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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*{differential cohesive (infinity,1)-topos} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes} [[!include cohesive infinity-toposes - contents]] \begin{quote}% This is a subsection of the entry [[cohesive (∞,1)-topos]]. See there for background and context. \end{quote} \vspace{.5em} \hrule \vspace{.5em} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{differential_cohesion}{Differential cohesion}\dotfill \pageref*{differential_cohesion} \linebreak \noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{PresentationOnInfinitesimalNeighbourhoodSites}{From $\infty$-sheaves over infinitesimal neighbourhood sites}\dotfill \pageref*{PresentationOnInfinitesimalNeighbourhoodSites} \linebreak \noindent\hyperlink{RelationToInfinitesimalCohesion}{Relation to infinitesimal cohesion}\dotfill \pageref*{RelationToInfinitesimalCohesion} \linebreak \noindent\hyperlink{StructuresInDifferentialCohesion}{Structures in a differential cohesive $(\infty,1)$-topos}\dotfill \pageref*{StructuresInDifferentialCohesion} \linebreak \noindent\hyperlink{InfinitesimalPaths}{Infinitesimal paths and de Rham spaces}\dotfill \pageref*{InfinitesimalPaths} \linebreak \noindent\hyperlink{infinitesimal_path_groupoid}{Infinitesimal path $\infty$-groupoid}\dotfill \pageref*{infinitesimal_path_groupoid} \linebreak \noindent\hyperlink{JetBundleObjects}{Jet $\infty$-bundles}\dotfill \pageref*{JetBundleObjects} \linebreak \noindent\hyperlink{formally_smoothtaleunramified_morphisms}{Formally smooth/\'e{}tale/unramified morphisms}\dotfill \pageref*{formally_smoothtaleunramified_morphisms} \linebreak \noindent\hyperlink{InfinitesimalA1Homotopy}{Infinitesimal $\mathbb{A}^1$-homotopy}\dotfill \pageref*{InfinitesimalA1Homotopy} \linebreak \noindent\hyperlink{StructureSheaves}{Structure sheaves}\dotfill \pageref*{StructureSheaves} \linebreak \noindent\hyperlink{SheavesOfModules}{Sheaves of (quasi-coherent) modules}\dotfill \pageref*{SheavesOfModules} \linebreak \noindent\hyperlink{PoincareCocycle}{Liouville-Poincar\'e{} cocycle}\dotfill \pageref*{PoincareCocycle} \linebreak \noindent\hyperlink{EtaleObjects}{Manifolds and \'e{}tale groupoids}\dotfill \pageref*{EtaleObjects} \linebreak \noindent\hyperlink{GLnTangentBundles}{Frame bundles}\dotfill \pageref*{GLnTangentBundles} \linebreak \noindent\hyperlink{structures}{$G$-Structures}\dotfill \pageref*{structures} \linebreak \noindent\hyperlink{StrucInfinitesimalLocalSystem}{Flat $\infty$-connections and infinitesimal local systems}\dotfill \pageref*{StrucInfinitesimalLocalSystem} \linebreak \noindent\hyperlink{FormalInfinityGroupoids}{Formal cohesive $\infty$-groupoids}\dotfill \pageref*{FormalInfinityGroupoids} \linebreak \noindent\hyperlink{LieTheory}{Lie theory}\dotfill \pageref*{LieTheory} \linebreak \noindent\hyperlink{StrucDeformationTheory}{Deformation theory}\dotfill \pageref*{StrucDeformationTheory} \linebreak \noindent\hyperlink{idelic_structure}{Idelic structure}\dotfill \pageref*{idelic_structure} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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 [[cohesive (∞,1)-topos]] is a context of [[∞-groupoid]]s that are equipped with a [[geometry|geometric]] notion of \emph{cohesion} on their collections of [[object]]s and [[k-morphism]]s, for instance [[Euclidean-topological ∞-groupoid|topological cohesion]] or [[smooth ∞-groupoid|smooth cohesion]]. While the axioms of cohesion do imply the intrinsic existence of \emph{exponentiated} [[infinitesimal spaces]], they do not admit access to an explicit [[synthetic differential geometry|synthetic]] notion of infinitesimal extension. Here we consider one extra axiom on a [[cohesive (∞,1)-topos]] that does imply a good intrinsic notion of synthetic differential extension, compatible with the given notion of cohesion. We speak of \emph{differential cohesion}. In a cohesive $(\infty,1)$-topos with differential cohesion there are for instance good intrinsic notions of [[formally smooth morphism|formal smoothness]] and of [[de Rham space]]s of objects. \hypertarget{differential_cohesion}{}\subsection*{{Differential cohesion}}\label{differential_cohesion} We discuss [[extra structure]] on a [[cohesive (∞,1)-topos]] that encodes a refinement of the corresponding notion of cohesion to \emph{[[infinitesimal object|infinitesimal]] cohesion} . More precisely, we consider inclusions $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$ of cohesive $(\infty,1)$-toposes that exhibit the objects of $\mathbf{H}_{th}$ as infinitesimal cohesive neighbourhoods of objects in $\mathbf{H}$. \hypertarget{Definition}{}\subsubsection*{{Definition}}\label{Definition} \begin{defn} \label{InfinitesimalCohesiveInfTopos}\hypertarget{InfinitesimalCohesiveInfTopos}{} Given a cohesive $(\infty,1)$-topos $\mathbf{H}$ we say that an \textbf{infinitesimal cohesive neighbourhood} of $\mathbf{H}$ is another cohesive $(\infty,1)$-topos $\mathbf{H}_{th}$ equipped with an [[adjoint quadruple]] of [[adjoint (∞,1)-functors]] of the form \begin{displaymath} (i_! \dashv i^* \dashv i_* \dashv i^!) : \mathbf{H} \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\hookrightarrow}}{\underset{i^!}{\leftarrow}}}} \mathbf{H}_{th} \end{displaymath} where $i_!$ is a [[full and faithful (∞,1)-functor|full and faithful]] and preserves [[finite products]]. Conversely we will say that data as in def. \ref{InfinitesimalCohesiveInfTopos} equips the cohesive $\infty$-topos $\mathbf{H}$ with \textbf{differential cohesion}. \end{defn} \begin{remark} \label{}\hypertarget{}{} This definition is an abstraction of similar situations considered in (\hyperlink{SimpsonTeleman}{SimpsonTeleman}) and in \hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg}. See also the section at [[Q-category]]. \end{remark} \begin{remark} \label{InfinitesimalInclusionIfFullAndFaithful}\hypertarget{InfinitesimalInclusionIfFullAndFaithful}{} This implies that also $i_*$ is a [[full and faithful (∞,1)-functor]]. \end{remark} \begin{proof} By the characterizaton of full and faithful [[adjoint (∞,1)-functor]]s the condition on $i_!$ is equivalent to $i^* i_! \simeq Id$. Since $(i^* i_! \dashv i^* i_*)$ it follows by essential uniqueness of [[adjoint (∞,1)-functor]]s that also $i^* i_* \simeq Id$. \end{proof} \begin{remark} \label{}\hypertarget{}{} This definition captures the characterization of an [[infinitesimal object]] as having a single [[global element|global point]] surrounded by an infinitesimal neighbourhood: as we shall see in more detail \hyperlink{InfinitesimalPathsAndReduction}{below}, the [[(∞,1)-functor]] $i^*$ may be thought of as contracting away any infinitesimal extension of an object. Thus $X$ being an \hyperlink{InfinitesimalObject}{infinitesimal object} amounts to $i^* X \simeq *$, and the [[adjoint (∞,1)-functor|(∞,1)-adjunction]] $(i_! \dashv i^*)$ then indeed guarantees that $X$ has only a single global point, since \begin{displaymath} \begin{aligned} \mathbf{H}_{th}(*, X) & \simeq \mathbf{H}_{th}(i_! *, X) \\ & \simeq \mathbf{H}(*, i^* X) \\ & \simeq \mathbf{H}(*, *) \\ & \simeq * \end{aligned} \,. \end{displaymath} \end{remark} \begin{prop} \label{InfinitesimalNeighbourhoodIsOverInfGroupoid}\hypertarget{InfinitesimalNeighbourhoodIsOverInfGroupoid}{} The inclusion into the infinitesimal neighbourhood is necessarily a morphism of [[(∞,1)-topos]]es over [[∞Grpd]]. \begin{displaymath} \itexarray{ \mathbf{H} && \stackrel{(i^* \dashv i_*)}{\to} && \mathbf{H}_{th} \\ & {}_{\mathllap{\Gamma}}\searrow && \swarrow_{\mathrlap{\Gamma}} \\ && \infty Grpd } \end{displaymath} as is the induced geometric morphism $(i_* \dashv i^!) : \mathbf{H}_{th} \to \mathbf{H}$ \begin{displaymath} \itexarray{ \mathbf{H}_{th} && \stackrel{(i_* \dashv i^!)}{\to} && \mathbf{H} \\ & {}_{\mathllap{\Gamma}}\searrow && \swarrow_{\mathrlap{\Gamma}} \\ && \infty Grpd } \,. \end{displaymath} Moreover $i_*$ is necessarily a [[full and faithful (∞,1)-functor]]. \end{prop} \begin{proof} By essential uniqueness of th [[global section]] [[geometric morphism]]: In both cases the [[direct image]] functor has as [[left adjoint]] that preserves the [[terminal object]]. Therefore \begin{displaymath} \begin{aligned} \Gamma_{\mathbf{H}_{th}}( i_* X ) & \simeq \mathbf{H}_{th}(*, i_* X) \\ & \simeq \mathbf{H}(i^* *, X) \\ & \simeq \mathbf{H}(*, X) \\ & \simeq \Gamma_{\mathbf{H}}(X) \end{aligned} \,. \end{displaymath} Analogously in the second case. \end{proof} We shall write \begin{displaymath} (\Pi_{inf} \dashv Disc_{inf} \dashv \Gamma_{inf}) := (i^* \dashv i_* \dashv i^!) \end{displaymath} so that the [[global section]] geometric moprhism of $\mathbf{H}_{th}$ factors as \begin{displaymath} (\Pi_{\mathbf{H}_{th}} \dashv Disc_{\mathbf{H}_{th}} \dashv \Gamma_{\mathbf{H}_{th}}) : \mathbf{H}_{th} \stackrel{\overset{\Pi_{inf}}{\longrightarrow}}{\stackrel{\overset{Disc_{inf}}{\leftarrow}}{\underset{\Gamma_{inf}}{\longrightarrow}}} \mathbf{H} \stackrel{\overset{\Pi_{\mathbf{H}}}{\longrightarrow}}{\stackrel{\overset{Disc_{\mathbf{H}}}{\longleftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\longrightarrow}}} \infty Grpd \,. \end{displaymath} We also consider the [[(∞,1)-monads]]/comonads induced from these reflections: \begin{enumerate}% \item the [[reduction modality]] $\Re \coloneqq i_! i^\ast$ ; \item the [[infinitesimal shape modality]] $\Im \coloneqq i_\ast i^\ast$; \item the [[infinitesimal flat modality]] ${\&} \coloneqq i_* i^!$. \end{enumerate} The above says that these interact with the modalities of the ambient cohesion, i.e. \begin{enumerate}% \item the [[shape modality]] $ʃ$; \item the [[flat modality]] $\flat$; \item the [[sharp modality]] $\sharp$ \end{enumerate} as follows: \begin{displaymath} \itexarray{ && && \Re \\ && && \bot \\ && ʃ & \subset & \Im \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & {\&} \\ \bot & & \bot && \\ \ast & \subset& \sharp } \end{displaymath} Here the inclusion sign $\subset$ is to mean that the [[modal types]] of the modality on the left are included in the modal types of the modality on the right. For more details on this, see at \emph{[[geometry of physics -- categories and toposes]]} the section \emph{\href{geometry+of+physics+--+categories+and+toposes#ElasticToposes}{Elastic toposes}}. Let for the remainder of this section an infinitesimal neighbourhood $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$ be fixed. \begin{remark} \label{SequenceOfOrdersOfInfinitesimals}\hypertarget{SequenceOfOrdersOfInfinitesimals}{} More generally we may ask for a sequence of differential inclusions of $\infty$-toposes as above, reflecting ever higher orders of infinitesimals, hence notably a progression \begin{displaymath} ʃ \lt \Im = \Im_{(\infty)} \lt \cdots \lt \Im_{(3)} \lt \Im_{(2)} \lt \Im_{(1)} \lt id \end{displaymath} of [[infinitesimal shape modalities]] of various order, yielding a further factorization of the shape unit as \begin{displaymath} X \to \Im_{(1)}X \to \Im_{(2)}X \to \Im_{(3)} X \to \cdots \to \Im X \to ʃ X \,. \end{displaymath} \end{remark} \hypertarget{Properties}{}\subsubsection*{{Properties}}\label{Properties} \hypertarget{PresentationOnInfinitesimalNeighbourhoodSites}{}\paragraph*{{From $\infty$-sheaves over infinitesimal neighbourhood