$ we get a fiber functor \begin{displaymath} F_x := \infty Func(i, \infty Grpd) : Func(\Pi(X), \infty Grpd) \to \infty Grpd \end{displaymath} which takes a [[local system]] $\rho : \Pi(X) \to \infty Grpd$ and evaluates it on $x$. By the [[(∞,1)-Yoneda lemma]] this means that $F_x$ is given by homming out of the local system $Y_{\Pi(X)^{op}} x$ [[representable functor|represented by]] $x$: \begin{displaymath} \infty Func(i, \infty Grpd) \simeq Hom_{PSh_{(\infty,1)}(\Pi(X)^{op})}(Y_{\Pi(X)^{op}}) x, -) \,. \end{displaymath} But this in turn means that $\infty Func(i,\infty Grpd) : \infty Func(\Pi(X),\infty Grod) \to \infty Grpd$ is itself a representable functor, in the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(PSh_{(\infty,1)}(\Pi(X)^{op})^{op})$: \begin{displaymath} \infty Func(i, \infty Grpd) \simeq Y_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \,. \end{displaymath} This way we find, by applying the [[(∞,1)-Yoneda lemma]] two more times, that the automorphism [[∞-group]] of the fiber functor is \begin{displaymath} \begin{aligned} Aut_{PSh_{(\infty,1)}((PSh_{(\infty,1)}(\Pi(X)^{op}))^{op})} \infty Func(i, \infty Grod) & = Aut_{PSh_{(\infty,1)}((PSh_{(\infty,1)}(\Pi(X)^{op}))^{op})} Y_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \\ & \simeq Aut_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \\ & \simeq Aut_{\Pi(X)^{op}} x \\ & \simeq \Omega_x \Pi(X) \\ & =: \Omega_x^{geom} X \,. \end{aligned} \end{displaymath} Now, the same is of course true even if we don't have $\Pi(X)$ in hands yet, but only know the [[∞-groupoid]] $CovBund(X)$ of covering $\infty$-bundles / [[locally constant ∞-stack]]s in $X$: in terms of this we may reconstruct the automorphism [[∞-group]]s of $\Pi(X)$ as \begin{displaymath} Aut( CovBund(X) \stackrel{F_x}{\to} \infty Grpd ) \simeq \Omega_x \Pi(X) =: \Omega^{geom}_x X \,. \end{displaymath} \hypertarget{InTermsOfMonodromyReferences}{}\subsubsection*{{References}}\label{InTermsOfMonodromyReferences} The idea that geometric homotopy groups of generalized [[space]]s given by [[sheaf|sheaves]], [[stack]]s, [[∞-stack]]s is detected and definable by the behaviour of locally constant sheaves, stacks, $\infty$-stacks on these objects goes back to [[Grothendieck's Galois theory]] and the notion of [[fundamental group of a topos]]. The state of the art treatment of the Galois theory of coverings in a topos is in \begin{itemize}% \item Marta Bunge, \emph{Galois groupoids and covering morphisms in topos theory}, Galois theory, Hopf algebras, and semiabelian categories, 131--161, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004, \href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.7071}{links}. \end{itemize} In [[Pursuing Stacks]] [[Alexander Grothendieck|Grothendieck]] talked about how this 1-categorical situation generalizes to [[∞-stack]]s. After \emph{Pursuing Stacks}, apparently the first to publish a detailed formalization and proof of how the [[homotopy group]]s of a [[topological space]] $X$ may be recovered from the behaviour of [[locally constant ∞-stack]]s on $X$ was \begin{itemize}% \item [[Bertrand Toen]], \emph{Toward a Galoisian interpretation of homotopy theory} (\href{http://arxiv.org/abs/math/0007157}{arXiv:0007157}) \end{itemize} This has a followup construction in \begin{itemize}% \item [[Bertrand Toen]] and [[Gabriele Vezzosi]], \emph{Segal topoi and stacks over Segal categories} , Proceedings of the Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 (\href{http://arxiv.org/abs/math/0212330}{arxiv:math/0212330}). \end{itemize} Very similar constructions and statement then appeared in \begin{itemize}% \item [[Pietro Polesello]] and [[Ingo Waschkies]], \emph{Higher