Spahn Pi-factorization system (Rev #5, changes)

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For the cohesive modality $\mathbf{\Pi}$
- External formulation
- Internal formulation
For the infinitesimal-cohesive modality $\mathbf{\Pi}_inf$
- External formulation
  - Cotangent bundles
  - Critical locus
- Internal formulation

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

We A ~~say~~ morphism an ~~object~~ $f : X \in \to H_{th} Y$ f \colon X ~~\in~~ \to ~~\mathbf{H}_{th}~~ Y is in~~formally smooth~~ $\mathbf{H}_{th}$ if is ~~the~~ called ~~constant~~ ~~infinitesimal~~ ~~path~~ ~~inclusion,~~ $X \to Π_{\inf} (X)$ X \mathbf{\Pi}_{inf} ~~\to~~ ~~\mathbf{\Pi}_{inf}(X)~~ , -closed ~~def.~~ if ~~\ref{~~ its~~Infinitesimal Paths And Reduction?~~ $\mathbf{\Pi}_{inf}$ }, -unit is naturality an square~~effective epimorphism~~.

\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.

~~In this form this is the evident $(\infty,1)$ -categorical analog of the conditions as they appear for instance in (SimpsonTeleman, page 7).~~

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.

Remark Proof

An This ~~object~~ is again given by the fact that $X Π_{\inf} \in = H_{i th *} i^{*}$ X \mathbf{\Pi}_{inf} ~~\in~~ = ~~\mathbf{H}_{th}~~ i_* i^* is by ~~formally~~ definition ~~smooth~~ and ~~according~~ that to ~~def.~~ ~~\ref{Formal Smoothness?~~ $i_!$ } ~~precisely~~ is if fully ~~the~~ faithful, ~~canonical~~ so ~~morphism~~ that

i_{!} \begin{matrix} i_{!} X & \to & Π_{\inf} (i_{!} X) ≃ i_{*} i^{*} i_{!} X & \overset{≃}{\to} & i_{*} X \\ ↓^{i_{!} f} & ↓^{i_{*} i^{*} i_{!} f} & ↓^{i_{*} f} \\ i_{!} Y & \to & Π_{\inf} (i_{!} Y) ≃ i_{*} i^{*} i_{!} Y & \overset{≃}{\to} & i_{*} Y \end{matrix} X \to . i_{*} X

\array{ i_! X ~~\to~~ &\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 } \,.

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

Proof Definition

~~The~~ We ~~canonical~~ say ~~morphism~~ an is object ~~the~~ ~~composite~~ $X \in \mathbf{H}_{th}$ 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.

~~$(i_! \to i_*) := i_! \stackrel{\eta i_!}{\to} \mathbf{\Pi}_{inf} i_! := i_* i^* i_! \stackrel{\simeq}{\to} i_* \,.$~~
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

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

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

is an effective epimorphism.

$f$ 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.

$f$ 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.

In this form this is the evident $(\infty,1)$ -categorical analog of the conditions as they appear for instance in (SimpsonTeleman, page 7).

Remark

An ~~order-(-2)~~ object ~~formally~~ ~~unramified~~ ~~morphism~~ is ~~equivalently~~ a~~formally étale morphism~~ $X \in \mathbf{H}_{th}$ . is formally smooth according to def. \ref{Formal Smoothness?} precisely if the canonical morphism

~~Only for 0-truncated $X$ does formal smoothness together with formal unramifiedness imply formal étaleness.~~

i_! X \to i_* X

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

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

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

A For ~~morphism~~ $f : X \to Y$ f ~~\colon~~ : X \to Y a morphism in $H_{th} H$ ~~\mathbf{H}_{th}~~ \mathbf{H} , is we say that~~formall étale~~ ~~if it is~~ ~~$\mathbf{\Pi}_{inf}$~~ ~~-closed, hence if its~~ ~~$\mathbf{\Pi}_{inf}$~~ ~~-unit naturality square~~

~~$\array{ X &\to& \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(y) }$~~

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

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

is an effective epimorphism.
$f$ 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.
$f$ 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.

~~is an (∞,1)-pullback.~~

Remark

A An order-(-2) formally unramified morphism is equivalently a ~~$f$~~ formally étale morphism . in ~~$\mathbf{H}$~~ ~~is formally etale in the sense of def. \ref{Formal Relative Smoothness By Canonical Morphism?} precisely if its image~~ ~~$i_!(f)$~~ in ~~$\mathbf{H}_{th}$~~ ~~is formally etale in the sense of def. \ref{Formally Etale In HTh?~~}.

Only for 0-truncated $X$ does formal smoothness together with formal unramifiedness imply formal étaleness.

Proof

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

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

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

Proposition

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

Every equivalence is formally étale.

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

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.

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

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 Remark

~~The~~ A ~~first~~ morphism ~~statement~~ ~~follows~~ ~~since~~ $∞ f$ ~~\infty~~ f ~~-pullbacks~~ ~~are~~ in ~~well~~ ~~defined~~ up to ~~quivalence.~~ $\mathbf{H}$ satisfies

~~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 \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }

~~$\array{ i_! X &\stackrel{i_! f }{\to}& i_! Y &\stackrel{i_! g}{\to}& Z \\ \downarrow && \downarrow && \downarrow \\ i_* X &\stackrel{i_* f }{\to}& i_* Y &\stackrel{i_* g}{\to}& i_* Z } \,.$~~

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

If Such an $f$ ~~and~~ is also called ~~$g$~~ formally étale morphism . ~~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)-limits are computed by homotopy limits in any presentable (∞,1)-category $C$ presenting $\mathbf{H}$ ;

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 Proof

~~The~~ This ~~properties~~ is ~~listed~~ again in given ~~prop.~~ by ~~\ref{~~ the fact that~~Properties Of Formally Etale Morphisms?~~ $\mathbf{\Pi}_{inf} = i_* i^*$ } ~~correspond~~ by to definition ~~the~~ and ~~axioms~~ that on ~~theopen maps~~ $i_!$ ~~(“admissible~~ is ~~maps”)~~ fully in faithful, a so that~~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~~

~~cohesive (∞,1)-topos -- structure ∞-sheaves.~~

\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.

Every equivalence is formally étale.
The composite of two formally étale morphisms is itself formally étale.
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.
Any retract of a formally étale morphisms is itself formally étale.
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)-limits are computed by homotopy limits in any presentable (∞,1)-category $C$ presenting $\mathbf{H}$ ;
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

cohesive (∞,1)-topos -- structure ∞-sheaves.

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.

Internal formulation

Revision on December 2, 2012 at 03:50:34 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn Pi-factorization system (Rev #5, changes)

Contents

For the cohesive modality Π\mathbf{\Pi}

External formulation

Definition

Lemma

Proof

Proposition

Proof

Proposition

Proof

Observation

Proof

Internal formulation

For the infinitesimal-cohesive modality Π inf\mathbf{\Pi}_inf

External formulation

Definition

Remark

Remark Proof

Proof Definition

Definition

Remark

Proof

Definition

Remark

Proof

Proposition

Proof Remark

Remark Proof

Proposition

Proof

Remark

Proposition

Proof

Observation

Proof

Cotangent bundles

Definition

Proposition

Proof

Definition

Remark

Critical locus

Definition

Internal formulation

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

For the infinitesimal-cohesive modality $\mathbf{\Pi}_inf$