Spahn Pi-factorization system (Rev #4, changes)

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category: cohesion

Contents

For the cohesive modality Π\mathbf{\Pi}

External formulation

We A discussorthogonal factorization systemsΠ\mathbf{\Pi} -factorization in system is a cohesive(∞,1)(\infty,1)reflective factorization system? -topos . that The characterize discussion or given are here characterized is by a the terminological variant thereof.reflective subcategory of dicrete objects, with reflector Π:H→Π∞Grpd↪DiscH\mathbf{\Pi} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{Disc}{\hookrightarrow} \mathbf{H}.

Definition

Let f:X→Yf : X \to Y be a morphism in H\mathbf{H}, write c Πf→Yc_{\mathbf{\Pi}} f \to Y for the (∞,1)-pullback in

c Πf → ΠX ↓ ↓ Πf Y → ΠY, \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 (Π⊣Disc)(\Pi \dashv Disc)-unit.

(1) We say that c Πfc_{\mathbf{\Pi}} f (respectively c Πf→Yc_{\mathbf{\Pi}} f\to Y) is the Π\mathbf{\Pi}-closure of ff.

(2) ff is called Π\mathbf{\Pi}-closed if X≃c ΠfX \simeq c_{\mathbf{\Pi}} f.

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

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

The LetΠf:X→Y \mathbf{\Pi} f : X \to Y -closure of be a morphism infH f \mathbf{H} , is writeΠc Πf→Y \mathbf{\Pi} c_{\mathbf{\Pi}} f \to Y -closed. for the(∞,1)-pullback in

c Πf → ΠX ↓ ↓ Πf Y → ΠY, \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 (Π⊣Disc)(\Pi \dashv Disc)-unit.

(1) We say that c Πfc_{\mathbf{\Pi}} f (respectively c Πf→Yc_{\mathbf{\Pi}} f\to Y) is the Π\mathbf{\Pi}-closure of ff.

(2) ff is called Π\mathbf{\Pi}-closed if X≃c ΠfX \simeq c_{\mathbf{\Pi}} f.

(3) ff is called a Π\mathbf{\Pi}-equivalence if Πf\mathbf{\Pi} f is an equivalence.
Proof Lemma

Apply TheΠ\mathbf{\Pi} -closure to of the defining Pullback square, the result is a pullback square sinceΠfY \mathbf{\Pi}Y f is discrete.Π\mathbf{\Pi} -closed. is idempotent. Hence by the Pullback pasting lemmaΠC Πf≃C Πf\mathbf{\Pi}C_\mathbf{\Pi} f\simeq C_\mathbf{\Pi} f.

Proposition Proof

If Apply H Π \mathbf{H} \mathbf{\Pi} has to an the defining Pullback square, the result is a pullback square since∞-cohesive siteΠY\mathbf{\Pi}Y of is definition, discrete. then every morphismfΠ:X→Y f \mathbf{\Pi} : X \to Y in is idempotent. Hence by the Pullback pasting lemma H ΠC Πf≃C Πf \mathbf{H} \mathbf{\Pi}C_\mathbf{\Pi} f\simeq C_\mathbf{\Pi} f . factors as

X →f Y ↘ ↗ c Πf, \array{ X &&\stackrel{f}{\to}&& Y \\ & \searrow && \nearrow \\ && c_{\mathbf{\Pi}}f } \,,

such that X→c ΠfX \to c_{\mathbf{\Pi}} f is a Π\mathbf{\Pi}-equivalence and CΠ→YC\mathbf{\Pi}\to Y is Π\mathbf{\Pi}-closed.
Proof Proposition

The If factorization is given by the naturality of Π H \mathbf{\Pi} \mathbf{H} and has the an universal property of the(∞,1)(\infty,1)∞-cohesive site -pullback in of def. definition, \ref{ then every morphismPi Closure?f:X→Yf : X \to Y }. inH\mathbf{H} factors as

