Spahn Pi-factorization system

category: cohesion

Contents

For the cohesive modality $\mathbf{\Pi}$

External formulation

A $\mathbf{\Pi}$-factorization system is a reflective factorization system?. The discussion given here is a terminological variant thereof.

We discuss orthogonal factorization systems in a cohesive $(\infty,1)$-topos that characterize or are characterized by the reflective subcategory of dicrete objects, with reflector $\mathbf{\Pi} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{Disc}{\hookrightarrow} \mathbf{H}$.

Definition

Let $f : X \to Y$ be a morphism in $\mathbf{H}$, write $c_{\mathbf{\Pi}} f \to Y$ for the (∞,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 $(\Pi \dashv Disc)$-unit.

(1) We say that $c_{\mathbf{\Pi}} f$ (respectively $c_{\mathbf{\Pi}} f\to Y$) is the $\mathbf{\Pi}$-closure of $f$.

(2) $f$ is called $\mathbf{\Pi}$-closed if $X \simeq c_{\mathbf{\Pi}} f$.

(3) $f$ is called a $\mathbf{\Pi}$-equivalence if $\mathbf{\Pi} f$ is an equivalence.

Lemma

The $\mathbf{\Pi}$-closure of $f$ is $\mathbf{\Pi}$-closed.

Proof

Apply $\mathbf{\Pi}$ to the defining Pullback square, the result is a pullback square since $\mathbf{\Pi}Y$ is discrete. $\mathbf{\Pi}$ is idempotent. Hence by the Pullback pasting lemma $\mathbf{\Pi}C_\mathbf{\Pi} f\simeq C_\mathbf{\Pi} f$.

Proposition

If $\mathbf{H}$ has an ∞-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 and $C_\mathbf{\Pi}\to Y$ is $\mathbf{\Pi}$-closed.

Proof

The factorization is given by the naturality of $\mathbf{\Pi}$ and the universal property of the $(\infty,1)$-pullback in def. \ref{Pi Closure?}.

$\array{ X &\to & c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X \\ &{}_{\mathllap{f}}\searrow & \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}} \\ && Y &\to& \mathbf{\Pi} Y } \,.$

Then by prop. \ref{Pi Preserves Pullbacks Over Discretes?} 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.

Proposition

The pair of classes

$(\mathbf{\Pi}-equivalences, \mathbf{\Pi}-closed morphisms)$

is an orthogonal factorization system in $\mathbf{H}$.

Proof

This follows by the general reasoning discussed at reflective factorization system:

By prop. \ref{Factorization Pi Equivalence Pi Closed?} 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 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 $\mathbf{\Pi} B$ and hence also this lift is essentially unique.

Finally by universality of the pullback, this induces an essentially unique lift $\sigma$ in

$\array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,.$
Observation

For $f : X \to Y$ a $\mathbf{\Pi}$-closed morphism and $y : * \to Y$ a global element, the homotopy fiber $X_y := y^* X$ is a discrete object.

Proof

By the def. \ref{Pi Closure?} and the pasting law we have that $y^* X$ is equivalently the $\infty$-pullback in

$\array{ y^* X &\to& X &\to& \mathbf{\Pi} X \\ \downarrow &&\downarrow && \downarrow \\ * &\stackrel{y}{\to}& Y &\stackrel{}{\to}& \mathbf{\Pi}Y } \,.$

Since the terminal object is discrete, and since the 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

Definition

A morphism $f \colon X \to Y$ in $\mathbf{H}_{th}$ is called $\mathbf{\Pi}_{inf}$-closed 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 (∞,1)-pullback.

Remark

A morphism $f$ in $\mathbf{H}$ satisfies

$\array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }$

is a pullback, iff $i_! f$ is $\mathbf{\Pi}_inf$-closed.

Such an $f$ is also called formally étale morphism.

Proof

This is established by the fact that $\mathbf{\Pi}_{inf} = i_* i^*$ by definition and that $i_!$ is fully faithful, so that

$\array{ i_! X &\to& \mathbf{\Pi}_{inf}(i_! X) \simeq i_* i^* i_! X &\stackrel{\simeq}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* i^* i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! Y &\to& \mathbf{\Pi}_{inf}(i_! Y) \simeq i_* i^* i_! Y &\stackrel{\simeq}{\to}& i_* Y } \,.$

{}

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{Infinitesimal Paths And Reduction?}, is an effective epimorphism.

Remark

An object $X \in \mathbf{H}_{th}$ is formally smooth according to def. \ref{Formal Smoothness?} precisely if the canonical morphism

$i_! X \to i_* X$

(induced from the adjoint quadruple $(i_! \dashv i^* \dashv i_* \dashv i^!)$, see there) is an effective epimorphism.

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 full and faithful (∞,1)-functor the second morphism here in an 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.

Definition

For $f : X \to Y$ a morphism in $\mathbf{H}$, we say that

1. $f$ is a formally smooth morphism if the canonical morphism

$i_! X \to i_! Y \prod_{i_* Y} i_* Y$

is an effective epimorphism.

2. $f$ is a formally unramified morphism if this is a (-1)-truncated morphism. More generally, $f$ is an order-$k$ formally unramified morphisms for $(-2) \leq k \leq \infty$ if this is a k-truncated morphism.

3. $f$ is a formally étale morphism if this morphism is an equivalence, hence if

$\array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }$

is an (∞,1)-pullback square.

Remark

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 étaleness.

Even more generally we can formulate formal smoothness in $\mathbf{H}_{th}$:

(—)

(—)

Remark

A morphism $f$ in $\mathbf{H}$ satisfies

$\array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }$

is a pullback, iff $i_! f$ is $\mathbf{\Pi}_inf$-closed.

