# Contents * table of contents {:toc} ## $\Box$-closed morphisms +-- {: .num_lemma} ###### Definition Let $\Box$ be an idempotent monad on a presentable $(\infty,1)$-category $C$. A morphism $f:X\to Y$ is called *$\Box$-closed* if $$\array{ X&\to &\Box X \\ \downarrow^f &&\downarrow^{\Box f} \\ Y&\to& \Box Y }$$ is a pullback square. =-- +-- {: .num_lemma #daggerclosed} ###### Theorem The class of $\Box$-closed morphisms $C$ satisfies the following closure properties: (1) Every equivalence is $\Box$-closed. (2) The composite of two $\Box$-closed morphisms is $\Box$-closed. (3) The left cancellation property is satisfied: If $h=g\circ f$ and $h$ and $g$ are $\Box$-closed, then so is $f$. (4) Any retract of a $\Box$-closed morphism is $\Box$-closed. (5) The class is closed under pullbacks which are preserved by $\Box$. =-- +-- {: .num_lemma} ###### Remark A class of $\Box$-closed morphism which is closed under pullback is an *admissible structure* defining a *geometry* in the sense of Lurie's DAG. =-- +-- {: .num_lemma} ###### Theorem (Formally étale subslices are reflective andcoreflecive) (1) Let $C/X$ be a slice of $C$. The full sub-$(\infty,1)$-category $(C/X)_\Box\stackrel{\iota}{\hookrightarrow} C/X$ on those morphisms into $X$ which are $\Box$-closed is reflective and coreflective; i.e. $\iota$ fits into an adjoint triple $$ (C/X)_\Box \stackrel{\overset{L}{\leftarrow}}{\stackrel{\hookrightarrow}{\underset{Et}{\leftarrow}}} (C/X) \,. $$ (2) By the Giraud-Lurie axioms $C/X$ and $(C/X)_\Box$ are $(\infty,1)$-toposes. (3) In particular $C_\Box:=(C/*)_\Box\hookrightarrow C$ is a reflective and coreflective subtopos. =-- +-- {: .num_lemma} ###### Remark (relation of reflective subcategories and reflective factorization systems) =-- +-- {: .num_lemma} ###### Example ($\mathbf{\Pi}_inf$-closed morphism) Let $H$ be a cohesive $(\infty,1)$-topos equipped with infinitesimal cohesion $$(i_!\dashv i^*\dashv i_*):H\stackrel{i_*}{\to} H_th$$ Then the class of formally étale morphisms in $H$ equals the class of $\mathbf{\Pi}_inf:=i_*i^*$-closed morphisms in $H_th$ which happen to lie in $H$. =-- +-- {: .num_lemma} ###### Theorem For the classs $E$ of $\mathbf{\Pi}_inf$-closed morphisms in $C$ we have in addition to the above closure properties also the following ones: (1) If in a pullback square in $C$ the left arrow is in $E$ and the bottom arrow is an effective epimorphism, then the right arrow is in $E$. (2) Every morphism $D\to *$ from a discrete object to the terminal object is in $E$. (3) $E$ is closed under colimit (taken in the arrow category). (4) $E$ is closed under forming diagonals. =-- +-- {: .num_lemma} ###### Definition ($\mathbf{\Pi}_inf$-closed object) An object of $H$ is called *formally étale object* if there is a formally étale (effective) epimorphism (called *atlas*) from a $0$-truncated object into $X$. =-- +-- {: .num_lemma} ###### Theorem (De Rham theorem for formally étale objects){#DR} The de Rham theorem holds for any formally étale object for which the de Rham theorem holds level-wise in regard to the Cech nerve induced from the atlas. =-- ## Derived structures and models ### $U$-modelled higher manifolds +-- {: .num_lemma} ###### Definition (Cover, Atlas) Let $U\to X$ be a morphism in an $(\infty,1)$-category. (1) We call $U\to X$ a *cover of $X$* if it is an effective epimorphism. (2) We call $U\to X$ a *relative cover wrt. a class $M$ of morphisms* if its pullback along every morphism in $M$ is a cover of $U$ and lies in $M$. (3) We call $U\to X$ a *$n$-atlas of $X$*, if it is a cover and $U$ is $n$-truncated. A $0$-atlas we call just an *atlas*. =-- +-- {: .num_lemma} ###### Theorem (Hausdorff manifold) (1) A manifold $X$ is a paracompact if there is a jointly epimorphic set of monomorphisms $\phi_i:\mathbb{R}^n\to X$ such that the corresponding Cech groupoid $\zeta_\phi$ is degree-wise a coproduct of copies of $\mathbb{R}^n$. (2) $X$ is hausdorff if $\zeta_\phi$ is moreover étale. =-- +-- {: .num_lemma} ###### Remark and Definition (1) In a cohesive $(\infty,1)$-topos $H$, $\mathbb{R}^n$ can be defined solely in terms of the internal logic of $H$. (2) The previous theorem suggests to call an object $X\in H$ an *$U$-modelled hausdorff $\infty$-manifold* or an *$U$-modelled étale $\infty$-manifold* if there is an étale atlas $U\to X$ of $X$. =-- ### $\infty$-orbifolds +-- {: .num_lemma} ###### Definition ($\kappa$-compact object, $\kappa$-compact cover, relatively $\kappa$-compact atlas) Let $C$ be an $(\infty,1)$-category. Let $U\to X$ be a morphism. (1) $U$ is called *$\kappa$-compact* if $C(X,-)$ preserves $\kappa$-filtered colimits. (2) $U\to X$ is called *$\kappa$-compact cover* if it is a cover and $U$ is $\kappa$-compact. (3) $U\to X$ is called *relative $\kappa$-compact cover* if it is a relative cover wrt. all morphisms with $\kappa$-compact domain. =-- +-- {: .num_lemma} ###### Remark The class of relative $\kappa$-compact covers is closed under composition, pullbacks, and contains all isomorphisms. =-- +-- {: .num_lemma} ###### Definition ($\infty$-orbifold) An *$\infty$-orbifold* is defined to be a groupoid object in $H$ posessing a relative $\kappa$-compact atlas which is also $\mathbf{\Pi}_inf$-closed. =-- +-- {: .num_lemma} ###### Corollary (De Rham theorem for $\infty$-orbifolds) As a corollary to the [De Rham Theorem for étale objects](#DR) we obtain the de Rham Theorem for $\infty$-orbifolds. =-- +-- {: .num_lemma} ###### Observation and Definition (Inertia $\infty$-orbifold) The free loop space object of an $\infty$-orbifold is an $\infty$-orbifold and is called the *inertia $\infty$-orbifold of $X$*. =-- ## Models ### $(\diamond\dashv \Box)$ What we described so far dualizes to the case of an *idempotent comonad*. Also the situation where we have adjoint modalities - i.e an adjoint pair $(\diamond\dashv \Box)$ where $\Box$ is an idempotent monad and $\diamond$ is an idempotent comonad is of interest. ### Synthetic differential geometry We will describe the foundations of the theory of *synthetic differential geometry* in terms of adjoint modalities. The $(\infty,1)$-topos $H_th$ of synthetic differential $\infty$-groupoids is an infinitesimal cohesive neighborhood of the $(\infty,1)$-topos $H$-of smoooth $\infty$-groupoids. In this context we call $\Box$-closed morphisms also étale morphisms. +-- {: .num_lemma} ###### Definition An object $X\in H_th$ we call to be an *infinitesimal smooth locus* if $\diamond X$ is contractible. $$\diamond X\simeq *$$ =-- +-- {: .num_lemma} ###### Lemma (Tangent space) For every $X\in H_th$ the$\Box$-unit $\eta^\Box_X$ evaluates in every infinitesimal smooth locus $D$ to the tangent bundle. =-- +-- {: .proof} ###### Proof $$(\eta_X^\Box:X(D)\to (\Box X)(D))=([D,X]\to [D,\Box X]\simeq( [\diamond D,X]\simeq [*,X]\simeq X)$$ =-- +-- {: .num_lemma} ###### Proposition (submersion -, immersion -, and étale map of smooth paracompact manifolds) A morphism $f:X\to Y$ of smooth paracompact manifolds in $H_th$ is a submersion / imersion / étale morphism iff the induced morphism $$X\to Y\times_{\Box Y} \Box X$$ is an epimorphism / monomorphism / isomorphism. =-- +-- {: .proof} ###### Proof Every $Z\in H_th$ can be written as $Z=U\times D$ where $U\in H$ and $D$ is an infinitesimal smooth locus. We compute: $$\array{ X(U\times D)=[U\times