X → c Π→ff → ΠYX f↘ ↓ ↓ Πf↗ Yc Πf → ΠY . , \array{ X &\to &&\stackrel{f}{\to}&& & Y c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X \\ &{}_{\mathllap{f}}\searrow & \downarrow \searrow && \downarrow^{\mathrlap{\mathbf{\Pi} \nearrow f}} \\ && Y c_{\mathbf{\Pi}}f &\to& \mathbf{\Pi} Y } \,. \,,

Then such by that prop. \ref{Pi Preserves Pullbacks Over Discretes?X→c ΠfX \to c_{\mathbf{\Pi}} f } the is functor aΠ\mathbf{\Pi} -equivalence preserves and the(C Π∞→,Y1) (\infty,1) C_\mathbf{\Pi}\to Y -pullback over is the discrete objectΠY \mathbf{\Pi}Y \mathbf{\Pi} -closed. and sinceΠ(X→ΠX)\mathbf{\Pi}(X \to \mathbf{\Pi}X) is an equivalence, it follows that Π(X→c Πf)\mathbf{\Pi}(X \to c_{\mathbf{\Pi}f}) is an equivalence.

Proposition Proof

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

(X → c Πf → ΠX f↘ ↓ ↓ Πf Y → ΠYΠ − .equivalences,Π−closedmorphisms) (\mathbf{\Pi}-equivalences, \array{ \mathbf{\Pi}-closed morphisms) X &\to & c_{\mathbf{\Pi}} f &\to & \mathbf{\Pi} X \\ &{}_{\mathllap{f}}\searrow & \downarrow && \downarrow^{\mathrlap{\mathbf{\Pi} f}} \\ && Y &\to& \mathbf{\Pi} Y } \,.

is Then an by prop. \ref{orthogonal factorization systemPi Preserves Pullbacks Over Discretes? } in the functor H Π \mathbf{H} \mathbf{\Pi} . preserves the(∞,1)(\infty,1)-pullback over the discrete object ΠY\mathbf{\Pi}Y and since Π(X→ΠX)\mathbf{\Pi}(X \to \mathbf{\Pi}X) is an equivalence, it follows that Π(X→c Πf)\mathbf{\Pi}(X \to c_{\mathbf{\Pi}f}) is an equivalence.

Proof Proposition

This The follows pair by of the classes general reasoning discussed atreflective factorization system:

By prop. \ref{Factorization Pi Equivalence Pi Closed?} we have the required factorization. It remains to check the orthogonality.

(Π−equivalences,Π−closedmorphisms) (\mathbf{\Pi}-equivalences, \mathbf{\Pi}-closed morphisms)

So is let anorthogonal factorization system in H\mathbf{H}.

A → X ↓ ↓ B → Y \array{ A &\to& X \\ \downarrow && \downarrow \\ B &\to& Y }

be a square diagram in H\mathbf{H} where the left morphism is a Π\mathbf{\Pi}-equivalence and the right morphism is Π\mathbf{\Pi}-closed. Then by assumption there is a pullback square on the right in
A → X → ΠX ↓ ↓ ↓ B → Y → ΠY.. \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
A → ΠA → ΠY ↓ ↓ ≃ ↓ B → ΠB → ΠX. \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 athbfΠB\athbf{\Pi} B and hence also this lift is essentially unique.

Finally by universality of the pullback, this induces an essentially unique lift σ\sigma in
A → X → ΠX ↓ σ↗ ↓ ↓ B → Y → ΠY.. \array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,.
Observation Proof

For This follows by the general reasoning discussed atf:X→Yf : X \to Yreflective factorization system : aΠ\mathbf{\Pi}-closed morphism and y:*→Yy : * \to Y a global element, the homotopy fiber X y:=y *XX_y := y^* X is a discrete object.

By prop. \ref{Factorization Pi Equivalence Pi Closed?} we have the required factorization. It remains to check the orthogonality.