monodromy} , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (\href{http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf}{pdf}) \item [[Mike Shulman]], \emph{Parametrized spaces model locally constant homotopy sheaves} (\href{http://arxiv.org/abs/0706.2874}{arXiv:0706.2874}) \end{itemize} and, building on that, in example 1.8 of \begin{itemize}% \item [[Denis-Charles Cisinski]], \emph{Locally constant functors} , Math. Proc. Camb. Phil. Soc. , 147 (\href{http://www.math.univ-toulouse.fr/~dcisinsk/lcmodcat3.pdf}{pdf} \end{itemize} Notably the article by [[Pietro Polesello]] and [[Ingo Waschkies]] makes fully explicit the observation that \emph{locally} constant $n$-stacks are precisely the sections of the \emph{constant} $(n+1)$-stack on the $(n+1)$-groupoid $n Grpd$. This is a key observation for bringing the full power of the adjunction $(\Pi \dashv LConst)$ into the picture, as we do here. It was pointed out to [[Urs Schreiber]] by [[Richard Williamson]] that these constructions should generalize from topological spaces to objects in any [[(∞,1)-topos]], maybe along the lines outlined above, and that this way suitable $(\infty,1)$-toposes $\mathbf{H}$ comes canonically equipped with a notion of [[schreiber:homotopy ∞-groupoid]] $\Pi(X)$ of every object $X \in \mathbf{H}$. \hypertarget{Paths}{}\subsection*{{In terms of concrete paths}}\label{Paths} \ldots{} \hypertarget{references_3}{}\subsubsection*{{References}}\label{references_3} The following references discuss fundamental groupoids of an entire [[topos]] constructed from concrete [[interval object]]s. In the context of the above discussion these toposes are to be thought of as \emph{[[petit topos|petit]]} over-toposes over a given object in an ambient [[gros topos]], and as such are concerned with the fundamental groupoid of that object, in our sense. The construction of the [[fundamental groupoid]] of a topos from [[interval object]]s is in \begin{itemize}% \item [[Ieke Moerdijk]], [[Gavin Wraith]], \emph{Connected locally connected toposes are path-connected} , Transactions of the AMS, volume 295, number 2, (1986) \end{itemize} The comparison of this construction with the one by monodromy/Galois theory is in \begin{itemize}% \item [[Marta Bunge]], [[Ieke Moerdijk]], \emph{On the construction of the Grothendieck fundamental group of a topos by paths} , J. Pure and Applied Algebra, 116 (1997) \end{itemize} \hypertarget{GeomExamples}{}\subsection*{{Examples}}\label{GeomExamples} \hypertarget{Pi0Ofsheafontopspace}{}\subsubsection*{{Geometric $\Pi_0$ of a sheaf on a locally connected topological space}}\label{Pi0Ofsheafontopspace} Here we discuss the 0-th geometric homotopy group $\Pi_0 : Sh(X) \to Set$ of objects in a [[Grothendieck topos|sheaf topos]] in terms of a [[left adjoint]] $\Pi_0$ of the [[constant sheaf]] functor. This is a special case of the more general situation discussed in \href{Pi0InLocConTop}{Pi0 of a general object in a locally connected topos} below. Let $X$ be a sufficiently nice [[topological space]]. \begin{uproposition} There is a triple of [[adjoint functor]]s \begin{displaymath} (\Pi_0 \dashv LConst \dashv \Gamma) \;\;\; : \;\;\; Sh(X) \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \end{displaymath} where \begin{itemize}% \item $(LConst \dashv \Gamma)$ is the usual [[global section]] [[geometric morphism]] with $LConst_S$ the [[constant sheaf]] of [[locally constant function]]s with values in $S \in Set$ and \item $\Pi : Sh(X) \to Set$ is [[left adjoint]] to $LConst$ and sends each sheaf $A$ to the set of connected components of the corresponding [[etale space]] $p_A : Et(A) \to X$: \begin{displaymath} \Pi_0(A) = \pi_0 Et(A) \,. \end{displaymath} \end{itemize} \end{uproposition} \begin{proof} The [[etale space]] of $LConst_S$ is $E(LConst_S) = X \times S$. By the