sites}}\label{PresentationOnInfinitesimalNeighbourhoodSites} We give a presentation of classes of infinitesimal neighbourhoods by [[simplicial presheaves]] on suitable [[sites]]. \begin{defn} \label{InfinitesimalNeighBourhoodSite}\hypertarget{InfinitesimalNeighBourhoodSite}{} Let $C$ be an [[∞-cohesive site]]. We say a [[site]] $C_{th}$ \begin{itemize}% \item equipped with a [[coreflective subcategory|coreflective embedding]] \begin{displaymath} (i \dashv p) : C \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\leftarrow}} C_{th} \end{displaymath} \item such that \begin{itemize}% \item $i$ preserves [[pullback]]s along morphisms in [[covering]] families; \item both $i$ and $p$ send [[covering]] families to covering families; \item for all $\mathbf{U}$ in $C_{th}$ and covering families $\{U_i \to p(\mathbf{U})\}$ there is a lift through $p$ to a covering family $\{\mathbf{U}_i \to \mathbf{U}\}$ \end{itemize} \end{itemize} is an \textbf{infinitesimal neighbourhood site} of $C$. \end{defn} \begin{prop} \label{InfinitesimalNeighbourhoodFromInfinitesimalSite}\hypertarget{InfinitesimalNeighbourhoodFromInfinitesimalSite}{} Let $C$ be an [[∞-cohesive site]] and $(i \dashv p) : C \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\leftarrow}} C_{th}$ an \hyperlink{InfinitesimalNeighBourhoodSite}{infinitesimal neighbourhood site}. Then the [[(∞,1)-category of (∞,1)-sheaves]] on $C_{th}$ is a cohesive $(\infty,1)$-topos and the restriction $i^*$ along $i$ exhibits it as an \hyperlink{InfinitesimalNeighbourhoodIsOverInfGroupoid}{infinitesimal neighbourhood} of the cohesive $(\infty,1)$-topos over $C$. \begin{displaymath} ( i_! \dashv i^* \dashv i_* \dashv i^! ) : Sh_{(\infty,1)}(C) \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\to}}{\stackrel{i^!}{\leftarrow}}}} Sh_{(\infty,1)}(C^{th}) \,. \end{displaymath} Moreover, $i_!$ restricts on representables to the [[(∞,1)-Yoneda embedding]] factoring through $i$: \begin{displaymath} \itexarray{ C &\hookrightarrow& Sh_{(\infty,1)}(C) \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{i_!}} \\ C_{th} &\hookrightarrow& Sh_{(\infty,1)}(C_{th}) } \,. \end{displaymath} \end{prop} \begin{proof} We [[presentable (∞,1)-category|present]] the [[(∞,1)-sheaf (∞,1)-category]] $Sh_{(\infty,1)}(C_{th})$ by the projective [[model structure on simplicial presheaves]] [[Bousfield localization of model categories|left Bousfield localized]] at the [[covering]] [[sieve]] inclusions \begin{displaymath} Sh_{(\infty,1)}(C_{th}) \simeq ([C_{th}^{op}, sSet]_{loc})^\circ \end{displaymath} (as discussed at [[models for ∞-stack (∞,1)-toposes|models for (∞,1)-sheaf (∞,1)-toposes]]). Consider the right [[Kan extension]] $Ran_i : [C^{op}, sSet] \to [C_{th}^{op},sSet]$ of [[simplicial presheaves]] along the functor $i$. On an object $K \times D \in C_{th}$ it is given by the [[end]]-expression \begin{displaymath} \begin{aligned} \mathrm{Ran}_{i} F : \mathbf{K} & \mapsto \int_{U \in C} \mathrm{sSet}( C_{\mathrm{th}}(i(U), \mathbf{K}) , F(U)) \\ & \simeq \int_{U \in C} \mathrm{sSet}( C(U, p(\mathbf{K})) , F(U)) \\ & \simeq F(p(\mathbf{K})) \\ & =: (p^* F)(\mathbf{K}) \end{aligned} \,, \end{displaymath} where in the last step we use the [[Yoneda reduction]]-form of the [[Yoneda lemma]]. This shows that the [[right adjoint]] to $(-)\circ i$ is itself given by precomposition with a functor, and hence has itself a further right adjoint, which gives us a total of four [[adjoint functor]]s \begin{displaymath} [C^{op}, sSet] \stackrel{\overset{Lan_i}{\longrightarrow}}{\stackrel{\overset{(-)\circ i}{\longleftarrow}}{\stackrel{\overset{(-)\circ p}{\longrightarrow}}{\underset{Ran_p}{\longleftarrow}}}} [C_{th}^{op}, sSet] \,. \end{displaymath} From this are directly induced the corresponding [[simplicial Quillen adjunction]]s on the global projective and injective [[model structure on simplicial presheaves]] \begin{displaymath} (Lan_i \dashv (-) \circ i) : [C^{op}, sSet]_{proj} \stackrel{\overset{Lan_i}{\to}}{\underset{(-)\circ i}{\leftarrow}} [C_{th}^{op}, sSet]_{proj} \,; \end{displaymath} \begin{displaymath} ((-)\circ i \dashv (-) \circ p) : [C^{op}, sSet]_{proj} \stackrel{\overset{(-)\circ i}{\longleftarrow}} {\underset{(-)\circ p}{\longrightarrow}} [C_{th}^{op}, sSet]_{proj} \,; \end{displaymath} \begin{displaymath} ((-) \circ p \dashv Ran_p) : [C^{op}, sSet]_{inj} \stackrel{\overset{(-)\circ p}{\longrightarrow}}{\underset{Ran_p}{\longleftarrow}} [C_{th}^{op}, sSet]_{inj} \,. \end{displaymath} Observe that $Lan_i$, being a left [[Kan extension]], sends representables to representables: we have \begin{displaymath} Lan_i C(-,T) : \mathbf{K} \mapsto \int^{U \in C} C_{th}(\mathbf{K}, i(U)) \cdot C(U,T) \end{displaymath} and by [[Yoneda reduction]] (more explicitly: observing that this is equivalently the formula for left [[Kan extension]] of the non-corepresentable $C_{th}(K \times D, i(-)) : C \to sSet$ along the identity functor) this is \begin{displaymath} \cdots \simeq C_{th}(\mathbf{K}, i(T)) \,. \end{displaymath} By the discussion at [[simplicial Quillen adjunction]] for the above Quillen adjunctions to descend to the Cech-local [[model structure on simplicial presheaves]] it suffices that the [[right adjoint]]s preserve locally fibrant objects. We first check that $(-) \circ i$ sends locally fibrant objects to locally fibrant objects. To that end, let $\{U_i \to U\}$ be a [[covering family]] in $C$. Write $\int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} (j(U_{i_0}) \times_{j(U)} j(U_{i_1}) \times_{j(U)} \cdots \times_{j(U)} j(U_k))$ for its [[Cech nerve]], where $j$ denotes the [[Yoneda embedding]]. Recall by the definition of the [[∞-cohesive site]] $C$ that all the [[fiber product]]s of representable presheaves here are again themselves representable, hence $\cdots = \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} (j(U_{i_0} \times_U U_{i_1} \times_U \cdots \times_U U_k))$. This means that the [[left adjoint]] $Lan_i$ preserves not only the [[coend]] and [[tensoring]], but by the remark in the previous paragraph and the assumption that $i$ preserves [[pullback]]s along covers we have that \begin{displaymath} \begin{aligned} Lan_i C(\{U_i \to U\}) & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} Lan_i (j(U_{i_0} \times_U U_{i_1} \times_U \cdots \times_U U_k)) \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} j i (U_{i_0} \times_U U_{i_1} \times_U \cdots \times_U U_k) \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} j (i(U_{i_0}) \times_{i(U)} i(U_{i_1}) \times_{i(U)} \cdots \times_{i(U)} i(U_k)) \end{aligned} \,. \end{displaymath} By the assumption that $i$ preserves covers, this is the [[Cech nerve]] of a [[covering family]] in $C_{th}$. Therefore for $F \in [C_{th}^{op}, sSet]_{proj,loc}$ fibrant we have for all [[covering]]s $\{U_i \to U\}$ in $C$ that the [[descent]] morphism \begin{displaymath} (i^* F)(U) = F(i(U)) \stackrel{}{\to} [C_{th}^{op}, sSet](C(\{i(U_i)\}), F) = [C^{op}, sSet](C(\{U_i\}), i^* F) \end{displaymath} is a weak equivalence, hence that $i^* F$ is locally fibrant. To see that $(-) \circ p$ preserves locally fibrant objects, we apply the analogous reasoning after observing that its [[left adjoint]] $(-)\circ i$ preserves all [[limit]]s and [[colimit]]s of [[simplicial presheaves]] (as these are computed objectwise) and by observing that for $\{\mathbf{U}_i \stackrel{p_i}{\to} \mathbf{U}\}$ a covering family in $C_{th}$ we have that its image under $(-) \circ i$ is its image under $p$, by the [[Yoneda lemma]]: \begin{displaymath} \begin{aligned} [C^{op}, sSet](K, ((-)\circ i) (\mathbf{U})) & \simeq C_{th}(i(K), \mathbf{U}) \\ & \simeq C(K, p(\mathbf{U})) \end{aligned} \end{displaymath} and using that $p$ preserves covers by assumption. Therefore $(-) \circ i$ is a left and right local [[Quillen adjunction|Quillen functor]] with left local Quillen adjoint $Lan_i$ and right local Quillen adjoint $(-)\circ p$. It follows that $i^* : Sh_{(\infty,1)}(C_{th}) \to Sh_{(\infty,1)}(C)$ is given by the left [[derived functor]] of restriction along $i$, and $i_* : Sh_{(\infty,1)}(C) \to Sh_{(\infty,1)}(C_{th})$ is given by the right [[derived functor]] of restriction along $p$. Finally to see that also $Ran_p$ preserves locally fibrant objects by the same reasoning as above, notice that for every [[covering]] family $\{U_i \to U\}$ in $C$ and every morphism $\mathbf{K} \to p^* U$ in $C_{th}$ we may find a covering $\{\mathbf{K}_j \to \mathbf{K}\}$ of $\mathbf{K}$ such that we find commuting diagrams on the left of \begin{displaymath} \itexarray{ \mathbf{K}_j &\to& p^* U_{i(j)} \\ \downarrow && \downarrow \\ \mathbf{K} &\to& p^* U } \;\;\; \leftrightarrow \;\;\; \itexarray{ p(\mathbf{K}_j) & =& i^*(\mathbf{K}_j) &\to& U_{i(j)} \\ \downarrow && \downarrow && \downarrow \\ p(\mathbf{K}) &= & i^*(\mathbf{K}) &\to& U } \,, \end{displaymath} because by adjunction these correspond to commuting diagrams as indicated on the right, which exist by definition of [[coverage]] on $C$ and lift through $p$ by assumption on $C_{th}$. This implies that $\{p^* U_i \to p^* U\}$ is a \emph{generalized cover} in the terminology at [[model structure on simplicial presheaves]], which by the discussion there implies that the corresponding [[Cech nerve]] equivalent to the [[sieve]] inclusion is a weak equivalence. This establishes the quadruple of [[adjoint (∞,1)-functor]]s as claimed. It remains to see that $i_!$ is full and faithful. For that notice the general fact that left [[Kan extension]] (see the properties discussed there) along a [[full and faithful functor]] $i$ satisfies $Lan_i \circ i \simeq id$. It remains to observe that since $(-)\circ i$ is not only right but also left Quillen by the above, we have that $i^* Lan_i$ applied to a cofibrant object is already the [[derived functor]] of the composite. \end{proof} \begin{remark} \label{}\hypertarget{}{} Conversely this implies that $Sh_{(\infty,1)}(C_{th})$ is an [[∞-connected (∞,1)-topos]] over [[Smooth∞Grpd]], exhibited by the triple of adjunctions \begin{displaymath} (i^* \dashv i_* \dashv i^!) : SynthDiff \infty Grpd \to Smooth \infty Grpd \,. \end{displaymath} \end{remark} \hypertarget{RelationToInfinitesimalCohesion}{}\paragraph*{{Relation to infinitesimal cohesion}}\label{RelationToInfinitesimalCohesion} We discuss how differential cohesion in the sense of def. \ref{InfinitesimalCohesiveInfTopos} relates to [[infinitesimal cohesion]]. \begin{quote}% under construction \end{quote} \begin{defn} \label{InducedRelativeShapeAndFlat}\hypertarget{InducedRelativeShapeAndFlat}{} Given differential cohesion, def. \ref{InfinitesimalCohesiveInfTopos}, \begin{displaymath} \itexarray{ \Re &\dashv& \Im &\dashv& {\&} \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp } \end{displaymath} define operations $ʃ^{rel}$ and $\flat^{rel}$ by \begin{displaymath} ʃ^{rel} X \coloneqq (ʃ X) \underset{\Re X}{\coprod} X \end{displaymath} \begin{displaymath} \flat^{rel} X \coloneqq (\flat X) \underset{\Im}{\times} X \,. \end{displaymath} Hence $ʃ^{rel} X$ makes a [[homotopy pushout]] square \begin{displaymath} \itexarray{ \Re X &\longrightarrow& X \\ \downarrow && \downarrow \\ ʃ X &\longrightarrow& ʃ^{rel} X } \end{displaymath} and $\flat^{rel}$ makes a [[homotopy pullback]] square \begin{displaymath} \itexarray{ \flat^{rel} X &\longrightarrow& X \\ \downarrow && \downarrow \\ \flat X &\longrightarrow& \Im X } \,. \end{displaymath} We call $ʃ^{rel}$ the \emph{[[relative shape modality]]} and $\flat^{rel}$ the \emph{[[relative flat modality]]}. \end{defn} \begin{prop} \label{}\hypertarget{}{} The relative shape and flat modalities of def. \ref{InducedRelativeShapeAndFlat} \begin{enumerate}% \item form an [[adjoint pair]] $(ʃ^{rel} \dashv \flat^{rel})$; \item whose (co-)[[modal types]] are precisely the properly infinitesimal types, hence those for which $\flat \to \Im$ is an [[equivalence]]; \item $ʃ^{rel}$ preserves the [[terminal object]]. \end{enumerate} \end{prop} It follows that when $\flat^{rel}$ has a further [[right adjoint]] $\sharp^{rel}$ with equivalent [[modal types]] containing the [[codiscrete object|codiscrete types]], then this defines a [[level of a topos|level]] \begin{displaymath} \itexarray{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv& \sharp \\ \vee && \vee \\ \emptyset &\dashv& \ast } \end{displaymath} hence an intermediate subtopos $\infty Grpd \hookrightarrow \mathbf{H}_{infinitesimal}\hookrightarrow \mathbf{H}_{th}$ which is [[infinitesimal cohesion|infinitesimally cohesive]]. This happens notably for the model of [[formal smooth ∞-groupoids]] and all its variants such as formal [[complex analytic ∞-groupoids]] etc. But in this case $(\flat^{rel} \dashv \sharp^{rel})$ does not provide [[Aufhebung]] for $(\flat \dashv \sharp)$. (\ldots{}) \begin{prop} \label{CounitOfFlatRelIsFormallyEtale}\hypertarget{CounitOfFlatRelIsFormallyEtale}{} The [[counit of a comonad|counit]] of the relative flat modality is a [[formally étale morphism]]. \end{prop} \begin{proof} From the fact that the [[infinitesimal shape modality]] is [[idempotent monad|idempotent]] and preserves [[homotopy pullbacks]]. \end{proof} \hypertarget{StructuresInDifferentialCohesion}{}\subsubsection*{{Structures in a differential cohesive $(\infty,1)$-topos}}\label{StructuresInDifferentialCohesion} We discuss structures that are canonically present in a cohesive $(\infty,1)$-topos equipped with differential cohesion. These structures parallel the \hyperlink{Structures}{structures in a general cohesive (∞,1)-topos}. \hypertarget{InfinitesimalPaths}{}\paragraph*{{Infinitesimal paths and de Rham spaces}}\label{InfinitesimalPaths} In the presence of \hyperlink{InfinitesimalCohesiveInfTopos}{differential cohesion} there is an infinitesimal analog of the \hyperlink{Paths}{geometric paths ∞-groupoids}. \hypertarget{infinitesimal_path_groupoid}{}\paragraph*{{Infinitesimal path $\infty$-groupoid}}\label{infinitesimal_path_groupoid} \begin{defn} \label{InfinitesimalPathsAndReduction}\hypertarget{InfinitesimalPathsAndReduction}{} Define the [[adjoint triple]] of [[adjoint (∞,1)-functor]]s corresponding to the [[adjoint quadruple]] $(i_! \dashv i^* \dashv i_* \dashv i^!)$: \begin{displaymath} (\Re \dashv \Im \dashv \& ) : (i_! i^* \dashv i_* i^* \dashv i_* i^! ) : \mathbf{H}_{th} \to \mathbf{H}_{th} \,. \end{displaymath} We say that \begin{itemize}% \item $\Re$ is the \textbf{[[reduction modality]]}. \item $\Im$ is the \textbf{[[infinitesimal shape modality]]}. \item $\&$ is the \textbf{[[infinitesimal flat modality]]}. \end{itemize} An object in the full sub-$\infty$-category \begin{itemize}% \item of $\Re$ we call a \textbf{[[reduced object]]} \item of $\Im$ we call a \textbf{[[coreduced object]]}. \end{itemize} For $X\in \mathbf{H}_{th}$ we say that \begin{itemize}% \item $\Im(X)$ is the \textbf{[[infinitesimal path ∞-groupoid]]} of $X$; The $(i^* \dashv i_*)$-[[unit of an adjunction|unit]] \begin{displaymath} X \to \Im(X) \end{displaymath} we call the \textbf{constant infinitesimal path inclusion}. \item $\Re(X)$ is the \textbf{[[reduced cohesive ∞-groupoid]]} underlying $X$. The $(i_* \dashv i^*)$-[[unit of an adjunction|counit]] \begin{displaymath} \Re X \to X \end{displaymath} we call the \textbf{inclusion of the reduced part} of $X$. \end{itemize} \end{defn} \begin{remark} \label{TerminologyDeRhamspace}\hypertarget{TerminologyDeRhamspace}{} In traditional contexts see (\hyperlink{SimpsonTeleman}{SimpsonTeleman, p. 7}) the object $\Im(X)$ is called the \textbf{[[de Rham space]] of $X$} or the \textbf{de Rham stack of $X$} . Here we may tend to avoid this terminology, since by the discussion at we have a good notion of intrinsic [[de Rham cohomology]] in any [[cohesive (∞,1)-topos]] already without equipping it with differential cohesion. From this point of view the object $\Im(X)$ is not primarily characterized by the fact that (in some models, see \hyperlink{Examples}{below}) it does co-represent de Rham cohomology -- because the object $\mathbf{\Pi}_{dR}(X)$ from \hyperlink{deRhamCohomology}{above} does, too -- but by the fact that it does so in an explicitly ([[synthetic differential geometry|synthetic]]) infinitesimal way. \end{remark} \begin{prop} \label{InclusionOfConstantIntoInfinitesimalIntoAllPaths}\hypertarget{InclusionOfConstantIntoInfinitesimalIntoAllPaths}{} There is a canonical [[natural transformation]] \begin{displaymath} \Im(X) \to \int(X) \end{displaymath} that factors the finite path inclusion through the infinitesimal one \begin{displaymath} \itexarray{ && \Im(X) \\ & \nearrow && \searrow \\ X &&\to&& \int(X) } \,. \end{displaymath} \end{prop} \begin{proof} This is the formula for the [[unit of an adjunction|unit]] of the composite adjunction $\mathbf{H}_{th} \stackrel{\overset{\Pi_{inf}}{\to}}{\underset{Disc_{inf}}{\leftarrow}} \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\underset{Disc}{\leftarrow}} \infty Grpd$: \begin{displaymath} X \stackrel{i_{inf}X}{\to} Disc_{inf}\Pi_{inf} X \stackrel{Disc_{inf}(i_{\mathbf{H}}\Pi_{inf}X)}{\to} Disc_{\mathbf{H}} Disc_{inf} \Pi_{\mathbf{H}} \Pi_{inf} X = Disc_{\mathbf{H}_{th}} \Pi_{\mathbf{H}_{th}} \,. \end{displaymath} \end{proof} \hypertarget{JetBundleObjects}{}\paragraph*{{Jet $\infty$-bundles}}\label{JetBundleObjects} Notice that for $f : X \to Y$ any [[morphism]] in any [[(∞,1)-topos]] $\mathbf{H}$, there is the corresponding [[base change geometric morphism]] between the [[over-(∞,1)-toposes]] \begin{displaymath} (f^* \dashv f_*) : \mathbf{H}/X \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathbf{H}/Y \,. \end{displaymath} \begin{defn} \label{}\hypertarget{}{} For any object $X \in \mathbf{H}$ write \begin{displaymath} Jet : \mathbf{H}/X \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}} \mathbf{H}/\Im(X) \end{displaymath} for the [[base change geometric morphism]] induced by the constant infinitesimal path inclusion $i : X \to \Im(X)$, def. \ref{InfinitesimalPathsAndReduction}. For $(E \to X) \in \mathbf{H}/X$ we call $Jet(E) \to \Im(X)$ as well as its pullback $i^* Jet(E) \to X$ (depending on context) the \textbf{[[jet bundle]]} of $E \to X$. \end{defn} \hypertarget{formally_smoothtaleunramified_morphisms}{}\paragraph*{{Formally smooth/\'e{}tale/unramified morphisms}}\label{formally_smoothtaleunramified_morphisms} \begin{defn} \label{FormalSmoothness}\hypertarget{FormalSmoothness}{} We say an object $X \in \mathbf{H}_{th}$ is \textbf{formally smooth} if the constant infinitesimal path inclusion, $X \to \Im(X)$, def. \ref{InfinitesimalPathsAndReduction}, is an [[effective epimorphism in an (∞,1)-category|effective epimorphism]]. \end{defn} In this form this is the evident $(\infty,1)$-categorical analog of the conditions as they appear for instance in (\hyperlink{SimpsonTeleman}{SimpsonTeleman, page 7}). \begin{remark} \label{FormalSmoothnessByCanonicalMorphism}\hypertarget{FormalSmoothnessByCanonicalMorphism}{} An object $X \in \mathbf{H}_{th}$ is formally smooth according to def. \ref{FormalSmoothness} precisely if the canonical morphism \begin{displaymath} i_! X \to i_* X \end{displaymath} (induced from the [[adjoint quadruple]] $(i_! \dashv i^* \dashv i_* \dashv i^!)$, see there) is an [[effective epimorphism in an (∞,1)-category|effective epimorphism]]. \end{remark} \begin{proof} The canonical morphism is the composite \begin{displaymath} (i_! \to i_*) := i_! \stackrel{\eta i_!}{\to} \Im i_! := i_* i^* i_! \stackrel{\simeq}{\to} i_* \,. \end{displaymath} By the condition that $i_!$ is a [[full and faithful (∞,1)-functor]] the second morphism here in an [[equivalence in an (∞,1)-category|equivalence]], as indicated, and hence the component of the composite on $X$ being an effective epimorphism is equivalent to the component $i_! X \to \mathbf{\Pi} i_! X$ being an effective epimorphism. \end{proof} \begin{remark} \label{RelationToRK}\hypertarget{RelationToRK}{} In this form this characterization of formal smoothness is the evident generalization of the condition given in (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, section 4.1}). See the section \emph{} at \emph{[[Q-category]]} for more discussion. Notice that by the notation there is related to the one used here by $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$. \end{remark} Therefore we have the following more general definition. \begin{defn} \label{FormalRelativeSmoothnessByCanonicalMorphism}\hypertarget{FormalRelativeSmoothnessByCanonicalMorphism}{} For $f : X \to Y$ a morphism in $\mathbf{H}$, we say that \begin{enumerate}% \item $f$ is a \textbf{[[formally smooth morphism]]} if the canonical morphism \begin{displaymath} i_! X \to i_! Y \prod_{i_* Y} i_* Y \end{displaymath} is an [[effective epimorphism in an (∞,1)-category|effective epimorphism]]. \item $f$ is a \textbf{[[formally unramified morphism]]} if this is a [[(-1)-truncated]] morphism. More generally, $f$ is an \emph{order-$k$ formally unramified morphisms} for $(-2) \leq k \leq \infty$ if this is a [[k-truncated]] morphism. \item $f$ is a \textbf{[[formally étale morphism]]} if this morphism is an [[equivalence in an (∞,1)-category|equivalence]], hence if \begin{displaymath} \itexarray{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y } \end{displaymath} is an [[(∞,1)-pullback]] square. \end{enumerate} \end{defn} \begin{remark} \label{MeaningOfFormallyUnramified}\hypertarget{MeaningOfFormallyUnramified}{} An order-(-2) formally unramified morphism is equivalently a [[formally étale morphism]]. Only for [[0-truncated]] $X$ does formal smoothness together with formal unramifiedness imply formal \'e{}taleness. \end{remark} Even more generally we can formulate formal smoothness in $\mathbf{H}_{th}$: \begin{defn} \label{FormallyEtaleInHTh}\hypertarget{FormallyEtaleInHTh}{} A morphism $f \colon X \to Y$ in $\mathbf{H}_{th}$ is \textbf{[[formally etale morphism|formally étale]]} if it is $\Im$-[[Pi-closed morphism|closed]], hence if its $\Im$-unit naturality square \begin{displaymath} \itexarray{ X &\to& \Im(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im(f)}} \\ Y &\to& \Im(y) } \end{displaymath} is an [[(∞,1)-pullback]]. \end{defn} \begin{remark} \label{}\hypertarget{}{} A morphism $f$ in $\mathbf{H}$ is formally etale in the sense of def. \ref{FormalRelativeSmoothnessByCanonicalMorphism} precisely if its image $i_!(f)$ in $\mathbf{H}_{th}$ is formally etale in the sense of def. \ref{FormallyEtaleInHTh}. \end{remark} \begin{proof} This is again given by the fact that $\Im = i_* i^*$ by definition and that $i_!