So let
A → X ↓ ↓ B → Y \array{ A &\to& X \\ \downarrow && \downarrow \\ B &\to& Y }

be a square diagram in H\mathbf{H} where the left morphism is a Π\mathbf{\Pi}-equivalence and the right morphism is Π\mathbf{\Pi}-closed. Then by assumption there is a pullback square on the right in
A → X → ΠX ↓ ↓ ↓ B → Y → ΠY.. \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
A → ΠA → ΠY ↓ ↓ ≃ ↓ B → ΠB → ΠX. \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 ΠB\mathbf{\Pi} B and hence also this lift is essentially unique.

Finally by universality of the pullback, this induces an essentially unique lift σ\sigma in
A → X → ΠX ↓ σ↗ ↓ ↓ B → Y → ΠY.. \array{ A &\to& X &\to& \mathbf{\Pi}X \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow \\ B &\to& Y &\to& \mathbf{\Pi}Y \,. } \,.
Proof Observation

By For the def. \ref{Pi Closure?f:X→Yf : X \to Y } and a thepasting lawΠ\mathbf{\Pi} -closed we morphism have and thaty *yX:*→Y y^* y X : * \to Y is a equivalently the∞\inftyglobal element -pullback , in thehomotopy fiber X y:=y *XX_y := y^* X is a discrete object.

y *X → → ΠX ↓ ↓ * →y Y → ΠY. \array{ y^* X &\to& &\to& \mathbf{\Pi} X \\ \downarrow && && \downarrow \\ * &\stackrel{y}{\to}& Y &\stackrel{}{\to}& \mathbf{\Pi}Y } \,.

Since the terminal object is discrete, and since the right adjoint DiscDisc preserves ∞\infty-pullbacks, this exhibits y *Xy^* X as the image under DiscDisc of an ∞\infty-pullback of ∞\infty-groupoids.
Proof

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

y *X → X → ΠX ↓ ↓ ↓ * →y Y → ΠY. \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 DiscDisc preserves ∞\infty-pullbacks, this exhibits y *Xy^* X as the image under DiscDisc 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)(E,M) is a reflective factorization system in HH.

  • There is a reflective subcategory C↪HC\hookrightarrow H with reflector ♯\sharp, EE is the class of morphisms whose ♯\sharp-image is invertible in CC, and C=M/1C=M/1.

  • (E,M)(E,M) is a factorization system and EE satisfies 2-out -of-3.

  • (E,M)(E,M) is a factorization system and MM is the class of fibrant morphisms P→AP\to A which as dependent types x:A⊣P(x):Typex:A\dashv P(x): Type satisfy forallxinRsc(P(x))forall\, x \,in Rsc(P(x)).

  • For every HH-morphism f:A→Bf:A\to B satisfying: ♯A\sharp A and ♯B\sharp B are contractible, also for all bb we have ♯hFiber(f,b)\sharp \, hFiber(f,b) is contractible.

isContr(♯A),isContr(♯B),f:A→B,b:B⊢isContr(♯hFiber(f,b))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)(E,M) is a factorization system in HH.

  • The class (E,M) ×:={M/x|x∈H,M/x↪H/xis.refl,refl.fact}(E,M)^\times:=\{M/x|x\in H,\M/x\hookrightarrow H/x\,is.refl,\,refl.fact\} is pullback-stable where refl.factrefl.fact means that each reflection is defined by (E,M)(E,M)-factorization.

  • (C.x⊆H/x) x∈H(C.x\subseteq H/x)_{x\in H} is a pullback-stable system of reflective subcategories of slices of HH, and for every xx the class of objects of C.xC.x is closed under composition.

  • The class of types BB satisfying inRsc(B)in Rsc (B) is closed under dependent sums.

inRsc(A),forallx,inRsc(P(x))⊢inRsc(∑ x:AP(x))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⊆H/x) x∈H(C.x\subseteq H/x)_{x\in H} is a pullback-stable system of reflective subcategories of slices of HH, for every xx the class of objects of C.xC.x is closed under composition, and all reflectors commute with pullbacks.

