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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*{formally étale morphism} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{tale_morphisms}{}\paragraph*{{\'E{}tale morphisms}}\label{tale_morphisms} [[!include etale morphisms - contents]] \hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry} [[!include higher geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{GeneralAbstractNotion}{General abstract notion}\dotfill \pageref*{GeneralAbstractNotion} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{ConcreteNotion}{Concrete notions}\dotfill \pageref*{ConcreteNotion} \linebreak \noindent\hyperlink{in_differential_geometry}{In differential geometry}\dotfill \pageref*{in_differential_geometry} \linebreak \noindent\hyperlink{in_algebraic_geometry}{In algebraic geometry}\dotfill \pageref*{in_algebraic_geometry} \linebreak \noindent\hyperlink{in_noncommutative_geometry}{In noncommutative geometry}\dotfill \pageref*{in_noncommutative_geometry} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak A [[morphisms]] of [[spaces]] $X \overset{p}{\longrightarrow} Y$ is called \emph{formally \'e{}tale} if it has a [[lifting property]] as [[étalé spaces]] do locally, but for \emph{[[infinitesimal space|infinitesmal]]} extensions: If for $Z \overset{f}{\to} Y$ any morphism and $\Re(Z) \to X$ a lift of its restriction along its [[reduction]] $\Re(Z) \to Z$, there is a unique extension to a complete lift. (If there exists at least one such infinitesimal extension, it is called a [[formally smooth morphism]]. If there exists at most one such extension, it is called a [[formally unramified morphism]]. The formally \'e{}tale morphisms are precisely those that are both formally smooth and formally unramified.) Traditionally this has been considered in the context of [[geometry]] over formal duals of [[ring]]s and [[associative algebra]]s. This we discuss in the section (\hyperlink{ConcreteNotion}{Concrete notion}). But generally the notion makes sense in any context of . This we discuss in the section \hyperlink{GeneralAbstractNotion}{General abstract notion}. \hypertarget{GeneralAbstractNotion}{}\subsection*{{General abstract notion}}\label{GeneralAbstractNotion} \hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition} Let \begin{displaymath} \mathbf{H} \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}} \mathbf{H}_{th} \end{displaymath} be an [[adjoint triple]] of [[functor]] with $u^*$ a [[full and faithful functor]] that preserves the [[terminal object]]. We may think of this as exhibiting (see there for details, but notice that in the notation used there we have $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$). We think of the objects of $\mathbf{H}$ as [[cohesive topos|cohesive space]]s and of the objects of $\mathbf{H}_{th}$ as such cohesive spaces possibly equipped with [[infinitesimal object|infinitesimal extension]]. As a class of examples that is useful to keep in mind consider a [[Q-category]] \begin{displaymath} (cod \dashv \epsilon \dashv dom) : \bar A \to A \end{displaymath} of and let \begin{displaymath} ((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) := ([dom,Set] \dashv [\epsilon, Set] \dashv [cod,Set] : [\bar A, Set] \to [A,Set]) \end{displaymath} be the corresponding . For any such setup there is a canonical [[natural transformation]] \begin{displaymath} \phi : u^* \to u^! \,. \end{displaymath} Details of this are in the section at [[cohesive topos]]. From this we get for every morphism $f : X \to Y$ in $\mathbf{H}$ a canonical morphism \begin{equation} u^* X \to u^* Y \prod_{u^! Y} u^! X \,. \label{MorphismIntoPullback}\end{equation} \begin{defn} \label{AbstractFormallyEtaleMorphism}\hypertarget{AbstractFormallyEtaleMorphism}{} A morphism $f : X \to Y$ in $\mathbf{H}$ is called \textbf{formally \'e{}tale} if \eqref{MorphismIntoPullback} is