relation of [[sheaves]] on $X$ with [[etale space]]s over $X$ we have \begin{displaymath} Hom_{Sh(X)}(A, LConst_S) \simeq Hom_{Et/X}(E(A), X \times S) \end{displaymath} For $\gamma : I \to E(A)$ any continuous path in $E(A)$, and for $f : E(A) \to X \times S$ a morphism in $Et/X$, the image of $\gamma$ in $X \times I$ is fixed by, say, the image $f(\gamma(0)) = (p_A(\gamma_0),s)$ to be $f(\gamma) : t \mapsto (p_A(\gamma(t)),s)$. This means that the value of $f$ on any path component of $E(A)$ is uniquely fixed by its value on any point in that path component. Choosing a basepoint in each path component therefore induces bijection \begin{displaymath} \simeq Hom_{Set}(\pi_0(Et(A)), S) = Hom_{Set}(\Pi_0(A),S) \,. \end{displaymath} \end{proof} \hypertarget{Pi0InLocConTop}{}\subsubsection*{{Geometric $\Pi_0$ of a general object in a locally connected topos}}\label{Pi0InLocConTop} More generally, if $E$ is a [[locally connected topos]] then the [[global section]]s [[geometric morphism]] $(LConst \dashv \Gamma) : E \stackrel{\leftarrow}{\to} Set$ has also a [[left adjoint]] $\Pi_0$ to $LConst$: \begin{displaymath} (\Pi_0 \dashv LConst \dashv \Gamma) : E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \,. \end{displaymath} For instance page 17 of \begin{itemize}% \item [[Ieke Moerdijk]], \emph{Classifying Spaces and Classifying Topoi} Lecture Notes in Mathematics 1616, Springer (1995) \end{itemize} \hypertarget{geometric__of_objects_in_a_1topos}{}\subsubsection*{{Geometric $\pi_1$ of objects in a 1-topos}}\label{geometric__of_objects_in_a_1topos} The general idea is that of \begin{itemize}% \item [[Grothendieck's Galois theory]]. \end{itemize} A discussion of of how this produces first homotopy groups of a 1-[[topos]] is at \begin{itemize}% \item [[fundamental group of a topos]]. \end{itemize} The general construction of the first geometric homotopy group of objects in a [[Grothendieck topos]] is for instance in section 8.4 of \begin{itemize}% \item [[Peter Johnstone]], \emph{Topos theory} . \end{itemize} \hypertarget{geometric__of_a_topological_space}{}\subsubsection*{{Geometric $\Pi_2$ of a topological space}}\label{geometric__of_a_topological_space} This case is discussed in \begin{itemize}% \item [[Pietro Polesello]] and [[Ingo Waschkies]], \emph{Higher monodromy} , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (\href{http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf}{pdf}) \end{itemize} We indicate briefly how the results stated in this article fit into the general abstract picture as indicated above: The authors consider locally constant 1-stacks and 2-stacks on sites of open subsets of sufficiently nice [[topological space]]s. Prop. 1.1.9 gives the [[adjunction]] \begin{displaymath} (LConst \dashv \Gamma) : Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Grpd \end{displaymath} between forming constant stacks and taking global sections. Then prop 1.2.5, 1.2.6, culminating in \href{http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf#page=13}{theorem 1.2.9, p. 121} gives (somewhat implicitly) the other adjunction \begin{displaymath} (\Pi_1\dashv LConst) : Op(X) \hookrightarrow Sh_{(2,1)}(X) \stackrel{\overset{\Pi_1}{\to}}{\underset{LConst}{\leftarrow}} Grpd \end{displaymath} with the [[right adjoint]] to $LConst$ being the [[fundamental groupoid]] functor on representables. (Where we change a bit the perspective on the results as presented there, to amplify the pattern indicated above. For instance where the authors write $\Gamma(X,C_X)$ we think of this here equivalently as $Sh_{(2,1)}(X)(X,LConst(C))$, so that the theorem then gives the adjunction equivalence $\cdots \simeq Grpd(\Pi_1(X),C)$). Then in essentially verbatim analogy, these results are lifted from stacks to 2-stacks