  • The (by (2)) to (C.x⊆H/x) x∈H(C.x\subseteq H/x)_{x\in H} corresponding factorization system (E,M)(E,M) is pullback stable.

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

External formulation

Definition

We say an object X∈H thX \in \mathbf{H}_{th} is formally smooth if the constant infinitesimal path inclusion, X→Π inf(X)X \to \mathbf{\Pi}_{inf}(X), def. \ref{Infinitesimal Paths And Reduction?}, is an effective epimorphism.

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

Remark

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

i !X→i *X i_! X \to i_* X

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

Proof

The canonical morphism is the composite

(i !→i *):=i !→ηi !Π infi !:=i *i *i !→≃i *. (i_! \to i_*) := i_! \stackrel{\eta i_!}{\to} \mathbf{\Pi}_{inf} i_! := i_* i^* i_! \stackrel{\simeq}{\to} i_* \,.

By the condition that i !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 XX being an effective epimorphism is equivalent to the component i !X→Πi !Xi_! X \to \mathbf{\Pi} i_! X being an effective epimorphism.

Remark Definition

In For this form this characterization of formal smoothness is the evident generalization of the condition given in (Kontsevich-Rosenberg, section 4.1f:X→Yf : X \to Y ). See a the morphism section inFormal smoothnessH\mathbf{H} , at we say thatQ-category for more discussion. Notice that by this remark the notation there is related to the one used here by u *=i !u^* = i_!, u *=i *u_* = i^* and u !=i *u^! = i_*.

  1. ff is a formally smooth morphism if the canonical morphism

    i !X→i !Y∏ i *Yi *Y i_! X \to i_! Y \prod_{i_* Y} i_* Y

    is an effective epimorphism.

  2. ff is a formally unramified morphism if this is a (-1)-truncated morphism. More generally, ff is an order-kk formally unramified morphisms for (−2)≤k≤∞(-2) \leq k \leq \infty if this is a k-truncated morphism.

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

    i !X →i !f i !Y ↓ ↓ i *X →i *f i *Y \array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }

    is an (∞,1)-pullback square.

Therefore we have the following more general definition.

Remark

An order-(-2) formally unramified morphism is equivalently a formally étale morphism.

Only for 0-truncated XX does formal smoothness together with formal unramifiedness imply formal étaleness.
Definition

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

  1. ff is a formally smooth morphism if the canonical morphism

    i !X→i !Y∏ i *Yi *Y i_! X \to i_! Y \prod_{i_* Y} i_* Y

    is an effective epimorphism.

  2. ff is a formally unramified morphism if this is a (-1)-truncated morphism. More generally, ff is an order-kk formally unramified morphisms for (−2)≤k≤∞(-2) \leq k \leq \infty if this is a k-truncated morphism.

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

    i !X →i !f i !Y ↓ ↓ i *X →i *f i *Y \array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }

    is an (∞,1)-pullback square.

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

An A order-(-2) formally unramified morphism is equivalently aformally étale morphismf:X→Yf \colon X \to Y . inH th\mathbf{H}_{th} is formall étale if it is Π inf\mathbf{\Pi}_{inf}-closed, hence if its Π inf\mathbf{\Pi}_{inf}-unit naturality square

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

X → Π inf(X) ↓ f ↓ Π inf(f) Y → Π inf(y) \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.