Such an $f$ is also called formally étale morphism.

Proof

This is again given by the fact that $\mathbf{\Pi}_{inf} = i_* i^*$ by definition and that $i_!$ is fully faithful, so that

$\array{ i_! X &\to& \mathbf{\Pi}_{inf}(i_! X) \simeq i_* i^* i_! X &\stackrel{\simeq}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* i^* i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! Y &\to& \mathbf{\Pi}_{inf}(i_! Y) \simeq i_* i^* i_! Y &\stackrel{\simeq}{\to}& i_* Y } \,.$
Proposition

The collection of formally étale morphisms in $\mathbf{H}$, def. \ref{Formal Relative Smoothness By Canonical Morphism?}, is closed under the following operations.

1. Every equivalence is formally étale.

2. The composite of two formally étale morphisms is itself formally étale.

3. If

$\array{ && Y \\ & {}^{\mathllap{f}}\nearrow &\swArrow_{\simeq}& \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z }$

is a diagram such that $g$ and $h$ are formally étale, then also $f$ is formally étale.

4. Any retract of a formally étale morphisms is itself formally étale.

5. The (∞,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). Notice that the extra assumption that $i_!$ preserves the pullback is implicit in their setup, by remark \ref{Relation To RK?}.

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)-pullbacks: let $f : X \to Y$ and $g : Y \to Z$ be two morphisms and consider the 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 retract in the 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 (∞,1)-pullback square, then so is its retract sqare. This implies the fourth claim.

To see this, we use that

1. (∞,1)-limits are computed by homotopy limits in any presentable (∞,1)-category $C$ presenting $\mathbf{H}$;

2. homotopy limits in $C$ may be computed by the left and right adjoints provided by the derivator $Ho(C)$ associated to $C$.

From this the claim follows as described in detail at retract in the section retracts of diagrams .

For the last claim, consider an (∞,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 natural transformation $\phi : i_! \to i_*$ to this yields a square of squares. Two sides of this are the 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 pasting law;

• the fourth square is a pullback since $i_*$ is 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 pasting law. Hence $p$ is formally étale.

Remark

The properties listed in prop. \ref{Properties Of Formally Etale Morphisms?} correspond to the axioms on the open maps (“admissible maps”) in a geometry (for structured (∞,1)-toposes) (Lurie, def. 1.2.1). This means that a notion of formally étale morphisms induces a notion of locally algebra-ed (∞,1)toposes/structured (∞,1)-toposes in a cohesive context. This is discuss in

In order to interpret the notion of formal smoothness, we turn now to the discussion of infinitesimal reduction.

Proposition

The operation $\mathbf{Red}$ is an 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

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{aligned} \mathbf{Red} \mathbf{Red} & \simeq i_! i^* i_! i^* \\ & \simeq i_! i^* \\ & \simeq \mathbf{Red} \end{aligned} \,.
Observation

For every $X \in \mathbf{H}_{th}$, we have that $\mathbf{\Pi}_{inf}(X)$ is formally smooth according to def. \ref{Formal Smoothness?}.

Proof

By prop. \ref{Red Is Idempotent?} we have that

$\mathbf{\Pi}_{inf}(X) \to \mathbf{\Pi}_{inf} \mathbf{\Pi}_{inf}X$

is an equivalence. As such it is in particular an effective epimorphism.

Cotangent bundles

Definition

For $X \in \mathbf{H}_{th}$ any object, write

$(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$

for the full sub-(∞,1)-category of the slice (∞,1)-topos over $X$ on those maps into $X$ which are formally étale, def. \ref{Formally Etale In HTh?}.

Proposition

The inclusion of def. \ref{Etale Slice?} is both reflective as well as coreflective, hence it fits into an 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

By the general discussion at 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 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 $I \to (\mathbf{H}_{th})_{/X}^{et}$ 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 $(\mathbf{H}_{th})_{/X}^{et}$ 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).

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 (∞,1)-pullback square. But since $\mathbf{\Pi}_{inf}$ is a 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 (∞,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 $\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally étale morphisms.

Using this étalification, we can now turn de Rham coefficient objects into genuine cotangent bundle.

Definition

Let $G \in Grp(\mathbf{H}_{th})$ be an ∞-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 projection $X \times \flat_{dR}\mathbf{B}G \to X$ as an object in the 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 étalifiation, the coreflection by prop. \ref{Etalification Is Coreflection?}. The sections of this object we call the flat sections of the $G$-valued cotangent bundle of $X$.

Remark

For $U \in \mathbf{H}_{th}$ a test object (say an object in a (∞,1)-site of definition, under the Yoneda embedding) a formall étale morphsim $U \to X$ 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

$\array{ U &&\to&& Et(X \times \flat_{dR} \mathbf{B}G) \\ & \searrow && \swarrow \\ && X } \,.$

By the fact that $Et(-)$ is 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 étale 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 é 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

Let $G$ be an ∞-group and write $\mathbf{d} \colon G \to \flat_{dR}\mathbf{B}G$ for its Maurer-Cartan form.

Definition

For $S \colon X \to G$ a morphism in $\mathbf{H}_{th}$, hence $G$-valued function, its 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 étale, This we may regard as a section of the $G$-valued cotangent bundle, def. \ref{Cotangent Bundle?},

$\array{ X &&\stackrel{(id, \mathbf{d}S)}{\to}&& Et(X \times \flat_{dR}\mathbf{B}G) \\ & {}_{\mathllap{id}}\searrow && \swarrow \\ && X } \,.$

The critical locus $\{x \in X | \mathbf{d}S = 0\}$ of $S$ is the 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 derived critical locus? for more discussion of this.