$ is fully faithful, so that \begin{displaymath} \itexarray{ i_! X &\to& \Im(i_! X) \simeq i_* i^* i_! X &\stackrel{\simeq}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* i^* i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! Y &\to& \Im(i_! Y) \simeq i_* i^* i_! Y &\stackrel{\simeq}{\to}& i_* Y } \,. \end{displaymath} \end{proof} \begin{prop} \label{PropertiesOfFormallyEtaleMorphisms}\hypertarget{PropertiesOfFormallyEtaleMorphisms}{} The collection of [[formally étale morphisms]] in $\mathbf{H}$, def. \ref{FormalRelativeSmoothnessByCanonicalMorphism}, is closed under the following operations. \begin{enumerate}% \item Every [[equivalence in an (∞,1)-category|equivalence]] is formally \'e{}tale. \item The composite of two formally \'e{}tale morphisms is itself formally \'e{}tale. \item If \begin{displaymath} \itexarray{ && Y \\ & {}^{\mathllap{f}}\nearrow &\swArrow_{\simeq}& \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z } \end{displaymath} is a [[diagram]] such that $g$ and $h$ are formally \'e{}tale, then also $f$ is formally \'e{}tale. \item Any [[retract]] of a formally \'e{}tale morphisms is itself formally \'e{}tale. \item The [[(∞,1)-pullback]] of a formally \'e{}tale morphisms is formally \'e{}tale if the pullback is preserved by $i_!$. \end{enumerate} \end{prop} The statements about closure under composition and pullback appears as(\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenberg, prop. 5.4, prop. 5.6}). Notice that the extra assumption that $i_!$ preserves the pullback is implicit in their setup, by remark \ref{RelationToRK}. \begin{proof} The first statement follows since $\infty$-pullbacks are well defined up to quivalence. The second two statements follow by the [[pasting law]] for [[(∞,1)-pullback]]s: let $f : X \to Y$ and $g : Y \to Z$ be two morphisms and consider the [[pasting diagram]] \begin{displaymath} \itexarray{ i_! X &\stackrel{i_! f }{\to}& i_! Y &\stackrel{i_! g}{\to}& Z \\ \downarrow && \downarrow && \downarrow \\ i_* X &\stackrel{i_* f }{\to}& i_* Y &\stackrel{i_* g}{\to}& i_* Z } \,. \end{displaymath} If $f$ and $g$ are formally \'e{}tale then both small squares are pullback squares. Then the pasting law says that so is the outer rectangle and hence $g \circ f$ is formally \'e{}tale. Similarly, if $g$ and $g \circ f$ are formally \'e{}tale then the right square and the total reactangle are pullbacks, so the pasting law says that also the left square is a pullback and so also $f$ is formally \'e{}tale. For the fourth claim, let $Id \simeq (g \to f \to g)$ be a [[retract]] in the [[arrow category|arrow (∞,1)-category]] $\mathbf{H}^I$. By applying the natural transformation $\phi : i_! \to I_*$ we obtain a retract \begin{displaymath} Id \simeq ((i_! g \to i_*g) \to (i_! f \to i_*f) \to (i_! g \to i_*g)) \end{displaymath} in the category of squares $\mathbf{H}^{\Box}$. We claim that generally, if the middle piece in a retract in $\mathbf{H}^\Box$ is an [[(∞,1)-pullback]] square, then so is its retract sqare. This implies the fourth claim. To see this, we use that \begin{enumerate}% \item [[(∞,1)-limit]]s are computed by [[homotopy limit]]s in any [[presentable (∞,1)-category]] $C$ presenting $\mathbf{H}$; \item homotopy limits in $C$ may be computed by the left and right adjoints provided by the [[derivator]] $Ho(C)$ associated to $C$. \end{enumerate} From this the claim follows as described in detail at \emph{[[retract]]} in the section \emph{} . For the last claim, consider an [[(∞,1)-pullback]] diagram \begin{displaymath} \itexarray{ A \times_Y X &\to& X \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{f}} \\ A &\to& Y } \end{displaymath} where $f$ is formally \'e{}tale. Applying the [[natural transformation]] $\phi : i_! \to i_*$ to this yields a square of squares. Two sides of this are the [[pasting]] composite \begin{displaymath} \itexarray{ i_! A \times_Y X &\to& i_! X &\stackrel{\phi_X}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! p}} && \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! A &\to& i_! Y &\stackrel{\phi_Y}{\to}& i_* Y } \end{displaymath} and the other two sides are the pasting composite \begin{displaymath} \itexarray{ i_! A \times_Y X &\stackrel{\phi_{A \times_Y X}}{\to}& i_* A \times_Y A &\stackrel{}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! p}} && \downarrow^{\mathrlap{i_* p}} && \downarrow^{\mathrlap{i_* f}} \\ i_! A &\stackrel{\phi_A}{\to}& i_* A &\to& i_* Y } \,. \end{displaymath} Counting left to right and top to bottom, we have that \begin{itemize}% \item the first square is a pullback by assumption that $i_!$ preserves the given pullback; \item the second square is a pullback, since $f$ is formally \'e{}tale. \item the total top rectangle is therefore a pullback, by the [[pasting law]]; \item the fourth square is a pullback since $i_*$ is [[right adjoint]] and so also preserves pullbacks; \item also the total bottom rectangle is a pullback, since it is equal to the top total rectangle; \item therefore finally the third square is a pullback, by the other clause of the [[pasting law]]. Hence $p$ is formally \'e{}tale. \end{itemize} \end{proof} \begin{remark} \label{AsOpenMaps}\hypertarget{AsOpenMaps}{} The properties listed in prop. \ref{PropertiesOfFormallyEtaleMorphisms} correspond to the axioms on the \emph{[[open map]]s} (``admissible maps'') in a [[geometry (for structured (∞,1)-toposes)]] (\hyperlink{LurieStSp}{Lurie, def. 1.2.1}). This means that a notion of formally \'e{}tale morphisms induces a notion of [[locally algebra-ed topos|locally algebra-ed (∞,1)toposes]]/[[structured (∞,1)-toposes]] in a cohesive context. This is discuss in \begin{itemize}% \item [[cohesive (∞,1)-topos -- structure ∞-sheaves]]. \end{itemize} \end{remark} In order to interpret the notion of formal smoothness, we turn now to the discussion of infinitesimal reduction. \begin{prop} \label{RedIsIdempotent}\hypertarget{RedIsIdempotent}{} The operation $\Re$ is an [[idempotent]] projection of $\mathbf{H}_{th}$ onto the image of $\mathbf{H}$ \begin{displaymath} \Re \Re \simeq \Re \,. \end{displaymath} Accordingly also \begin{displaymath} \Im \Im \simeq \Im \end{displaymath} \end{prop} \begin{proof} By definition of infinitesimal neighbourhood we have that $i_!$ is a [[full and faithful (∞,1)-functor]]. It follows that $i^* i_! \simeq Id$ and hence \begin{displaymath} \begin{aligned} \Re \Re & \simeq i_! i^* i_! i^* \\ & \simeq i_! i^* \\ & \simeq \Re \end{aligned} \,. \end{displaymath} \end{proof} \begin{cor} \label{PiInfXIsFormallySmooth}\hypertarget{PiInfXIsFormallySmooth}{} For every $X \in \mathbf{H}_{th}$, we have that $\Im(X)$ is formally smooth according to def. \ref{FormalSmoothness}. \end{cor} \begin{proof} By prop. \ref{RedIsIdempotent} we have that \begin{displaymath} \Im(X) \to \Im \Im X \end{displaymath} is an [[equivalence in an (∞,1)-category|equivalence]]. As such it is in particular an [[effective epimorphism in an (∞,1)-category|effective epimorphism]]. \end{proof} \hypertarget{InfinitesimalA1Homotopy}{}\paragraph*{{Infinitesimal $\mathbb{A}^1$-homotopy}}\label{InfinitesimalA1Homotopy} \begin{defn} \label{}\hypertarget{}{} A set of objects $\{D_\alpha \in \mathbf{H}_{th}\}_\alpha$ is said to \textbf{exhibit the differential structure} or \textbf{exhibit the infinitesimal thickening} if the [[localization of an (∞,1)-category|localization]] \begin{displaymath} L_{\{D_\alpha\}_\alpha} \mathbf{H}_{th} \stackrel{\leftarrow}{\hookrightarrow} \mathbf{H}_{th} \end{displaymath} of $\mathbf{H}_{th}$ at the morphisms of the form $D_\alpha \times X \to X$ is exhibited by the [[infinitesimal shape modality]] $\Im$. \end{defn} \begin{Remark} \label{}\hypertarget{}{} This is the [[infinitesimal cohesion|infinitesimal]] analog of the notion of objects exhibiting [[cohesion]], see at \hyperlink{cohesive+%28infinity,1%29-topos+--+structures#A1HomotopyContinuum}{structures in cohesion -- A1-homotopy and the continuum}. \end{Remark} For more see at \emph{[[Lie differentiation]]}. \hypertarget{StructureSheaves}{}\paragraph*{{Structure sheaves}}\label{StructureSheaves} We discuss how in differential cohesion $\mathbf{H}_{th}$ every object $X$ canonically induces its [[étale (∞,1)-topos]] $Sh_{\mathbf{H}_{th}}(X)$. For $X \in \mathbf{H}_{th}$ any object in a differential cohesive $\infty$-topos, we formulate \begin{enumerate}% \item the [[(∞,1)-topos]] denoted $\mathcal{X}$ or $Sh_\infty(X)$ of [[(∞,1)-sheaves]] over $X$, or rather of formally \'e{}tale maps into $X$; \item the [[structure (∞,1)-sheaf]] $\mathcal{O}_{X}$ of $X$. \end{enumerate} The resulting structure is essentially that discussed (\hyperlink{Lurie}{Lurie, Structured Spaces}) if we regard $\mathbf{H}_{th}$ equipped with its formally \'e{}tale morphisms, def. \ref{FormallyEtaleInHTh}, as a ([[large category|large]]) [[geometry for structured (∞,1)-toposes]]. One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let $G \in Grp(\mathbf{H}_{th})$ a differential cohesive [[∞-group]] with \href{cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology}{de Rham coefficient object} $\flat_{dR}\mathbf{B}G$ and for $X \in \mathbf{H}_{th}$ any differential homotopy type, the product projection \begin{displaymath} X \times \flat_{dR} \mathbf{B}G \to X \end{displaymath} regarded as an object of the [[slice (∞,1)-topos]] $(\mathbf{H}_{th})_{/X}$ \emph{almost} qualifies as a ``bundle of flat $\mathfrak{g}$-valued differential forms'' over $X$: for $U \to X$ an cover (a [[1-epimorphism]]) regarded in $(\mathbf{H}_{th})_{/X}$, a $U$-plot of this product projection is a $U$-plot of $X$ together with a flat $\mathfrak{g}$-valued de Rham cocycle on $X$. This is indeed what the sections of a corresponding bundle of differential forms over $X$ are supposed to look like -- but only \emph{if} $U \to X$ is sufficiently \emph{spread out} over $X$, hence sufficiently [[étale map|étale]]. Because, on the extreme, if $X$ is the point (the [[terminal object]]), then there should be no non-trivial section of differential forms relative to $U$ over $X$, but the above product projection instead reproduces all the sections of $\flat_{dR} \mathbf{B}G$. In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be \emph{restricted} to plots out of suficiently \'e{}tale maps into $X$. In order to correctly test differential form data, ``suitable'' here should be ``formally'', namely infinitesimally. Hence the restriction should be along the full inclusion \begin{displaymath} (\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X} \end{displaymath} of the formally \'e{}tale maps of def. \ref{FormallyEtaleInHTh} into $X$. Since on formally \'e{}tale covers the sections should be those given by $\flat_{dR}\mathbf{B}G$, one finds that the corresponding ``cotangent bundle'' must be the [[coreflective subcategory|coreflection]] along this inclusion. The following proposition establishes that this coreflection indeed exists. \begin{defn} \label{EtaleSlice}\hypertarget{EtaleSlice}{} For $X \in \mathbf{H}_{th}$ any object, write \begin{displaymath} (\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X} \end{displaymath} for the full [[sub-(∞,1)-category]] of the [[slice (∞,1)-topos]] over $X$ on those maps into $X$ which are formally \'e{}tale, def. \ref{FormallyEtaleInHTh}. We also write $FEt_{\mathbf{X}}$ or $Sh_{\mathbf{H}}(X)$ for $(\mathbf{H}_{th})_{/X}^{fet}$. \end{defn} \begin{prop} \label{EtalificationIsCoreflection}\hypertarget{EtalificationIsCoreflection}{} The inclusion $\iota$ of def. \ref{EtaleSlice} is both [[reflective sub-(∞,1)-category|reflective]] as well as [[coreflective subcategory|coreflective]], hence it fits into an [[adjoint triple]] of the form \begin{displaymath} (\mathbf{H}_{th})_{/X}^{fet} \stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{\iota}{\hookrightarrow}}{\underset{Et}{\leftarrow}}} (\mathbf{H}_{th})_{/X} \,. \end{displaymath} \end{prop} \begin{proof} By the general discussion at \emph{[[reflective factorization system]]}, the reflection is given by sending a morphism $f \colon Y \to X$ to $X \times_{\Im(X)} \Im(Y) \to Y$ and the reflection [[unit of an adjunction|unit]] is the left horizontal morphism in \begin{displaymath} \itexarray{ Y &\to& X \times_{\Im(Y)} \Im(Y) &\to& \Im(Y) \\ & \searrow & \downarrow^{} && \downarrow^{\mathrlap{\Im(f)}} \\ && X &\to& \Im(X) } \,. \end{displaymath} Therefore $(\mathbf{H}_{th})_{/X}^{fet}$, being a reflective subcategory of a [[locally presentable (∞,1)-category]], is (as discussed there) itself locally presentable. Hence by the [[adjoint (∞,1)-functor theorem]] it is now sufficient to show that the inclusion preserves all small [[(∞,1)-colimits]] in order to conclude that it also has a right [[adjoint (∞,1)-functor]]. So consider any [[diagram]] [[(∞,1)-functor]] $I \to (\mathbf{H}_{th})_{/X}^{fet}$ out of a [[small (∞,1)-category]]. Since the inclusion of $(\mathbf{H}_{th})_{/X}^{fet}$ is full, it is sufficient to show that the $(\infty,1)$-colimit over this diagram taken in $(\mathbf{H}_{th})_{/X}$ lands again in $(\mathbf{H}_{th})_{/X}^{fet}$ in order to have that $(\infty,1)$-colimits are preserved by the inclusion. Moreover, colimits in a slice of $\mathbf{H}_{th}$ are computed in $\mathbf{H}_{th}$ itself (this is discussed at \emph{\href{overcategory#LimitsAndColimits}{slice category - Colimits}}). Therefore we are reduced to showing that the square \begin{displaymath} \itexarray{ \underset{\to_i}{\lim} Y_i &\to& \Im \underset{\to_i}{\lim} Y_i \\ \downarrow && \downarrow \\ X &\to& \Im(X) } \end{displaymath} is an [[(∞,1)-pullback]] square. But since $\Im$ is a [[left adjoint]] it commutes with the $(\infty,1)$-colimit on objects and hence this diagram is equivalent to \begin{displaymath} \itexarray{ \underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \Im Y_i \\ \downarrow && \downarrow \\ X &\to& \Im(X) } \,. \end{displaymath} This diagram is now indeed an [[(∞,1)-pullback]] by the fact that we have [[universal colimits]] in the [[(∞,1)-topos]] $\mathbf{H}_{th}$, hence that on the left the component $Y_i$ for each $i \in I$ is the [[(∞,1)-pullback]] of $\Im(Y_i) \to \Im(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally \'e{}tale morphisms. \end{proof} \begin{example} \label{EtalificationOverThePoint}\hypertarget{EtalificationOverThePoint}{} For the case that $X \simeq \ast$ in prop. \ref{EtalificationIsCoreflection}, then the proof there shows that the \'e{}talification operation over the point is just ${\&}$: \begin{displaymath} {\&} \simeq Et_{/\ast} \,. \end{displaymath} Indeed, for any $X$ then ${\&} X \to \ast$ is a [[formally étale morphism]] since \begin{displaymath} \itexarray{ {\&} X &\longrightarrow& \Im{\&} X \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \Im \ast } \;\;\; \simeq \;\;\; \itexarray{ {\&} X &\stackrel{\simeq}{\longrightarrow}& {\&} X \\ \downarrow && \downarrow \\ \ast &\stackrel{\simeq}{\longrightarrow}& \ast } \end{displaymath} is a [[homotopy pullback]]. \end{example} \begin{prop} \label{}\hypertarget{}{} The $\infty$-category $(\mathbf{H}_{th})_{/X}^{fet}$ is an [[(∞,1)-topos]] and the canonical inclusion into $(\mathbf{H}_{th})_{/X}$ is a [[geometric embedding]]. \end{prop} \begin{proof} By prop. \ref{EtalificationIsCoreflection} the inclusion $(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ is [[reflective sub-(infinity,1)-category|reflective]] with reflector given by the $(\Im-equivalences , \Im-closed)$ factorization system. Since $\Im$ is a [[right adjoint]] and hence in particular preserves [[(∞,1)-pullbacks]], the $\Im$-equivalences are stable under pullbacks. By the discussion at \emph{[[stable factorization system]]} this is the case precisely if the corresponding reflector preserves [[finite (∞,1)-limits]]. Hence the embedding is a [[geometric embedding]] which exhibits a [[sub-(∞,1)-topos]] inclusion. \end{proof} \begin{defn} \label{TheStructureSheafOfX}\hypertarget{TheStructureSheafOfX}{} For $X \in \mathbf{H}_{th}$ we speak of \begin{displaymath} \mathcal{X} \coloneqq Sh_{\mathbf{H}_{th}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{fet} \end{displaymath} also as the ([[petit (∞,1)-topos|petit]]) [[(∞,1)-topos]] of $X$ or the \emph{[[étale (∞,1)-topos]]} of $X$. Write \begin{displaymath} \mathcal{O}_X \colon \mathbf{H}_{th} \stackrel{(-) \times X}{\to} (\mathbf{H}_{th})_{/X} \stackrel{Et}{\to} (\mathbf{H}_{th})_{/X}^{fet} = Sh_{\mathbf{H}_{th}}(X) \end{displaymath} for the composite [[(∞,1)-functor]] that sends any $A \in \mathbf{H}_{th}$ to the etalification, prop. \ref{EtalificationIsCoreflection}, of the projection $A \times X \to X$. We call $\mathcal{O}_X$ the \textbf{[[structure sheaf]]} of $X$. \end{defn} \begin{remark} \label{}\hypertarget{}{} For $X, A \in \mathbf{H}_{th}$ and for $U \to X$ a [[formally étale morphism]] in $\mathbf{H}_{th}$ (hence like an [[open subset]] of $X$), we have that \begin{displaymath} \begin{aligned} \mathcal{O}_{X}(A)(U) & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , \mathcal{O}_{X}(A) ) \\ & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , Et(X \times A) ) \\ & \simeq (\mathbf{H}_{th})_{/X}(U, X \times A) \\ & \simeq \mathbf{H}_{th}(U,A) \\ & \simeq A(U) \end{aligned} \,, \end{displaymath} where we used the [[adjoint (∞,1)-functor|∞-adjunction]] $(\iota \dashv Et)$ of prop. \ref{EtalificationIsCoreflection} and the [[(∞,1)-Yoneda lemma]]. This means that $\mathcal{O}_{X}(A)$ behaves as the \emph{sheaf of $A$-valued functions over $X$}. \end{remark} \begin{prop} \label{StructureSheafIsIndeedStructureSheaf}\hypertarget{StructureSheafIsIndeedStructureSheaf}{} The functor $\mathcal{O}_X$ of def. \ref{StructureSheafRestrictedToH} is indeed an $\mathbf{H}_{th}$-[[structure sheaf]] in the sense of [[structured (∞,1)-toposes]], for $\mathbf{H}_{th}$ regarded as a (large) [[geometry (for structured (∞,1)-toposes)]] with the [[formally étale morphisms]] being the ``admissible morphisms''. \end{prop} This is the analog of ([[Structured Spaces|Lurie, Structured Spaces, prop. 2.2.11]]). \begin{proof} We need to check that $\mathcal{O}_{X}$ preserves [[finite (∞,1)-limits]] and [[formally étale morphism|formally étale]] [[covers]] (where covers here in the [[canonical topology]] on the given toposes are [[1-epimorphisms]]). The first statement follows since $\mathcal{O}_{X}$ is [[right adjoint]] to the forgetful functor \begin{displaymath} Sh_{\mathbf{H}}(X) \simeq (\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X} \stackrel{\underset{X}{\sum}}{\to} \mathbf{H}_{th} \end{displaymath} For the second statement, let $p \colon \widehat Y \longrightarrow Y$ be any [[1-epimorphism]] which is also a [[formally étale morphism|formally étale]]. We need to show that also $Et(X \times p)$ is a [[1-epimorphism]]. By the discussion at [[effective epimorphism in an (∞,1)-category]] for this it is sufficient that the [[0-truncated|0-truncation]] $\tau_0 Et(X \times p)$ is an [[epimorphism]] in the underlying [[sheaf topos]], hence that every [[generalized element]] of $Et(X \times Y)$ has a lift to $Et(X \times \widehat{Y})$ after refinement along a cover. By the fact that $Et$ is [[right adjoint]] to the inclusion, by construction, this means that it is sufficient to show that given a [[diagram]] in $\mathbf{H}_{th}$ of the form \begin{displaymath} \itexarray{ && && X \times \widehat{Y} \\ && && \downarrow \\ U && \longrightarrow && X \times Y \\ & \searrow && \swarrow \\ && X } \end{displaymath} with $U \to X$ formally \'e{}tale, this can be completed to a diagram of the form \begin{displaymath} \itexarray{ \widehat{U} && \longrightarrow && X \times \widehat{Y} \\ \downarrow && && \downarrow \\ U && \longrightarrow && X \times Y \\ & \searrow && \swarrow \\ && X } \end{displaymath} with $\widehat{U} \to U$ a formally \'e{}tale 1-epimorphism. But since both [[1-epimorphisms]] as well as [[formally étale morphisms]] are stable under [[(∞,1)-pullback]] we can take $\widehat U \coloneqq \widehat{Y} \times_Y U$. \end{proof} \begin{remark} \label{}\hypertarget{}{} In the case that $\mathbf{H}_{th}$ happens to have an [[(∞,1)-site]] of definition whose [[covers]] are ([[Grothendieck pretopology|generated]]) from [[formally étale morphisms]] (a small [[geometry (for structured (∞,1)-toposes)]]), then $\mathbf{H}_{th}$ is the [[classifying topos|classifying (∞,1)-topos]] for [[structure sheaves]] and $\mathcal{O}_X$ in prop. \ref{StructureSheafIsIndeedStructureSheaf} may be regarded as the [[inverse image]] of the classifying [[geometric morphism]] \begin{displaymath} Sh_{\mathbf{H}_{th}}(X) \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{}{\longrightarrow}} \mathbf{H}_{th} \,. \end{displaymath} \end{remark} \begin{example} \label{CotangentBundle}\hypertarget{CotangentBundle}{} Let $G \in Grp(\mathbf{H}_{th})$ be an [[∞-group]] and write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object. Then \begin{displaymath} \mathcal{O}_X(\flat_{dR}\mathbf{B}G) \in Sh_{\mathbf{H}}(X) \end{displaymath} we may call the \textbf{$G$-valued flat cotangent sheaf} of $X$. \end{example} \begin{remark} \label{}\hypertarget{}{} For $U \in \mathbf{H}_{th}$ a test object (say an object in a [[(∞,1)-site]] of definition, under the [[Yoneda embedding]]) a formally \'e{}tale morphism $U \to X$ is like an [[open map]]/open embedding. Regarded as an object in $(\mathbf{H}_{th})_{/X}^{fet}$ we may consider the sections over $U$ of the cotangent bundle as defined above, which in $\mathbf{H}_{th}$ are diagrams \begin{displaymath} \itexarray{ U &&\to&& \mathcal{O}_X(\flat_{dR}\mathbf{B}G) \\ & \searrow && \swarrow \\ && X } \,. \end{displaymath} By the fact that $Et(-)$ is [[right adjoint]], such diagrams are in bijection to diagrams \begin{displaymath} \itexarray{ U &&\to&& X \times \flat_{dR} \mathbf{B}G \\ & \searrow && \swarrow \\ && X } \end{displaymath} where we are now simply including on the left the formally \'e{}tale map $(U \to X)$ along $(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$. In other words, the sections of the $G$-valued flat cotangent sheaf $\mathcal{O}_X(\flat_{dR}\mathbf{B}G)$ are just the sections of $X \times \flat_{dR}\mathbf{B}G \to X$ itself, only that the \emph{domain} of the section is constrained to be a formally \'e{}tale patch of $X$. But then by the very nature of $\flat_{dR}\mathbf{B}G$ it follows that the flat sections of the $G$-valued cotangent bundle of $X$ are indeed nothing but the flat $G$-valued differential forms on $X$. \end{remark} \begin{prop} \label{StructuredPetitToposesAreLocallyContractible}\hypertarget{StructuredPetitToposesAreLocallyContractible}{} For $X \in \mathbf{H}_{th}$ an object in a differentially cohesive $\infty$-topos, then its petit structured $\infty$-topos $Sh_{\mathbf{H}_{th}}(X)$, according to def. \ref{TheStructureSheafOfX}, is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]]. \end{prop} \begin{proof} We need to check that the composite \begin{displaymath} \infty Grpd \stackrel{Disc}{\longrightarrow} \mathbf{H}_{th} \stackrel{(-) \times X}{\longrightarrow} (\mathbf{H}_{th})_{/X} \stackrel{L}{\longrightarrow} Sh_{\mathbf{H}}(X) \end{displaymath} preserves [[(∞,1)-limits]], so that it has a further [[left adjoint]]. Here $L$ is the reflector from prop. \ref{EtalificationIsCoreflection}. Inspection shows that this composite sends an object $A \in \infty Grpd$ to $\Im(Disc(A)) \times X \to X$: \begin{displaymath} \itexarray{ \Im(Disc(A)) \times X &\longrightarrow& \Im(Disc(A) \times X) & \simeq \Im(Disc(A)) \times \Im(X) \\ \downarrow &{}^{(pb)}& \downarrow \\ X &\longrightarrow& \Im(X) } \,. \end{displaymath} By the discussion at \href{slice+infinity-category#LimitsAndColimits}{slice (∞,1)-category -- Limits and colimits} an [[(∞,1)-limit]] in the slice $(\mathbf{H}_{th})_{/X}$ is computed as an [[(∞,1)-limit]] in $\mathbf{H}$ of the [[diagram]] with the slice [[cocone]] adjoined. By [[right adjoint|right adjointness]] of the inclusion $Sh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X}$ the same is then true for $Sh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}$. Now for $A \colon J \to \infty Grpd$ a [[diagram]], it is taken to the diagram $j \mapsto \Im(Disc(A_j)) \times X \to X$ in $Sh_{\mathbf{H}}(X)$ and so its $\infty$-limit is computed in $\mathbf{H}$ over the diagram locally of the form \begin{displaymath} \itexarray{ X \times \Im(Disc(A_{j})) &&\longrightarrow&& X \times \Im(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X } \simeq \itexarray{ X \times \Im(Disc(A_{j})) &&\longrightarrow&& X \times \Im(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X \times \ast } \,. \end{displaymath} Since $\infty$-limits commute with each other this limit is the product of \begin{enumerate}% \item $\underset{\leftarrow}{\lim}_j \Im(Disc(A_j))$ \item $\underset{\leftarrow}{\lim}_{J \star \Delta^0} X$ (over the co-coned diagram constant on $X$). \end{enumerate} For the first of these, since the [[infinitesimal shape modality]] $\Im$ is in particular a [[right adjoint]] (with [[left adjoint]] the [[reduction modality]]), and since $Disc$ is also [[right adjoint]] by [[cohesion]], we have a [[natural equivalence]] \begin{displaymath} \underset{\leftarrow}{\lim}_j \Im(Disc(A_j)) \simeq \Im(Disc(\underset{\leftarrow}{\lim}_j(A_j))) \,. \end{displaymath} For the second, the $\infty$-limit over an $\infty$-category $J \star \Delta^0$ of a functor constant on $X$ is \begin{displaymath} \begin{aligned} \underset{\leftarrow}{\lim}_{J \star \Delta^0} X & \simeq \underset{\leftarrow}{\lim}_{J \star \Delta^0} [\ast, X] \\ & \simeq [\underset{\rightarrow}{\lim}_{J \star \Delta^0} \ast, X] \\ & \simeq [{\vert {J \star \Delta^0}\vert}, X] \\ & \simeq [\ast, X] \simeq X \end{aligned} \,, \end{displaymath} where the last line follows since ${J \star \Delta^0}$ has a terminal object and hence contractible geometric realization. In conclusion this shows that $\infty$-limits are preserved by $L \circ (-)\times X\circ Disc$. \end{proof} \begin{prop} \label{FormallyEtaleMorphismInducesEtaleMorphismOfStructuredToposes}\hypertarget{FormallyEtaleMorphismInducesEtaleMorphismOfStructuredToposes}{} Let $f \;\colon\; Y \longrightarrow X$ be a [[formally étale morphism]] in a differentially cohesive $\infty$-topos $\mathbf{H}$. Then pullback $f^\ast$ is the [[inverse image]] of an [[étale morphism]] of [[structured (∞,1)-toposes]] between the corresponding [[étale toposes]], def. \ref{TheStructureSheafOfX}, hence there is an [[étale geometric morphism]] \begin{displaymath} Sh_{\mathbf{H}}(Y) \simeq Sh_{\mathbf{H}}(X)/_{f} \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}} Sh_{\mathbf{H}}(X) \end{displaymath} and an equivalence of [[structure sheaves]] \begin{displaymath} \mathcal{O}_Y \simeq f^\ast \mathcal{O}_X \,. \end{displaymath} \end{prop} \begin{proof} Since the inclusion of the point into the interval is an op-[[final (∞,1)-functor]] we have (by \href{over-%28infinity%2C1%29-category#FinalFunctorsInduceEquivalentSlices}{this proposition}) an [[equivalence]] of [[over-(∞,1)-categories]] \begin{displaymath} \mathbf{H}/_{Y} \simeq \mathbf{H}/_{f} \simeq (\mathbf{H}/_{X})/_f \,. \end{displaymath} Since $f$ is formally \'e{}tale by assumption and since [[formally étale morphisms]] are closed under [[composition]], this restricts to an [[equivalence]] $Sh_{\mathbf{H}}(Y) \simeq (Sh_{\mathbf{H}}(X))/_f$. For the equivalence of structure sheaves it is sufficient to show for each [[coefficient]] $A \in \mathbf{H}_{th}$ an equivalence \begin{displaymath} \mathcal{O}_Y(A) \simeq (f^\ast \mathcal{O}_X(A)) \end{displaymath} in $Sh_{\mathbf{H}}(Y)$. But by definition (\ref{TheStructureSheafOfX}) $\mathcal{O}_Y(A) \coloneqq Et(A \times Y)$ and similarly for $\mathcal{O}_X$ and since $Et$ is [[right adjoint]] to the inclusion $Sh_{\mathbf{H}}(Y) \hookrightarrow \mathbf{H}_{Y}$ we have \begin{displaymath} f^\ast \mathcal{O}_X(A) = f^\ast Et(A \times X) \simeq Et(f^\ast(A \times X)) \simeq Et(A \times Y) = \mathcal{O}_Y(A) \,. \end{displaymath} \end{proof} \hypertarget{SheavesOfModules}{}\paragraph*{{Sheaves of (quasi-coherent) modules}}\label{SheavesOfModules} We discuss the abstract formulation of sheaves of [[modules]] and of [[quasicoherent sheaves]] on petit $\infty$-toposes in differential cohesion. \begin{quote}% under construction -- check \end{quote} \begin{defn} \label{}\hypertarget{}{} For $X \in \mathbf{H}_{th}$ an object and $(Sh_{\mathbf{H}}(X), \mathcal{O}_X)$ its $\mathbf{H}_{th}$-[[structured (infinity,1)-topos]], according to def. \ref{TheStructureSheafOfX}, consider the composite functor \begin{displaymath} \mathcal{O}_X^{\mathbf{H}} \;\colon\; \mathbf{H} \stackrel{i_!}{\hookrightarrow} \mathbf{H}_{th} \stackrel{Et X^\ast}{\longrightarrow} Sh_{\mathbf{H}}(X) \,, \end{displaymath} then an \textbf{$\mathcal{O}_X$-module} $\mathcal{F}$ on $X$ is an extension of $\mathcal{O}_X^{\mathbf{H}}$ by a limit-preserving functor through $i_!$ \begin{displaymath} \itexarray{ \mathbf{H} &\stackrel{\mathcal{O}_X^{\mathbf{H}}}{\longrightarrow}& Sh_{\mathbf{H}}(X) \\ {}^{\mathllap{i_!}}\downarrow & \nearrow_{\mathrlap{\mathcal{F}}} \\ \mathbf{H}_{th} } \,. \end{displaymath} \end{defn} \begin{example} \label{}\hypertarget{}{} In particular $\mathcal{O}_X$ is canonically a module over itself by setting $\mathcal{F} = \mathcal{O}_X$. \end{example} This is a slight abstraction of the definition in (\href{quasicoherent+sheaf#LurieQC}{Lurie QC, section 2.3}). See at \emph{\href{quasicoherent+sheaf#HigherGeometryAsExtensionsOfStructureSheaf}{quasicoherent sheaf -- In higher geometry -- As extension of the structure sheaf}}. \hypertarget{PoincareCocycle}{}\paragraph*{{Liouville-Poincar\'e{} cocycle}}\label{PoincareCocycle} \begin{defn} \label{TheLiouvillePoincareCocycle}\hypertarget{TheLiouvillePoincareCocycle}{} For $X,A \in \mathbf{H}_{th}$ and with $\mathcal{O}_X(A) \in Sh_{\mathbf{H}}(X)$ as in def. \ref{TheStructureSheafOfX}, write \begin{displaymath} \theta_X(A) \;\colon\; \underset{X}{\sum} \iota \mathcal{O}_X(A) \to A \end{displaymath} for the [[morphism]] in $\mathbf{H}$ which is the $(\underset{X}{\sum} \dashv X^*)$-adjunct $\underset{X}{\sum}\iota Et(X^* A) \to A$ of the [[counit of a comonad|counit]] $\iota Et(X^* A) \to X^* A$ of the $(\iota \dashv Et)$-coreflection of def. \ref{TheStructureSheafOfX}. This $\theta_X(A)$ we call the \textbf{Liouville-Poincar\'e{} $A$-cocycle} on $\underset{X}{\sum} \iota \mathcal{O}_X(A)$. \end{defn} \begin{example} \label{}\hypertarget{}{} Consider the model of differential cohesion given by $\mathbf{H}_{th} =$ [[SynthDiff∞Grpd]]. Write $\Omega^1 \in \mathbf{H }\stackrel{i_!}{\hookrightarrow} \mathbf{H}_{th}$ for the abstract [[sheaf]] of [[differential 1-forms]]. Then for $X \in SmthMfd \hookrightarrow \mathbf{H}$ a [[smooth manifold]], we have that \begin{displaymath} \underset{X}{\sum} \iota \mathcal{O}_X(\Omega^1) \to X \end{displaymath} is the [[cotangent bundle]] \begin{displaymath} T^* X \to X \end{displaymath} of the manifold: because for $i_U \colon U \to X$ an open subset of the manifold regarded as an object of $Sh_{\mathbf{H}}(X)$, a section $\iota(\sigma_U)$ of $T^* X|_U \to U$ is equivalently a map $\sigma \colon i_U \to \mathcal{O}_X(\Omega^1)$ in $Sh_{\mathbf{H}_{th}}(X)$, which by the $(\iota \dashv Et)$-[[adjunction]] is a map $\iota(i_U) \to X \times \Omega^1$ in $(\mathbf{H}_{th})_{/X}$ which finally is equivalently a map $U \to \Omega^1$ in $\mathbf{H}_{th}$ hence an element in $\Omega^1(U)$. So the Liouville-Poincar\'e{} $\Omega^1$-cocycle according to \ref{TheLiouvillePoincareCocycle} is a [[differential 1-form]] \begin{displaymath} \theta \;\colon\; \underset{X}{\sum}\iota \mathcal{O}_X(\Omega^1) \to \Omega^1 \end{displaymath} on the total space of the cotangent bundle. For \begin{displaymath} \sigma \;\colon\; X \to \mathcal{O}_X(\Omega^1) \;\;\; \in Sh_{\mathbf{H}}(X) \end{displaymath} a [[section]] of the [[cotangent bundle]], the [[pullback of a differential form|pullback form]] $\sigma^* \theta$ on $X$ is the composite \begin{displaymath} \underset{X}{\sum}\iota X \stackrel{\underset{X}{\sum}\iota(\sigma)}{\to} \underset{X}{\sum}\iota \mathcal{O}_X(\Omega^1) \stackrel{\theta}{\to} \Omega^1 \,, \end{displaymath} hence the [[adjunct]] \begin{displaymath} \iota X \stackrel{\iota(\sigma)}{\to} \iota \mathcal{O}_X(\Omega^1) \stackrel{}{\to} X^* \Omega^1 \,, \end{displaymath} hence by definition \begin{displaymath} \iota(X) \stackrel{\iota(\sigma)}{\to} \iota(Et(X \times \Omega^1)) \stackrel{}{\to} X^*\Omega^1 \,, \end{displaymath} hence the adjunct \begin{displaymath} X \stackrel{\sigma}{\to} Et(X \times \Omega^1) \stackrel{id}{\to} Et(X \times \Omega^1) \end{displaymath} hence the original $\sigma$. This is the defining property which identifies $\that$ as the traditional [[Liouville-Poincaré 1-form]]. \end{example} \hypertarget{EtaleObjects}{}\paragraph*{{Manifolds and \'e{}tale groupoids}}\label{EtaleObjects} An ordinary [[topological groupoid|topological]]/[[Lie groupoid|Lie]] [[étale groupoid]] is one whose source/target map is an [[étale map]]. We consider now a notion that can be formulated in the presence of infinitesimal cohesion which generalizes this. \begin{defn} \label{FormalEtaleGroupoid}\hypertarget{FormalEtaleGroupoid}{} A [[groupoid object in an (∞,1)-category|groupoid object]] $\mathcal{G}_\bullet$ is an \textbf{[[étale ∞-groupoid]]} if the equivalent (via the higher [[Giraud theorem]]) [[effective epimorphism]] (the [[atlas]]) $\mathcal{G}_0 \longrightarrow \mathcal{G}$ is a [[formally étale morphism]]. \end{defn} Let now $V$ be any object (For instance let $\mathbb{A}^1 \in \mathbf{H}$ be a canonical line object that exhibits the cohesion of $\mathbf{H}$ in the sense of \emph{\href{cohesive+%28infinity%2C1%29-topos+--+structures#A1HomotopyContinuum}{structures in a differential infinity-topos -- A1 homotopy / The continuum}} and take $V = \coprod_n \mathbb{A}^n$.) \begin{defn} \label{Manifold}\hypertarget{Manifold}{} A \textbf{$V$-manifold} is an object $X$ such that there exists a \textbf{$V$-cover} $U$, namely a [[correspondence]] from $V$ to $X$ \begin{displaymath} \itexarray{ && U \\ & \swarrow && \searrow \\ V && && X } \end{displaymath} such that both morphisms are [[formally étale morphisms]] and such that $U \to X$ is in addition an [[effective epimorphism]]. \end{defn} See also at \emph{\href{smooth%20manifold#GeneralAbstractCharacterization}{smooth manifold -- general abstract geometric formulation}} \hypertarget{GLnTangentBundles}{}\paragraph*{{Frame bundles}}\label{GLnTangentBundles} We discuss how each manifold $X$ in differential cohesion as in def. \ref{Manifold} is associated a canonical [[frame bundle]] classified by a morphism $X \to \mathbf{B}GL(V)$. \begin{defn} \label{FormalDiskBundle}\hypertarget{FormalDiskBundle}{} For $X$ any object in differential cohesion, its \emph{[[infinitesimal disk bundle]]} $T_{inf} X \to X$ is the [[homotopy pullback]] \begin{displaymath} \itexarray{ T_{inf} X &\stackrel{ev}{\longrightarrow}& X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\longrightarrow& \Im X } \end{displaymath} of the [[unit of a monad|unit]] of its [[infinitesimal shape modality]] along itself. More generally, given a filtration of differential cohesion by orders of infinitesimals, remark \ref{SequenceOfOrdersOfInfinitesimals}, then the order-$k$ [[infinitesimal disk bundle]] is the [[homotopy pullback]] in \begin{displaymath} \itexarray{ T_{inf} X &\stackrel{ev}{\longrightarrow}& \Im_{(k)}X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\longrightarrow& \Im X } \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} The [[Atiyah groupoid]] of $T_{inf} X$ is the [[jet groupoid]] of $X$ \end{remark} \begin{remark} \label{RelationOfInfinitesimalDiskBundleToJetBundle}\hypertarget{RelationOfInfinitesimalDiskBundleToJetBundle}{} With respect to the [[base change]] [[geometric morphism]] \begin{displaymath} \mathbf{H}_{/X} \stackrel{\overset{p_!}{\longrightarrow}}{\stackrel{\overset{p^\ast}{\longleftarrow}}{\underset{p_\ast}{\longrightarrow}}} \mathbf{H}_{/\Im X} \end{displaymath} then then [[infinitesimal disk bundle]] of $X$ is \begin{displaymath} T_{inf} X \simeq p^\ast p_! X \,, \end{displaymath} where on the right $X$ is regarded as sitting by the identity morphism over itself. Written in this form it follows from the [[adjoint triple]] above that bundle morphisms \begin{displaymath} \itexarray{ T_{inf}X && \longrightarrow && E \\ & \searrow && \swarrow \\ && X } \end{displaymath} are equivalently [[sections]] of $p^\ast p_\ast E$. But such bundle morphisms are equivalently [[jets]] of $E$ and hence $p^\ast p_\ast E$ is the [[jet bundle]] of $E$. See there for more. \end{remark} \begin{lemma} \label{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle}\hypertarget{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle}{} If $\iota \colon U \to X$ is a [[formally étale morphism]], def. \ref{FormallyEtaleInHTh}, then \begin{displaymath} \iota^\ast T_{inf} X \simeq T_{inf}U \,. \end{displaymath} \end{lemma} \begin{proof} By the definition of formal \'e{}taleness and using the [[pasting law]] we have an equivalence of [[pasting diagrams]] of [[homotopy pullbacks]] of the following form: \begin{displaymath} \itexarray{ \iota^\ast T_{inf} X &\longrightarrow& T_{inf} X &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& X &\longrightarrow& \Im X } \;\;\;\; \simeq \;\;\;\; \itexarray{ T_{inf} U &\longrightarrow& U &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& \Im U &\longrightarrow& \Im X } \end{displaymath} \end{proof} \begin{defn} \label{Framing}\hypertarget{Framing}{} For $V$ an object, a \textbf{[[framing]]} on $V$ is a trivialization of its infinitesimal disk bundle, def. \ref{FormalDiskBundle}, i.e. an object $\mathbb{D}^V$ -- the typical [[infinitesimal disk]] or [[formal disk]] -- and a (chosen) [[equivalence]] \begin{displaymath} \itexarray{ T_{inf} V && \stackrel{\simeq}{\longrightarrow} && V \times \mathbb{D}^n \\ & \searrow && \swarrow_{\mathrlap{p_1}} \\ && V } \,. \end{displaymath} \end{defn} \begin{defn} \label{GeneralLinearGroup}\hypertarget{GeneralLinearGroup}{} For $V$ a framed object, def. \ref{Framing}, we write \begin{displaymath} GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V) \end{displaymath} for the [[automorphism ∞-group]] of its typical [[infinitesimal disk]]/[[formal disk]]. \end{defn} \begin{remark} \label{OrderOfInfinitesimalDisks}\hypertarget{OrderOfInfinitesimalDisks}{} When the [[infinitesimal shape modality]] exhibits first-order infinitesimals, such that $\mathbb{D}(V)$ is the first order [[infinitesimal neighbourhood]] of a point, then $\mathbf{Aut}(\mathbb{D}(V))$ indeed plays the role of the [[general linear group]]. When $\mathbb{D}^n$ is instead a higher order or even the whole [[formal neighbourhood]], then $GL(n)$ is rather a [[jet group]]. For order $k$-jets this is sometimes written $GL^k(V)$ We nevertheless stick with the notation ``$GL(V)$'' here, consistent with the fact that we have no index on the [[infinitesimal shape modality]]. More generally one may wish to keep track of a whole tower of infinitesimal shape modalities and their induced towers of concepts discussed here. \end{remark} This class of examples of framings is important: \begin{prop} \label{DifferentialCohesiveInfinityGroupIsCanonicallyFramed}\hypertarget{DifferentialCohesiveInfinityGroupIsCanonicallyFramed}{} Every differentially cohesive [[∞-group]] $G$ is canonically framed (def. \ref{Framing}) such that the horizontal map in def. \ref{FormalDiskBundle} is given by the left action of $G$ on its [[infinitesimal disk]] at the neutral element: \begin{displaymath} ev \colon T_{inf}G \simeq G \times \mathbb{D}^G_e \stackrel{\cdot}{\longrightarrow} G \,. \end{displaymath} \end{prop} \begin{proof} By the discussion at \emph{[[Mayer-Vietoris sequence]]} in the section \emph{\href{Mayer-Vietoris%20sequence#OverAGroupObject}{Over an ∞-group}} and using that the [[infinitesimal shape modality]] preserves group structure, the defining [[homotopy pullback]] of $T_{inf} G$ is equivalent to the pasting of pullback diagrams of the form \begin{displaymath} \itexarray{ T_{inf} G &\stackrel{}{\longrightarrow}& \mathbb{D}^G_e &\stackrel{}{\longrightarrow}& \ast \\ \downarrow && \downarrow && \downarrow \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G &\stackrel{}{\longrightarrow}& \Im G } \,, \end{displaymath} where the right square is the defining pullback for the [[infinitesimal disk]] $\mathbb{D}^G$. For the left square we find by \href{Mayer-Vietoris%20sequence#HTTArgumentForPullback}{this proposition} that $T_{inf} G \simeq G\times \mathbb{D}^G$ and that the top horizontal morphism is as claimed. \end{proof} By lemma \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} it follows that: \begin{prop} \label{FormalDiskBundleOfRegularManifoldsTrivializesOverCover}\hypertarget{FormalDiskBundleOfRegularManifoldsTrivializesOverCover}{} For $V$ a framed object, def. \ref{Framing}, let $X$ be a $V$-manifold, def. \ref{Manifold}. Then the infinitesimal disk bundle, def. \ref{FormalDiskBundle}, of $X$ canonically trivializes over any $V$-cover $V \leftarrow U \rightarrow X$ , i.e. there is a [[homotopy pullback]] of the form \begin{displaymath} \itexarray{ U \times \mathbb{D}^V &\longrightarrow& T_{inf} X \\ \downarrow && \downarrow \\ U &\longrightarrow& X } \,. \end{displaymath} This exhibits $T_{inf} X\to X$ as a $\mathbb{D}^V$-[[fiber ∞-bundle]]. \end{prop} \begin{prop} \label{ModulatingMapOfFormalDiskBundle}\hypertarget{ModulatingMapOfFormalDiskBundle}{} By \href{fiber+infinity-bundle#Properties}{this discussion} this fiber [[fiber ∞-bundle]] is the [[associated ∞-bundle]] of an essentially uniquely determined $\mathbf{Aut}(\mathbb{D}^V)$-[[principal ∞-bundle]]. \end{prop} \begin{defn} \label{FrameBundleMap}\hypertarget{FrameBundleMap}{} Given a $V$-manifold $X$, def. \ref{Manifold}, for framed $V$, def. \ref{Framing}, then its \emph{[[frame bundle]]} $Fr(X)$ is the $GL(V)$-[[principal ∞-bundle]] given by prop. \ref{FormalDiskBundleOfRegularManifoldsTrivializesOverCover} via remark \ref{ModulatingMapOfFormalDiskBundle}. \end{defn} \begin{remark} \label{}\hypertarget{}{} As in remark \ref{OrderOfInfinitesimalDisks}, this really axiomatizes in general [[higher order frame bundles]] with the order implicit in the nature of the [[infinitesimal shape modality]]. \end{remark} \begin{remark} \label{FrameBundlesFunctorial}\hypertarget{FrameBundlesFunctorial}{} By prop. \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} the construction of frame bundles in def. \ref{FrameBundleMap} is functorial in [[formally étale maps]] between $V$-manifolds. \end{remark} \hypertarget{structures}{}\paragraph*{{$G$-Structures}}\label{structures} We discuss the formalization of [[G-structures]] and [[integrability of G-structures]] in differential cohesion Let $V$ be framed, def. \ref{Framing}, let $G$ be an [[∞-group]] and $G \to GL(V)$ a homomorphism to the general linear group of $V$, def. \ref{GeneralLinearGroup}, hence \begin{displaymath} G\mathbf{Struc}\colon \mathbf{B}G \longrightarrow \mathbf{B}GL(V) \end{displaymath} a morphism between the [[deloopings]]. \begin{defn} \label{GStructure}\hypertarget{GStructure}{} For $X$ a $V$-manifold, def. \ref{Manifold}, a \textbf{[[G-structure]]} on $X$ is a lift of the structure group of its [[frame bundle]], def. \ref{FrameBundleMap}, to $G$, hence a diagram \begin{displaymath} \itexarray{ X && \longrightarrow&& \mathbf{B}G \\ & {}_{\mathllap{\tau_X}}\searrow && \swarrow_{\mathrlap{G\mathbf{Struc}}} \\ && \mathbf{B}GL(V) } \end{displaymath} hence a morphism \begin{displaymath} \mathbf{c} \colon \tau_X \longrightarrow G\mathbf{Struc} \end{displaymath} in the [[slice (∞,1)-topos]]. \end{defn} In fact $G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(n)}$ is the [[moduli ∞-stack]] of such $G$-structures. The double [[slice (∞,1)-topos|slice]] $(\mathbf{H}_{/\mathbf{B}GL(n)})_{/G\mathbf{Struc}}$ is the [[(∞,1)-category]] of such $G$-structures. \begin{example} \label{TrivialGStructure}\hypertarget{TrivialGStructure}{} If $V$ is framed, def. \ref{Framing}, then it carries the trivial $G$-structure, which we denote by \begin{displaymath} \mathbf{c}_0 \colon \tau_{V} \longrightarrow G\mathbf{Struc} \,. \end{displaymath} \end{example} \begin{defn} \label{IntegrableGStructure}\hypertarget{IntegrableGStructure}{} For $V$ framed, def. \ref{Framing}, and $X$ a $V$-manifold, def. \ref{Manifold}, then a $G$-structure $\mathbf{c}$ on $X$, def. \ref{GStructure}, is \emph{[[integrability of G-structures|integrable]]} (or \emph{locally flat}) if there exists a $V$-cover \begin{displaymath} \itexarray{ && U \\ & \swarrow && \searrow \\ V && && X } \end{displaymath} such that the [[correspondence]] of [[frame bundles]] induced via remark \ref{FrameBundlesFunctorial} \begin{displaymath} \itexarray{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && && \tau_X } \end{displaymath} (a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$) extends to a sliced correspondence between $\mathbf{c}$ and the trivial $G$-structure $\mathbf{c}_0$ on $V$, example \ref{TrivialGStructure}, hence to a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$ of the form \begin{displaymath} \itexarray{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && \swArrow_{\mathrlap{\simeq}} && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G\mathbf{Struct} } \end{displaymath} On the other hand, $\mathbf{c}$ is called \emph{infinitesimally integrable} (or \emph{torsion-free}) if such an extension exists (only) after restriction to all [[infinitesimal disks]] in $X$ and $U$, hence after composition with the [[counit of a comonad|counit]] \begin{displaymath} \flat^{rel} U \longrightarrow U \end{displaymath} of the [[relative flat modality]], def. \ref{InducedRelativeShapeAndFlat} (using that by prop. \ref{CounitOfFlatRelIsFormallyEtale} this is also formally \'e{}tale and hence induces map of frame bundles): \begin{displaymath} \itexarray{ && \tau_{\flat^{rel} U} \\ & \swarrow && \searrow \\ \tau_V && \swArrow_{\mathrlap{\simeq}} && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G\mathbf{Struct} } \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} As before, if the given [[reduction modality]] encodes order-$k$ infinitesimals, then the infinitesimal integrability in def. \ref{IntegrableGStructure} is order-$k$ integrability. For $k = 1$ this is [[torsion of a G-structure|torsion-freeness]]. \end{remark} \hypertarget{StrucInfinitesimalLocalSystem}{}\paragraph*{{Flat $\infty$-connections and infinitesimal local systems}}\label{StrucInfinitesimalLocalSystem} We discuss the in an infinitesimal neighbourhood. \begin{defn} \label{InfinitesimalFlatCohomology}\hypertarget{InfinitesimalFlatCohomology}{} For $X, A \in \mathbf{H}_{th}$ we say that \begin{displaymath} H_{infflat}(X,A) := \pi_0 \mathbf{H}(\Im(X), A) \simeq \pi_0 \mathbf{H}(X, \mathbf{\flat}_{inf}A) \end{displaymath} (where $(\Im \dashv \mathbf{\flat}_{inf})$ is given by def. \ref{InfinitesimalPathsAndReduction}) is the \textbf{infinitesimal flat cohomology} of $X$ with coefficient in $A$. \end{defn} \begin{note} \label{CrystallineCohomology}\hypertarget{CrystallineCohomology}{} In traditional contexts this is also called \emph{[[crystalline cohomology]]} or just \emph{[[de Rham cohomology]]} . Since we already have an in any [[cohesive (∞,1)-topos]], which is similar to but may slightly differ from infinitesimal flat differential cohomology, we shall say \textbf{[[synthetic differential geometry|synthetic]] de Rham cohomology} for the notion of def. \ref{InfinitesimalFlatCohomology} if we wish to honor traditional terminology. In this case we shall write \begin{displaymath} H_{dR,synth}(X,A) := \pi_0 \mathbf{H}_{th}(\Im(X), A) \,. \end{displaymath} \end{note} \begin{note} \label{FiniteAndInfinitesimalFlatConnections}\hypertarget{FiniteAndInfinitesimalFlatConnections}{} By the \hyperlink{InclusionOfConstantIntoInfinitesimalIntoAllPaths}{above observation} we have canonical morphisms \begin{displaymath} \mathbf{H}_{flat}(X,A) \to \mathbf{H}_{infflat}(X,A) \to \mathbf{H}(X,A) \end{displaymath} The objects on the left are \textbf{[[principal ∞-bundle]]s equipped with flat [[connection on an ∞-bundle|∞-connection]]} . The first morphism forgets their [[higher parallel transport]] along finite volumes and just remembers the parallel transport along infinitesimal volumes. The last morphism finally forgets also this connection information. \end{note} \begin{defn} \label{deRhamTheorem}\hypertarget{deRhamTheorem}{} For $A \in \mathbf{H}_{th}$ an abelian [[∞-group]] object we say that the \textbf{[[de Rham theorem]]} for $A$-coefficients holds in $\mathbf{H}_{th}$ if for all $X \in \mathbf{H}_{th}$ the \hyperlink{InclusionOfConstantIntoInfinitesimalIntoAllPaths}{infinitesimal path inclusion} \begin{displaymath} \Im(X) \to \mathbf{\Pi}(X) \end{displaymath} is an equivalence in $A$-[[cohomology]], hence if for all $n \in \mathbb{N}$ we have that \begin{displaymath} \pi_0 \mathbf{H}_{th}(\mathbf{\Pi}(X), \mathbf{B}^n A) \to \pi_0 \mathbf{H}_{th}(\Im(X), \mathbf{B}^n A) \end{displaymath} is an [[isomorphism]]. \end{defn} If we follow the notation of note \ref{CrystallineCohomology} and moreover write $\vert X \vert = \vert \Pi X \vert$ for the , then this becomes \begin{displaymath} H^{\bullet}_{dR, synth}(X,A) \simeq H^\bullet(|X|, A_{disc}) \,, \end{displaymath} where on the right we have ordinary cohomology in [[Top]] (for instance realized as [[singular cohomology]]) with coefficients in the [[discrete group]] $A_{disc} := \Gamma A$ underlying the cohesive group $A$. In certain contexts of infinitesimal neighbourhoods of cohesive $\infty$-toposes the de Rham theorem in this form has been considered in (\hyperlink{SimpsonTeleman}{SimpsonTeleman}). \hypertarget{FormalInfinityGroupoids}{}\paragraph*{{Formal cohesive $\infty$-groupoids}}\label{FormalInfinityGroupoids} Recall that a [[groupoid object in an (infinity,1)-category]] is equivalently an [[1-epimorphism]] $X \longrightarrow \mathcal{G}$, thought of as exhibiting an [[atlas]] $X$ for the groupoid $\mathcal{G}$. Now an $\infty$-Lie algebroid is supposed to be an $\infty$-groupoid which is only infinitesimally extended over its base space $X$. Hence: A groupoid object $p \colon X \longrightarrow \mathcal{G}$ is \emph{infinitesimal} if under the [[reduction modality]] $\Re$ (equivalently under the [[infinitesimal shape modality]] $\Im$) the [[atlas]] becomes an [[equivalence in an (infinity,1)-category|equivalence]]: $\Re(p), \Im(p) \in Equiv$. For example the tangent $\infty$-Lie algebroid $T X$ of any $X$ is the [[unit of a monad|unit]] of the [[infinitesimal shape modality]]. \begin{displaymath} \eta^{\Im}_X \;\colon\; X \stackrel{}{\longrightarrow} \Im X \,. \end{displaymath} It follows that every such $\infty$-Lie algebroid $X \to \mathcal{G}$ canonically maps to the tangent $\infty$-Lie algebroid of $X$ -- the \emph{anchor map}. The naturality square of the unit $\eta^{\Im}_{p}$ exhibits the morphism: \begin{displaymath} \itexarray{ X & \stackrel{id}{\longrightarrow} & X \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{\eta^\Im_X}} \\ && \Im X \\ \downarrow && \downarrow^{\mathrlap{\Im p}}_\simeq \\ \mathcal{G} &\stackrel{\eta^{\Im}_{\mathcal{G}}}{\longrightarrow}& \Im \mathcal{G} } \end{displaymath} \hypertarget{LieTheory}{}\paragraph*{{Lie theory}}\label{LieTheory} (\ldots{}) The discussion at \emph{\href{synthetic+differential+infinity-groupoid#LieDifferentiation}{synthetic differential ∞-groupoid -- Lie differentiation}} immediately generalizes to produce a concept of Lie differentiation in any differentially cohesive context. This Lie differentiation is just the flat modality of the differential cohesion but regarded as cohesive over its induced [[infinitesimal cohesion]]. As such, there is a left adjoint to Lie differentiation, given by the corresponding shape modality. However, the substance of Lie theory here will be to restrict this adjunction to [[geometric ∞-stacks]]. On the geometric $\infty$-stacks the Lie differentiation via passage to inffinitesimal cohesion will yield actual $L_\infty$-algebras, but some structure is required to make the formal [[Lie integration]] of these lang indeed in [[geometric ∞-stacks]]. (\ldots{}) \hypertarget{StrucDeformationTheory}{}\paragraph*{{Deformation theory}}\label{StrucDeformationTheory} For $C^{op}$ any [[(∞,1)-site]] the construction of the [[tangent (∞,1)-category]] $T_{C} \to C$ provides a canonical infinitesimal thickening of $C$: \begin{displaymath} C \stackrel{\overset{cod}{\leftarrow}}{\stackrel{\overset{\Delta}{\to}}{\underset{dom}{\leftarrow}}} C^{\Delta[1]} \stackrel{\overset{L}{\to}}{\underset{\Omega^\infty}{\leftarrow}} \,, \end{displaymath} where the $\infty$-functor pair on the right forms a $cod$-[[relative (∞,1)-adjunction]]. The composite $L \circ i$ is the [[cotangent complex]] functor for $C$ and $\Omega^\infty$ is fiberwise the canonical map out of the [[stabilization]]. The image of $i$ is contained in that of $\Omega^\infty$. Therefore we may restrict the $(cod \dashv i)$-adjunction on the right to the [[full sub-(∞,1)-category]] $\tilde T_C$ of $C^{\Delta[1]}$ on thise objects in the image of $\Omega^\infty$. This yields an infinitesimal neighbourhood of [[(∞,1)-sites]] \begin{displaymath} C^{op} \stackrel{\overset{i}{\hookrightarrow}}{\underset{cod}{\leftarrow}} (\tilde T_C)^{op} \,. \end{displaymath} (\ldots{}) \hypertarget{idelic_structure}{}\paragraph*{{Idelic structure}}\label{idelic_structure} See at \emph{[[differential cohesion and idelic structure]]}. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item [[formal smooth ∞-groupoid]] \item [[infinitesimally thickened Sierpinski topos]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[infinitesimal cohesive (infinity,1)-topos]] \item [[differential homotopy type theory]] \end{itemize} [[!include cohesion - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The material discussed here corresponds to the most part to sections 3.5 and 3.10 of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} For references on the general notion of \emph{[[cohesive (∞,1)-topos]]}, see there. The following literature is related to or subsumes by the discussion here. Something analogous to the notion of [[infinity-connected (infinity,1)-site|∞-connected site]] and the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] is the content of section 2.16. of \begin{itemize}% \item [[Carlos Simpson]], [[Constantin Teleman]], \emph{deRham theorem for $\infty$-stacks} (\href{http://math.berkeley.edu/~teleman/math/simpson.pdf}{pdf}) \end{itemize} The \hyperlink{LieTheory}{infinitesimal path ∞-groupoid adjunction} $(\Re \dashv \Im \dashv \&)$ is essentially discussed in section 3 there. The characterization of infinitesimal extensions and formal smoothness by adjoint functors (in 1-[[category theory]]) is considered in \begin{itemize}% \item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative spaces}, preprint MPI-2004-35 (\href{http://nlab.mathforge.org/nlab/files/KontsevichRosenbergNCSpaces.pdf}{pdf}, \href{http://www.mpim-bonn.mpg.de/preblob/2331}{ps}, \href{http://www.mpim-bonn.mpg.de/preblob/2303}{dvi}) \end{itemize} in the context of \emph{[[Q-categories]]} . The notion of forming [[petit topos|petit]] $(\infty,1)$-toposes of \'e{}tale objects over a given object appears in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Structured Spaces]]} \item [[David Carchedi]], \emph{The Meaning of \'E{}tale Stacks} (\href{http://arxiv.org/abs/1212.2282}{arXiv:1212.2282}) \end{itemize} [[!redirects differential cohesion]] [[!redirects differential cohesive (∞,1)-topos]] [[!redirects differential cohesive ∞-topos]] [[!redirects differential cohesive (∞,1)-toposes]] [[!redirects differential cohesive ∞-toposes]] [[!redirects differential cohesive (infinity,1)-topos]] [[!redirects differential cohesive infinity-topos]] [[!redirects differential cohesive (infinity,1)-toposes]] [[!redirects differential cohesive infinity-toposes]] [[!redirects cohesive (infinity,1)-topos -- infinitesimal cohesion]] [[!redirects differentially cohesive topos]] [[!redirects elastic topos]] [[!redirects elastic toposes]] [[!redirects elastic topoi]] [[!redirects elastic (∞,1)-topos]] [[!redirects elastic (∞,1)-toposes]] [[!redirects elastic (infinity,1)-topos]] [[!redirects elastic (infinity,1)-toposes]] [[!redirects ∞-elastic site]] [[!redirects ∞-elastic sites]] [[!redirects elastic model topos]] [[!redirects elastic model toposes]] \end{document}