an [[isomorphism]]. \end{defn} This appears as (\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenberg, def. 5.1, prop. 5.3.1.1}). In other words, $f$ is formally \'e{}tale if the $f$-component naturality square \begin{displaymath} \itexarray{ u^* X &\stackrel{u^* f}{\to}& u^* Y \\ {}^{\mathllap{\phi_X}}\downarrow && \downarrow^{\mathrlap{\phi_Y}} \\ u^! X &\stackrel{u^! f}{\to}& u^! Y } \end{displaymath} of the [[natural transformation]] $\phi$ is a [[pullback]] [[diagram]]. \begin{remark} \label{FormallySmooth}\hypertarget{FormallySmooth}{} The partial notions of this condition are: if the above morphism is a [[monomorphism]] then $f$ is a [[formally unramified morphism]], if it is an [[epimorphism]] then $f$ is a [[formally smooth morphism]]. \end{remark} \begin{defn} \label{AbstractFormallyEtaleObject}\hypertarget{AbstractFormallyEtaleObject}{} An object $X \in \mathbf{H}$ is called \textbf{formally \'e{}tale} if the morphism $X \to *$ to the [[terminal object]] is formally \'e{}tale. \end{defn} \begin{prop} \label{AbstractFormallyEtaleObjectDirect}\hypertarget{AbstractFormallyEtaleObjectDirect}{} The object $X$ is formally \'e{}tale precisely if \begin{displaymath} u^* X \to u^! X \end{displaymath} is an [[isomorphism]]. \end{prop} This appears as (\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenberg, def. 5.3.2}). \hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties} \begin{prop} \label{}\hypertarget{}{} Formally \'e{}tale morphisms are closed under composition. \end{prop} This appears as (\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenberg, prop. 5.4}). \begin{proof} This follows by the [[pasting law]] for pullbacks: let $f : X \to Y$ and $g : Y \to Z$ be two formally \'e{}tale morphisms. Then by definition both of the small squares in \begin{displaymath} \itexarray{ u^* X &\stackrel{u^* f }{\to}& u^* Y &\stackrel{u^* g}{\to}& u^* Z \\ \downarrow && \downarrow && \downarrow \\ u^! X &\stackrel{u^! f }{\to}& u^! Y &\stackrel{u^! g}{\to}& u^! Z } \end{displaymath} are pullback squares. Hence so is the total outer square. \end{proof} Using also the other case of the [[pasting law]], the above proof shows more: \begin{prop} \label{}\hypertarget{}{} If \begin{displaymath} \itexarray{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z } \end{displaymath} is a [[commuting diagram]] such that $g$ and $h$ are formally \'e{}tale, then also $f$ is formally \'e{}tale. \end{prop} \begin{prop} \label{}\hypertarget{}{} Formally \'e{}tale morphisms are closed under [[retract]]s. \end{prop} This means that if $f : X \to Y$ is formally \'e{}tale and \begin{displaymath} \itexarray{ A &\to & X &\to& A \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{p}} \\ B &\to& Y &\to& B } \end{displaymath} is a [[commuting diagram]] such that the two horizontal composites are [[identities]], then also $p$ is formally \'e{}tale. \begin{proof} By applying the [[natural transformation]] $\phi : u^* \to u^!$ to this diagram we obtain a retract diagram in the category of squares, given by the naturality squares of $\phi$ on $f$ and $p$, where the middle square is a pullback square. By at \emph{[[retract]]} this implies that also the retracting square is a pullback, which means that $p$ is formally \'e{}tale. \end{proof} \begin{prop} \label{}\hypertarget{}{} If $u^*$ preserves [[pullbacks]], then formally \'e{}tale morphisms are stable under pullback. \end{prop} \begin{proof} Consider a [[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]] $\phi : u^* \to u^!$ to this yields a square of squares. Two sides of this are the [[pasting]] composite \begin{displaymath} \itexarray{ u^* A \times_Y X &\to& u^* X &\stackrel{\phi_X}{\to}& u^! X \\ \downarrow && \downarrow^{\mathrlap{u^* f}} && \downarrow^{\mathrlap{u^! f}} \\ u^* A &\to& u^* Y &\stackrel{\phi_Y}{\to}& u^! Y } \end{displaymath} and the other two sides are the pasting composite \begin{displaymath} \itexarray{ u^* A \times_Y X &\stackrel{\phi_{A \times_Y X}}{\to}& u^! A \times_Y A &\stackrel{}{\to}& u^! X \\ \downarrow^{} && \downarrow && \downarrow^{\mathrlap{u^! f}} \\ u^* A &\stackrel{\phi_A}{\to}& u^! A &\to& u^! 