D, X]=[U,[D,X]] \\ \Box X(U\times D)=[U\times D,\Box X]=[U,[D,\Box X]]=[U,[\diamond D,X]]=[U,X]} $$ and analogously for $Y$. =-- ### Étale groupoids +-- {: .num_lemma} ###### Theorem Let $X$ be an étale simplicial manifold being a Kan fibrant object in the projective model structure on simplicial presheaves equipped with an atlas $f:U\to X$. Then $f$ is $\Box$-closed. =-- +-- {: .proof} ###### Proof $f$ is $\Box$-closed precisely if $U$ is equivalent to the homotopy pullback of $$\array{ &&\Box U \\ &&\downarrow^{\Box f} \\ X&\stackrel{\eta^\Box_X}{\to}&\Box X }$$ We compute this pullback equivalently by a ($1$-categorical) pullback by giving a fibration replacement of $\Box f$. Décalage $d_last:Dec (X)\to X$ is a fibration replacement of $f$ and since $\Box$ is a right Quillen functor $\Box d_last$ is a fibration replacement of $f$. Since ($1$-categorial) pullbacks of simplicial objects are level-wise pullbacks it remains to show that for all $n$ $$U\to X_n\times_{(\Box X)_n}(\Box Dec(X))_n$$ is an isomorphism. By the previous Proposition, this is the case precisely if $d_lstn:Dec(X)_n\to X_n$ is an étale map of manifolds. But this map is étale by the definition of "étale manifold" which we assumed. =-- +-- {: .num_lemma} ###### Corollary (Classical étale groupoids) An étale Lie groupoid possesses an étale atlas. =-- +-- {: .proof} ###### Proof We compute the same pullback as in the proof of the previous Theorem but with the fibration replacement of $\Box f$ obtained by the factorization lemma. The fibration replacement of $f$ is the left vertical composite in (diagram) where $X\stackrel{e}{\to}X^I\stackrel{d_0,d_1}{\to}X\times X$ is the factorization of the codiagonal of $X$ through the pathspace object $X^I=X^{\Delta[1]}$ of $X$. Spelled out the pullback (diagram) which is by assumption a $1$-truncated groupoid interprets as (list) The factorization lemma says that $$\array{ U&\to&\Box U \\ \downarrow&&\downarrow^{\Box f} \\ X&\stackrel{\eta^\Box_X}{\to}&\Box X }$$ is a homotopy pullback iff $$\array{ X^I\times_X U&\to&\Box X^I\times_X U \\ \downarrow&&\downarrow^{\Box \tilde{f}} \\ X&\stackrel{\eta^\Box_X}{\to}&\Box X }$$ is a $1$-categorial pullback. This means precisely that $f$ is $\Box$-closed iff $\tilde{f}$. Since the objects here are simplicial, the latter pullback is computed level-wise. By the previous Proposition the $1$-truncation of $f$ is $\Box$-closed iff the $1$ truncation of $\tilde{f}$ is a classical Lie groupoid. $X$ being a classical Lie Groupoid means that (description) =-- +-- {: .num_lemma} ###### Remark (Étale simplicial manifold) If $Mfd_et$ denotes a convenient category of manifolds with only étale maps as morphisms, and $G_\Delta^{op}$ is a simplicial object in this category, then Yoneda embedding of this object followed by sheafification yields an étale simplicial manifold. =-- ### Terminological remark on étaleness in algebraic geometry One can refine the notion of $\Box$-closedness in the following way. +-- {: .num_lemma} ###### Definition For some integer $k\ge -2$, a morphism $f:X\to Y$ in $H_th$ is called *$\Box$-$k$-subclosed* / *$\Box$-closed* / *$\Box$-$k$-supclosed* if the characterizing morphism $$X\to Y\times_{\Box Y} \Box X$$ is $k$-truncted / isomorphism / $k$-connected. =-- +-- {: .num_lemma} ###### Definition An object $X\in H_th$ is called *algebraically formally $k$-smooth* / *algebraically formally étale* if the morphism $X\to *$ is $\Box$-$k$-supclosed / $\Box$-closed. =-- +-- {: .num_lemma} ###### Remark (1) Every $n$-truncated object is algebraically formally $k$-smooth. (2) Every $n$-truncated object is algebraically formally étale if it is contractible. =--