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

Remark

A morphism ff in H\mathbf{H} is formally etale in the sense of def. \ref{Formal Relative Smoothness By Canonical Morphism?} precisely if its image i !(f)i_!(f) in H th\mathbf{H}_{th} is formally etale in the sense of def. \ref{Formally Etale In HTh?}.
Definition Proof

A This morphism is again given by the fact thatfΠ inf : =Xi *→i *Y f \mathbf{\Pi}_{inf} \colon = X i_* \to i^* Y in by definition and thatHi th! \mathbf{H}_{th} i_! is fully faithful, so thatformall étale if it is Π inf\mathbf{\Pi}_{inf}-closed, hence if its Π inf\mathbf{\Pi}_{inf}-unit naturality square

i !X → Π inf(i !X)≃i *i *i !X →≃ i *X ↓ fi !f ↓ Πi inf*(i *i !f) ↓ i *f i !Y → Π inf(yi !Y)≃i *i *i !Y →≃ i *Y. \array{ i_! X &\to& \mathbf{\Pi}_{inf}(X) \mathbf{\Pi}_{inf}(i_! X) \simeq i_* i^* i_! X &\stackrel{\simeq}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! f}} \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \downarrow^{\mathrlap{i_* i^* i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! Y &\to& \mathbf{\Pi}_{inf}(y) \mathbf{\Pi}_{inf}(i_! Y) \simeq i_* i^* i_! Y &\stackrel{\simeq}{\to}& i_* Y } \,.

is an (∞,1)-pullback.
Remark Proposition

A The morphism collection offfformally étale morphisms in H\mathbf{H} , is formally etale in the sense of def. \ref{Formal Relative Smoothness By Canonical Morphism? } }, precisely is if closed its under image the following operations.i !(f)i_!(f) in H th\mathbf{H}_{th} is formally etale in the sense of def. \ref{Formally Etale In HTh?}.

  1. Every equivalence is formally étale.

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

  3. If

    Y f↗ ⇙ ≃ ↘ g X →h Z \array{ && Y \\ & {}^{\mathllap{f}}\nearrow &\swArrow_{\simeq}& \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z }

    is a diagram such that gg and hh are formally étale, then also ff 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 !i_!.
Proof

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

i !X → Π inf(i !X)≃i *i *i !X →≃ i *X ↓ i !f ↓ i *i *i !f ↓ i *f i !Y → Π inf(i !Y)≃i *i *i !Y →≃ i *Y. \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 } \,.

The statements about closure under composition and pullback appears as(KontsevichRosenberg, prop. 5.4, prop. 5.6). Notice that the extra assumption that i !i_! preserves the pullback is implicit in their setup, by remark \ref{Relation To RK?}.
Proposition Proof

The collection first of statement follows sinceformally étale morphisms∞\infty -pullbacks in are well defined up to quivalence.H\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

    Y f↗ ⇙ ≃ ↘ g X →h Z \array{ && Y \\ & {}^{\mathllap{f}}\nearrow &\swArrow_{\simeq}& \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z }

    is a diagram such that gg and hh are formally étale, then also ff 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 !i_!.

The second two statements follow by the pasting law for (∞,1)-pullbacks: let f:X→Yf : X \to Y and g:Y→Zg : Y \to Z be two morphisms and consider the pasting diagram
i !X →i !f i !Y →i !g Z ↓ ↓ ↓ i *X →i *f i *Y →i *g i *Z. \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 ff and gg are formally étale then both small squares are pullback squares. Then the pasting law says that so is the outer rectangle and hence g∘fg \circ f is formally étale. Similarly, if gg and g∘fg \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 ff is formally étale.

For the fourth claim, let Id≃(g→f→g)Id \simeq (g \to f \to g) be a retract in the arrow (∞,1)-category H I\mathbf{H}^I. By applying the natural transformation ϕ:i !→I *\phi : i_! \to I_* we obtain a retract
Id≃((i !g→i *g)→(i !f→i *f)→(i !g→i *g)) Id \simeq ((i_! g \to i_*g) \to (i_! f \to i_*f) \to (i_! g \to i_*g))

in the category of squares H □\mathbf{H}^{\Box}. We claim that generally, if the middle piece in a retract in H □\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 CC presenting H\mathbf{H};

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

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
A× YX → X p↓ ↓ f A → Y \array{ A \times_Y X &\to& X \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{f}} \\ A &\to& Y }

where ff is formally étale.

Applying the natural transformation ϕ:i !→i *\phi : i_! \to i_* to this yields a square of squares. