[[!redirects pi-factorization systems]] category: cohesion #Contents# * table of contents {:toc} ## For the cohesive modality $\mathbf{\Pi}$ {#FactorizationSystemsForPi} ### External formulation We discuss [[nLab:orthogonal factorization system in an (infinity,1)-category|orthogonal factorization systems]] in a cohesive $(\infty,1)$-topos that characterize or are characterized by the [[nLab:reflective sub-(infinity,1)-category|reflective subcategory]] of dicrete objects, with reflector $\mathbf{\Pi} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{Disc}{\hookrightarrow} \mathbf{H}$. +-- {: .num_defn #PiClosure} ###### Definition For $f : X \to Y$ a morphism in $\mathbf{H}$, write $c_{\mathbf{\Pi}} f \to Y$ for the [[nLab:(∞,1)-pullback]] in $$ \array{ c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X \\ \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}} \\ Y &\to& \mathbf{\Pi} Y } \,, $$ where the bottom morphism is the $(\Pi \dashv Disc)$-[[nLab:unit of an adjunction|unit]]. We say that $c_{\mathbf{\Pi}} f$ is the **$\mathbf{\Pi}$-closure** of $f$, and that $f$ is **$\mathbf{\Pi}$[[nLab:Pi-closed morphism|-closed]]** if $X \simeq c_{\mathbf{\Pi}} f$. =-- +-- {: .num_prop #FactorizationPiEquivalencePiClosed} ###### Proposition If $\mathbf{H}$ has an [[nLab:∞-cohesive site]] of definition, then every morphism $f : X \to Y$ in $\mathbf{H}$ factors as $$ \array{ X &&\stackrel{f}{\to}&& Y \\ & \searrow && \nearrow \\ && c_{\mathbf{\Pi}}f } \,, $$ such that $X \to c_{\mathbf{\Pi}} f$ is a _$\mathbf{\Pi}$-equivalence_ in that it is inverted by $\mathbf{\Pi}$. =-- +-- {: .proof} ###### Proof The factorization is given by the naturality of $\mathbf{\Pi}$ and the universal property of the $(\infty,1)$-pullback in def. \ref{PiClosure}. $$ \array{ X &\to & c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X \\ &{}_{\mathllap{f}}\searrow & \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}} \\ && Y &\to& \mathbf{\Pi} Y } \,. $$ Then by prop. \ref{PiPreservesPullbacksOverDiscretes} the functor $\mathbf{\Pi}$ preserves the $(\infty,1)$-pullback over the discrete object $\mathbf{\Pi}Y$ and since $\mathbf{\Pi}(X \to \mathbf{\Pi}X)$ is an equivalence, it follows that $\mathbf{\Pi}(X \to c_{\mathbf{\Pi}f})$ is an equivalence. =-- +-- {: .num_prop #PiEquivalencePiClosedFactorizationSystem} ###### Proposition The pair of classes $$ (\mathbf{\Pi}-equivalences, \mathbf{\Pi}-closed morphisms) $$ is an [[nLab:orthogonal factorization system in an (infinity,1)-category|orthogonal factorization system]] in $\mathbf{H}$. =-- +-- {: .proof} ###### Proof This follows by the general reasoning discussed at [[nLab:reflective factorization system]]: By prop. \ref{FactorizationPiEquivalencePiClosed} we have the required factorization. It remains to check the orthogonality. So let $$ \array{ A &\to& X \\ \downarrow && \downarrow \\ B &\to& Y } $$ be a square diagram in $\mathbf{H}$ where the left morphism is a $\mathbf{\Pi}$-equivalence and the right morphism is $\mathbf{\Pi}$-closed. Then by assumption there is a pullback square on the right in $$ \array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow && \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,. $$ By naturality of the [[nLab:unit of an adjunction|adjunction unit]], the total rectangle is equivalent to $$ \array{ A &\to& \mathbf{\Pi} A &\to & \mathbf{\Pi} Y \\ \downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow \\ B &\to& \mathbf{\Pi} B &\to& \mathbf{\Pi}X } \,. $$ Here by assumption the middle morphism is an equivalence. Therefore there is an essentially unique lift in the square on the right and hence a lift in the total square. Again by the universality of the adjunction, any such lift factors through $\athbf{\Pi} B$ and hence also this lift is essentially unique. Finally by universality of the pullback, this induces an essentially unique lift $\sigma$ in $$ \array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,. $$ =-- +-- {: .num_prop } ###### Observation For $f : X \to Y$ a $\mathbf{\Pi}$-closed morphism and $y : * \to Y$ a [[nLab:global element]], the [[nLab:homotopy fiber]] $X_y := y^* X$ is a discrete object. =-- +-- {: .proof} ###### Proof By the def. \ref{PiClosure} and the [[nLab:pasting law]] we have that $y^* X$ is equivalently the $\infty$-pullback in $$ \array{ y^* X &\to& &\to& \mathbf{\Pi} X \\ \downarrow && && \downarrow \\ * &\stackrel{y}{\to}& Y &\stackrel{}{\to}& \mathbf{\Pi}Y } \,. $$ Since the [[nLab:terminal object in an (infinity,1)-category|terminal object]] is discrete, and since the [[nLab:right adjoint]] $Disc$ preserves $\infty$-pullbacks, this exhibits $y^* X$ as the image under $Disc$ of an $\infty$-pullback of $\infty$-groupoids. =-- ### Internal formulation There is also an [[internal formulation of cohesion]]. The following is a translation of the previous section in this language: (1) The following statements are equivalent: * $(E,M)$ is a *reflective factorization system* in $H$. * There is a reflective subcategory $C\hookrightarrow H$ with reflector $\sharp$, $E$ is the class of morphisms whose $\sharp$-image is invertible in $C$, and $C=M/1$. * $(E,M)$ is a factorization system and $E$ satisfies 2-out -of-3. * $(E,M)$ is a factorization system and $M$ is the class of fibrant morphisms $P\to A$ which as dependent types $x:A\dashv P(x): Type$ satisfy $forall\, x \,in Rsc(P(x))$. * For every $H$-morphism $f:A\to B$ satisfying: $\sharp A$ and $\sharp B$ are contractible, also for all $b$ we have $\sharp \, hFiber(f,b)$ is contractible. $$is Contr(\sharp A),is Contr(\sharp B), f:A\to B, b:B\vdash is Contr (\sharp h Fiber(f,b))$$ (2) The following statements are equivalent: * $(E,M)$ is a factorization system in $H$. * The class $(E,M)^\times:=\{M/x|x\in H,\M/x\hookrightarrow H/x\,is.refl,\,refl.fact\}$ is pullback-stable where $refl.fact$ means that each reflection is defined by $(E,M)$-factorization. * $(C.x\subseteq H/x)_{x\in H}$ is a pullback-stable system of reflective subcategories of slices of $H$, and for every $x$ the class of objects of $C.x$ is closed under composition. * The class of types $B$ satisfying $in Rsc (B)$ is closed under dependent sums. $$in Rsc(A), forall \, x,in Rsc(P(x))\vdash in Rsc (\sum_{x:A} P(x))$$ (3) The following statements are equivalent: * $(C.x\subseteq H/x)_{x\in H}$ is a pullback-stable system of reflective subcategories of slices of $H$, for every $x$ the class of objects of $C.x$ is closed under composition, and all reflectors commute with pullbacks. * The (by (2)) to $(C.x\subseteq H/x)_{x\in H}$ corresponding factorization system $(E,M)$ is pullback stable. ## For the infinitesimal-cohesive modality $\mathbf{\Pi}_inf$ ### External formulation {#FormallySmoothEtaleUnramified} +-- {: .num_defn #FormalSmoothness} ###### Definition We say an object $X \in \mathbf{H}_{th}$ is **formally smooth** if the constant infinitesimal path inclusion, $X \to \mathbf{\Pi}_{inf}(X)$, def. \ref{InfinitesimalPathsAndReduction}, is an [[nLab:effective epimorphism in an (∞,1)-category|effective epimorphism]]. =-- In this form this is the evident $(\infty,1)$-categorical analog of the conditions as they appear for instance in ([SimpsonTeleman, page 7](#SimpsonTeleman)). +-- {: .num_remark #FormalSmoothnessByCanonicalMorphism} ###### Remark An object $X \in \mathbf{H}_{th}$ is formally smooth according to def. \ref{FormalSmoothness} precisely if the canonical morphism $$ i_! X \to i_* X $$ (induced from the [[nLab:adjoint quadruple]] $(i_! \dashv i^* \dashv i_* \dashv i^!)$, see there) is an [[nLab:effective epimorphism in an (∞,1)-category|effective epimorphism]]. =-- +-- {: .proof} ###### Proof The canonical morphism is the composite $$ (i_! \to i_*) := i_! \stackrel{\eta i_!}{\to} \mathbf{\Pi}_{inf} i_! := i_* i^* i_! \stackrel{\simeq}{\to} i_* \,. $$ By the condition that $i_!$ is a [[nLab:full and faithful (∞,1)-functor]] the second morphism here in an [[nLab: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. =-- +-- {: .num_remark #RelationToRK} ###### Remark In this form this characterization of formal smoothness is the evident generalization of the condition given in ([Kontsevich-Rosenberg, section 4.1](#KontsevichRosenbergSpaces)). See the section _<a href="http://nlab.mathforge.org/nlab/show/Q-category#FormalSmoothness">Formal smoothness</a>_ at _[[nLab:Q-category]]_ for more discussion. Notice that by <a href="http://nlab.mathforge.org/nlab/show/Q-category#DiscussionOfTheInfinitesimalThickeningFormalization">this remark</a> the notation there is related to the one used here by $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$. =-- Therefore we have the following more general definition. +-- {: .num_defn #FormalRelativeSmoothnessByCanonicalMorphism} ###### Definition For $f : X \to Y$ a morphism in $\mathbf{H}$, we say that 1. $f$ is a **[[nLab:formally smooth morphism]]** if the canonical morphism $$ i_! X \to i_! Y \prod_{i_* Y} i_* Y $$ is an [[nLab:effective epimorphism in an (∞,1)-category|effective epimorphism]]. 1. $f$ is a **[[nLab:formally unramified morphism]]** if this is a [[nLab:(-1)-truncated]] morphism. More generally, $f$ is an _order-$k$ formally unramified morphisms_ for $(-2) \leq k \leq \infty$ if this is a [[nLab:k-truncated]] morphism. 1. $f$ is a **[[nLab:formally étale morphism]]** if this morphism is an [[nLab:equivalence in an (∞,1)-category|equivalence]], hence if $$ \array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y } $$ is an [[nLab:(∞,1)-pullback]] square. =-- +-- {: .num_remark #MeaningOfFormallyUnramified} ###### Remark An order-(-2) formally unramified morphism is equivalently a [[nLab:formally étale morphism]]. Only for [[nLab:0-truncated]] $X$ does formal smoothness together with formal unramifiedness imply formal étaleness. =-- Even more generally we can formulate formal smoothness in $\mathbf{H}_{th}$: +-- {: .num_defn #FormallyEtaleInHTh} ###### Definition A morphism $f \colon X \to Y$ in $\mathbf{H}_{th}$ is **formall étale** if it is $\mathbf{\Pi}_{inf}$-closed, hence if its $\mathbf{\Pi}_{inf}$-unit naturality square $$ \array{ X &\to& \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(y) } $$ is an [[nLab:(∞,1)-pullback]]. =-- +-- {: .num_remark} ###### Remark 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}. =-- +-- {: .proof} ###### Proof This is again given by the fact that $\mathbf{\Pi}_{inf} = i_* i^*$ by definition and that $i_!$ is fully faithful, so that $$ \array{ i_! X &\to& \mathbf{\Pi}_{inf}(i_! X) \simeq i_* i^* i_! X &\stackrel{\simeq}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* i^* i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! Y &\to& \mathbf{\Pi}_{inf}(i_! Y) \simeq i_* i^* i_! Y &\stackrel{\simeq}{\to}& i_* Y } \,. $$ =-- +-- {: .num_prop #PropertiesOfFormallyEtaleMorphisms} ###### Proposition The collection of [[nLab:formally étale morphisms]] in $\mathbf{H}$, def. \ref{FormalRelativeSmoothnessByCanonicalMorphism}, is closed under the following operations. 1. Every [[nLab:equivalence in an (∞,1)-category|equivalence]] is formally étale. 1. The composite of two formally étale morphisms is itself formally étale. 1. If $$ \array{ && Y \\ & {}^{\mathllap{f}}\nearrow &\swArrow_{\simeq}& \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z } $$ is a [[nLab:diagram]] such that $g$ and $h$ are formally étale, then also $f$ is formally étale. 1. Any [[nLab:retract]] of a formally étale morphisms is itself formally étale. 