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 on $u^*$; \item the second square is a pullback, since $f$ is formally \'e{}tale. \item the fourth square is a pullback since $u^!$ is [[right adjoint]] and so also preserves pullbacks; \item also the total bottom rectangle is a pullback, since it is equal to the bottom total rectangle; \item therefore finally the third square is a pullback, by the [[pasting law]], hence also $p$ is formally \'e{}tale. \end{itemize} \end{proof} \hypertarget{ConcreteNotion}{}\subsection*{{Concrete notions}}\label{ConcreteNotion} We discuss realizations of the above general abstract definition in concrete models of the axioms. See also the concrete notions of [[formally smooth morphism]] and [[formally unramified morphism]]. \hypertarget{in_differential_geometry}{}\subsubsection*{{In differential geometry}}\label{in_differential_geometry} The [[category]] [[SmoothMfd]] of [[smooth manifold]]s may naturally be thought of as sitting inside the more general context of the [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s. This is canonically equipped with a notion of [[differential cohesion|infinitesimal cohesion]] exhibited by its inclusion into [[SynthDiff∞Grpd]]. This implies that there is an intrinsic notion of [[formally étale morphism]]s of smooth $\infty$-groupoids in general and of smooth manifolds in particular \begin{prop} \label{}\hypertarget{}{} A [[smooth function]] is a formally \'e{}tale morphism in this sense precisely if it is a [[local diffeomorphism]] in the traditional sense. \end{prop} See for more details. \hypertarget{in_algebraic_geometry}{}\subsubsection*{{In algebraic geometry}}\label{in_algebraic_geometry} \begin{itemize}% \item [[formally étale morphism of schemes]] \end{itemize} \hypertarget{in_noncommutative_geometry}{}\subsubsection*{{In noncommutative geometry}}\label{in_noncommutative_geometry} See (\hyperlink{RosenbergKontsevich}{RosenbergKontsevich, section 5.8}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[formally smooth morphism]] and [[formally unramified morphism]] $\Rightarrow$ \textbf{formally \'e{}tale morphism} [[!include cohesion - table]] \hypertarget{References}{}\subsection*{{References}}\label{References} The idea of defining \'e{}tale morphisms $f$ as those that get send to a pullback square by a natural transformation goes back to lectures by [[André Joyal]] in the 1970s. See the introduction and see section 4 of \begin{itemize}% \item [[Eduardo Dubuc]], \emph{Axiomatic etal maps and a theory of spectrum}, Journal of pure and applied algebra, 149 (2000) \end{itemize} The identification of the natural transformation in question with that induced by an [[adjoint triple]] (``[[Q-categories]]'') and the relation to \emph{formal} \'e{}taleness is observed (apparently independently?) in \begin{itemize}% \item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative spaces}, preprint MPI-2004-35 ([[KontsevichRosenbergNCSpaces.pdf:file]], \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2331}{ps}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2303}{dvi}) \end{itemize} Formalization and discussion in the context of [[cohesive (∞,1)-toposes]] is in section 2.5.3 (and defn 5.3.19) of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} [[!redirects formally etale morphism]] [[!redirects formally etale]] [[!redirects formally étale morphism]] [[!redirects formally étale morphisms]] [[!redirects formally etale morphisms]] [[!redirects formally étale map]] [[!redirects formally étale maps]] [[!redirects formally etale map]] [[!redirects formally etale maps]] \end{document}