in section 2, where now prop 2.2.2, 2.2.3, culminating in \href{http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf#page=24}{theorem 2.2.5, p. 132} gives (somewhat implicitly) the adjunction \begin{displaymath} (\Pi_2\dashv LConst) : Op(X) \hookrightarrow Sh_{(3,1)}(X) \stackrel{\overset{\Pi_2}{\to}}{\underset{LConst}{\leftarrow}} Grpd \end{displaymath} now with the [[path n-groupoid|path 2-groupoid]] operation (locally) left adjoint to forming constant 2-stacks. \hypertarget{ooStackOnTopSpace}{}\subsubsection*{{Geometric $\Pi_\infty$ of a topological space}}\label{ooStackOnTopSpace} Let $X$ be a sufficiently nice (I think this should be locally (relatively) contractible. -DR) ([[paracompact space|paracompact]]) [[topological space]]. The canonical map $X \to {*}$ induces the [[geometric morphism]] \begin{displaymath} Sh_{(\infty,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \end{displaymath} where the [[right adjoint]] $\Gamma$ is taking [[global section]]s and the [[left adjoint]] is forming the [[constant ∞-stack]] on an $\infty$-groupoid $K$. If $K = Core (\infty Grpd)$ then $LConst_K$ is the [[constant ∞-stack]] of [[locally constant ∞-stack]]s and we write \begin{displaymath} LConst(X) := Sh_{(\infty,1)}(X, LConst_{\infty Grpd})= \Gamma LConst_{\infty Grpd} \end{displaymath} for the $\infty$-groupoid of locally constant $\infty$-stacks on $X$. Write $\Pi(X) := Sing X$ for the [[fundamental ∞-groupoid]] of $X$. \begin{utheorem} There is an equivalence of $\infty$-groupoids \begin{displaymath} LConst(X) \simeq \infty Grpd(\Pi(X), \infty Grpd) \,. \end{displaymath} \end{utheorem} \begin{quote}% [[Urs Schreiber]]: I think this is proven in the literature, if maybe slightly implicitly so. I'll now go through the available references to discuss this. \end{quote} After old ideas by [[Alexander Grothendieck]] from [[Pursuing Stacks]], it seems that the first explicit formalization and proof of this statement is given in \begin{itemize}% \item [[Bertrand Toen]], \emph{Toward a Galoisian interpretation of homotopy theory} (\href{http://arxiv.org/abs/math/0007157}{arXiv:0007157}) \end{itemize} In \href{http://arxiv.org/PS_cache/math/pdf/0007/0007157v4.pdf#page=25}{theorem 2.13, p. 25} the author proves an equivalence of [[(∞,1)-categories]] (modeled there as [[Segal category|Segal categories]]) \begin{displaymath} LConst(X) \simeq Fib(\Pi(X)) \end{displaymath} of [[locally constant ∞-stack]]s on $X$ and [[Kan fibration]]s over the [[fundamental ∞-groupoid]] $\Pi(X) = Sing(X)$. But Kan fibrations over a Kan complex such as $\Pi(X)$ are equivalently [[left fibration]]s (as discussed there) and by by the [[(∞,1)-Grothendieck construction]] these are equivalent to [[(∞,1)-functor]]s $\Pi(X) \to \infty Grpd$. So under the [[(∞,1)-Grothendieck construction]] To\"e{}n's result does actually produce an equivalence \begin{displaymath} LConst(X) \simeq Func(\Pi(X), \infty Grpd) \,. \end{displaymath} In \begin{itemize}% \item [[Bertrand Toen]] and [[Gabriele Vezzosi]], \emph{Segal topoi and stacks over Segal categories} , Proceedings of the Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 (\href{http://arxiv.org/abs/math/0212330}{arxiv:math/0212330}). \end{itemize} this is discussed in the context of [[Segal-topos]]es. Very similar statements are discussed in \begin{itemize}% \item [[Mike Shulman]], \emph{Parametrized spaces model locally constant homotopy sheaves} (\href{http://arxiv.org/abs/0706.2874}{arXiv:0706.2874}) \end{itemize} and, building on that, in example 1.8 of \begin{itemize}% \item [[Denis-Charles Cisinski]], \emph{Locally constant functors} , Math. Proc. Camb. Phil. Soc. , 147 (\href{http://www-math.univ-paris13.fr/~cisinski/lcmodcat3.pdf}{pdf}) \end{itemize} A variant of this statement -- more general in one respect, less general in another -- appears in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]} \end{itemize} as \href{http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=545}{theorem 7.1.0.1}. There it is shown that for any $K \in \infty Grpd$ there is a bijection of homotopy sets of morphisms \begin{displaymath} \pi_0 Top(X, |K|) \simeq \pi_0(p_* p^* K) \,, \end{displaymath} where $(p^* \dashv p_*) : Sh_{(\infty,1)}(X) \to \infty Grpd$ is the geometric morphism we denoted $(LConst \dashv \Gamma)$ above. If we also rewrite the left using [[homotopy hypothesis|the equivalence]] of $Top$ with $sSet$, this reads \begin{displaymath} \pi_0 \infty Grpd(\Pi(X), K) \simeq \pi_0(\Gamma LConst_K) = \pi_0 Sh_{(\infty,1)}(X,LConst_K) \,, \end{displaymath} For $K = Core(\infty Grpd)$ this is the $\pi_0$-[[decategorification]] of the above statement. \hypertarget{GeomPiOfTermObj}{}\subsubsection*{{Geometric $\Pi_\infty$ of the terminal object in a locally $\infty$-connected $(\infty,1)$-topos}}\label{GeomPiOfTermObj} The geometric $\Pi_\infty$ of the terminal object in a locally ∞-connected (∞,1)-topos can be called the [[fundamental ∞-groupoid of an (∞,1)-topos|fundamental ∞-groupoid]] of the topos. It [[representable functor|represents]] the [[shape of an (∞,1)-topos|shape]] of the topos. On page 18-19 of \begin{itemize}% \item [[Ieke Moerdijk]], \emph{Classifying Spaces and Classifying Topoi} Lecture Notes in Mathematics 1616, Springer (1995) \end{itemize} is described the construction of $\Pi(X) \in \infty Grpd$ for $X$ the [[terminal object]] in $Sh_{(\infty,1)}(C)$ on an ordinary [[site]] $C$ with $\Pi(X)$ as described above in \href{PiConstruction}{Geometric fundamental oo-groupoid}. This reviews in particular (slightly implicitly) \begin{theorem} \label{}\hypertarget{}{} Let $X$ be a [[topological space]] that has a basis of contractible open subsets. Write $X$ also for $X$ regarded as the terminal object in $Sh_{(\infty,1)}(X)$. Then the image of $X$ under $\Pi : Sh_{(\infty,1)}(X) \to \infty Grpd$ has the same homtopy groups as $X$ regarded as an object in [[Top]]: \begin{displaymath} \pi_n \Pi(X) \simeq \pi_n(X) \,. \end{displaymath} \end{theorem} \begin{proof} This is a slight reformulation of the statement in M. Artin, B. Mazur, \emph{Etale homotopy} , Springer lecture notes in mathematics 100, Berlin 1969 \end{proof} Notice the local contractibility assumption. This is necessary in general for $\Pi(X)$ to make sense. \hypertarget{GeneralExamples}{}\subsection*{{Examples}}\label{GeneralExamples} Let $C =$ [[Diff]] and consider in $Sh_{(\infty,1)}(Diff)$ the two objects \begin{itemize}% \item $S^1$, the $\infty$-stack represented by the standard circle in $Diff$; \item $\mathbf{B}\mathbb{Z}$ -- the $\infty$-stack constant on the [[delooping]] [[groupoid]] of the additive group $\mathbb{Z}$. \end{itemize} Then \begin{itemize}% \item the \emph{categorical} homotopy groups of $S^1$ are all trivial \begin{displaymath} \pi_n^{cat}(S^1) = {*} \end{displaymath} \item the \emph{geometric} homotopy groups of $S^1$ are the usual ones obtained from regarding $S^1$ as an object in [[Top]]: \begin{displaymath} \pi^{geom}_0(S^1) = * \end{displaymath} \begin{displaymath} \pi^{geom}_1(S^1) = \mathbb{Z} \end{displaymath} etc. \end{itemize} For $\mathbf{B}\mathbb{Z}$ it is the other way round: \begin{itemize}% \item the \emph{categorical} homotopy groups of $\mathbf{B}\mathbb{Z}$ are \begin{displaymath} \pi_n^{cat}(\mathbf{B}\mathbb{Z}) = \left\{ \itexarray{ \mathbb{Z} & | if\; n=1 \\ * & | otherwise } \right. \,. \end{displaymath} \end{itemize} [[!redirects geometric homotopy groups in an (infinity,1)-topos]] [[!redirects geometric homotopy groups in an (∞,1)-topos]] \end{document}