1. The [[nLab:(∞,1)-pullback]] of a formally étale morphisms is formally étale if the pullback is preserved by $i_!$. =-- The statements about closure under composition and pullback appears as([KontsevichRosenberg, prop. 5.4, prop. 5.6](#KontsevichRosenbergSpaces)). Notice that the extra assumption that $i_!$ preserves the pullback is implicit in their setup, by remark \ref{RelationToRK}. +-- {: .proof} ###### Proof The first statement follows since $\infty$-pullbacks are well defined up to quivalence. The second two statements follow by the [[nLab:pasting law]] for [[nLab:(∞,1)-pullback]]s: let $f : X \to Y$ and $g : Y \to Z$ be two morphisms and consider the [[nLab:pasting diagram]] $$ \array{ i_! X &\stackrel{i_! f }{\to}& i_! Y &\stackrel{i_! g}{\to}& Z \\ \downarrow && \downarrow && \downarrow \\ i_* X &\stackrel{i_* f }{\to}& i_* Y &\stackrel{i_* g}{\to}& i_* Z } \,. $$ If $f$ and $g$ are formally étale 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 étale. Similarly, if $g$ and $g \circ f$ are formally étale 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 étale. For the fourth claim, let $Id \simeq (g \to f \to g)$ be a [[nLab:retract]] in the [[nLab:arrow category|arrow (∞,1)-category]] $\mathbf{H}^I$. By applying the natural transformation $\phi : i_! \to I_*$ we obtain a retract $$ Id \simeq ((i_! g \to i_*g) \to (i_! f \to i_*f) \to (i_! g \to i_*g)) $$ 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 [[nLab:(∞,1)-pullback]] square, then so is its retract sqare. This implies the fourth claim. To see this, we use that 1. [[nLab:(∞,1)-limit]]s are computed by [[nLab:homotopy limit]]s in any [[nLab:presentable (∞,1)-category]] $C$ presenting $\mathbf{H}$; 1. homotopy limits in $C$ may be computed by the left and right adjoints provided by the [[nLab:derivator]] $Ho(C)$ associated to $C$. From this the claim follows as described in detail at _[[nLab:retract]]_ in the section _<a href="http://nlab.mathforge.org/nlab/show/retract#RetractsOfDiagrams">retracts of diagrams</a>_ . For the last claim, consider an [[nLab:(∞,1)-pullback]] diagram $$ \array{ A \times_Y X &\to& X \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{f}} \\ A &\to& Y } $$ where $f$ is formally étale. Applying the [[nLab:natural transformation]] $\phi : i_! \to i_*$ to this yields a square of squares. Two sides of this are the [[nLab:pasting]] composite $$ \array{ i_! A \times_Y X &\to& i_! X &\stackrel{\phi_X}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! p}} && \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! A &\to& i_! Y &\stackrel{\phi_Y}{\to}& i_* Y } $$ and the other two sides are the pasting composite $$ \array{ i_! A \times_Y X &\stackrel{\phi_{A \times_Y X}}{\to}& i_* A \times_Y A &\stackrel{}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! p}} && \downarrow^{\mathrlap{i_* p}} && \downarrow^{\mathrlap{i_* f}} \\ i_! A &\stackrel{\phi_A}{\to}& i_* A &\to& i_* Y } \,. $$ Counting left to right and top to bottom, we have that * the first square is a pullback by assumption that $i_!$ preserves the given pullback; * the second square is a pullback, since $f$ is formally étale. * the total top rectangle is therefore a pullback, by the [[nLab:pasting law]]; * the fourth square is a pullback since $i_*$ is [[nLab:right adjoint]] and so also preserves pullbacks; * also the total bottom rectangle is a pullback, since it is equal to the top total rectangle; * therefore finally the third square is a pullback, by the other clause of the [[nLab:pasting law]]. Hence $p$ is formally étale. =-- +-- {: .num_remark #AsOpenMaps} ###### Remark The properties listed in prop. \ref{PropertiesOfFormallyEtaleMorphisms} correspond to the axioms on the _[[nLab:open map]]s_ ("admissible maps") in a [[nLab:geometry (for structured (∞,1)-toposes)]] ([Lurie, def. 1.2.1](#LurieStSp)). This means that a notion of formally étale morphisms induces a notion of [[nLab:locally algebra-ed topos|locally algebra-ed (∞,1)toposes]]/[[nLab:structured (∞,1)-topos]]es in a cohesive context. This is discuss in * [[nLab:cohesive (∞,1)-topos -- structure ∞-sheaves]]. =-- In order to interpret the notion of formal smoothness, we turn now to the discussion of infinitesimal reduction. +-- {: .num_prop #RedIsIdempotent} ###### Proposition The operation $\mathbf{Red}$ is an [[nLab:idempotent]] projection of $\mathbf{H}_{th}$ onto the image of $\mathbf{H}$ $$ \mathbf{Red} \mathbf{Red} \simeq \mathbf{Red} \,. $$ Accordingly also $$ \mathbf{\Pi}_{inf} \mathbf{\Pi}_{inf} \simeq \mathbf{\Pi}_{inf} $$ and $$ \mathbf{\flat}_{inf} \mathbf{\flat}_{inf} \simeq \mathbf{\flat}_{inf} \,. $$ =-- +-- {: .proof} ###### Proof By definition of infinitesimal neighbourhood we have that $i_!$ is a [[nLab:full and faithful (∞,1)-functor]]. It follows that $i^* i_! \simeq Id$ and hence $$ \begin{aligned} \mathbf{Red} \mathbf{Red} & \simeq i_! i^* i_! i^* \\ & \simeq i_! i^* \\ & \simeq \mathbf{Red} \end{aligned} \,. $$ =-- +-- {: .num_cor #PiInfXIsFormallySmooth} ###### Observation For every $X \in \mathbf{H}_{th}$, we have that $\mathbf{\Pi}_{inf}(X)$ is formally smooth according to def. \ref{FormalSmoothness}. =-- +-- {: .proof} ###### Proof By prop. \ref{RedIsIdempotent} we have that $$ \mathbf{\Pi}_{inf}(X) \to \mathbf{\Pi}_{inf} \mathbf{\Pi}_{inf}X $$ is an [[nLab:equivalence in an (∞,1)-category|equivalence]]. As such it is in particular an [[nLab:effective epimorphism in an (∞,1)-category|effective epimorphism]]. =-- #### Cotangent bundles {#CotangentBundles} +-- {: .num_defn #EtaleSlice} ###### Definition For $X \in \mathbf{H}_{th}$ any object, write $$ (\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X} $$ for the full [[nLab:sub-(∞,1)-category]] of the [[nLab:slice (∞,1)-topos]] over $X$ on those maps into $X$ which are formally &eacute;tale, def. \ref{FormallyEtaleInHTh}. =-- +-- {: .num_prop #EtalificationIsCoreflection} ###### Proposition The inclusion of def. \ref{EtaleSlice} is both [[nLab:reflective sub-(∞,1)-category|reflective]] as well as [[nLab:coreflective subcategory|coreflective]], hence it fits into an [[nLab:adjoint triple]] of the form $$ (\mathbf{H}_{th})_{/X}^{th} \stackrel{\overset{L}{\leftarrow}}{\stackrel{\hookrightarrow}{\underset{Et}{\leftarrow}}} (\mathbf{H}_{th})_{/X}^{fet} \,. $$ =-- +-- {: .proof} ###### Proof By the general discussion at _[[nLab:reflective factorization system]]_, the reflection is given by sending a morphism $f \colon Y \to X$ to $X \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y$ and the reflection unit is the left horizontal morphism in $$ \array{ Y &\to& X \times_{\mathbf{\Pi}_{inf}(Y)} \mathbf{\Pi}_{inf}(Y) &\to& \mathbf{\Pi}_{inf}(Y) \\ & \searrow & \downarrow^{} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ && X &\to& \mathbf{\Pi}_{inf}(X) } \,. $$ Therefore $(\mathbf{H}_{th})_{/X}^{et}$, being a reflective subcategory of a [[nLab:locally presentable (∞,1)-category]], is (as discussed there) itself locally presentable. Hence by the [[nLab:adjoint (∞,1)-functor theorem]] it is now sufficient to show that the inclusion preserves all small [[nLab:(∞,1)-colimits]] in order to conclude that it also has a right [[nLab:adjoint (∞,1)-functor]]. So consider any [[nLab:diagram]] [[nLab:(∞,1)-functor]] $I \to (\mathbf{H}_{th})_{/X}^{et}$ out of a [[nLab: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 $(\mathbf{H}_{th})_{/X}^{et}$ 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 _[slice category - Colimits](overcategory#LimitsAndColimits)_). Therefore we are reduced to showing that the square $$ \array{ \underset{\to_i}{\lim} Y_i &\to& \mathbf{\Pi}_{inf} \underset{\to_i}{\lim} Y_i \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_{inf}(X) } $$ is an [[nLab:(∞,1)-pullback]] square. But since $\mathbf{\Pi}_{inf}$ is a [[nLab:left adjoint]] it commutes with the $(\infty,1)$-colimit on objects and hence this diagram is equivalent to $$ \array{ \underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \mathbf{\Pi}_{inf} Y_i \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_{inf}(X) } \,. $$ This diagram is now indeed an [[nLab:(∞,1)-pullback]] by the fact that we have [[nLab:universal colimits]] in the [[nLab:(∞,1)-topos]] $\mathbf{H}_{th}$, hence that on the left the component $Y_i$ for each $i \in I$ is the [[nLab:(∞,1)-pullback]] of $\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally &eacute;tale morphisms. =-- Using this &eacute;talification, we can now turn [de Rham coefficient objects](cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology) into genuine [[nLab:cotangent bundle]]. +-- {: .num_defn #CotangentBundle} ###### Definition Let $G \in Grp(\mathbf{H}_{th})$ be an [[nLab:∞-group]] and let write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object. For $X \in \mathbf{H}_{th}$ any object, consider the [[nLab:projection]] $X \times \flat_{dR}\mathbf{B}G \to X$ as an object in the [[nLab:slice (∞,1)-topos]] $(\mathbf{H}_{th})_{/X}$. Then write $$ Et(X \times \flat_{dR}\mathbf{B}G) \in (\mathbf{H}_{th})^{fet}_{/X} $$ for its &eacute;talifiation, the coreflection by prop. \ref{EtalificationIsCoreflection}. The sections of this object we call the flat sections of the **$G$-valued [[nLab:cotangent bundle]]** of $X$. =-- +-- {: .num_remark } ###### Remark For $U \in \mathbf{H}_{th}$ a test object (say an object in a [[nLab:(∞,1)-site]] of definition, under the [[nLab:Yoneda embedding]]) a formall &eacute;tale morphsim $U \to X$ is like an [[nLab: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 $$ \array{ U &&\to&& Et(X \times \flat_{dR} \mathbf{B}G) \\ & \searrow && \swarrow \\ && X } \,. $$ By the fact that $Et(-)$ is [[nLab:right adjoint]], such diagrams are in bijection to diagrams $$ \array{ U &&\to&& X \times \flat_{dR} \mathbf{B}G \\ & \searrow && \swarrow \\ && X } $$ where we are now simply including on the left the formally &eacute;tale map $(U \to X)$ along $(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$. In other words, the (flat) sections of the $G$-valued cotangent bundle $Et(X \times \flat_{dR}\mathbf{B}G) \to X$ are just the sections of $X \times \flat_{dR}\mathbf{B}G \to X$ itself, only that the _domain_ of the section is constrained to be a formally &eacute; 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 $X$. =-- #### Critical locus {#CriticalLocus} Let $G$ be an [[nLab:∞-group]] and write $\mathbf{d} \colon G \to \flat_{dR}\mathbf{B}G$ for its [Maurer-Cartan form](cohesive+%28infinity,1%29-topos+--+structures#CurvatureCharacteristics). +-- {: .num_defn #FormalEtaleGroupoid} ###### Definition For $S \colon X \to G$ a morphism in $\mathbf{H}_{th}$, hence $G$-valued function, its **[[nLab:derivative]]** is the composite $$ \mathbf{d}S \colon X \stackrel{S}{\to} G \stackrel{\mathbf{d}}{\to} \flat_{dR}\mathbf{B}G \,. $$ Since the identity on $X$ is formally &eacute;tale, This we may regard as a section of the $G$-valued cotangent bundle, def. \ref{CotangentBundle}, $$ \array{ X &&\stackrel{(id, \mathbf{d}S)}{\to}&& Et(X \times \flat_{dR}\mathbf{B}G) \\ & {}_{\mathllap{id}}\searrow && \swarrow \\ && X } \,. $$ The **[[nLab:critical locus]]** $\{x \in X | \mathbf{d}S = 0\}$ of $S$ is the [[nLab:homotopy fiber]] of this section in $(\mathbf{H}_{th})_{/X}^{fet}$, hence the $(\infty,1)$-pullback $$ \array{ \{x \in X | \mathbf{d}S = 0\} &\to& X \\ \downarrow && \downarrow^{\mathrlap{0}} \\ X &\stackrel{\mathbf{d}S}{\to}& Et(X \times \flat_{dR}\mathbf{B}G) } \,. $$ =-- See at _[[nLab:schreiber:derived critical locus]]_ for more discussion of this. ### Internal formulation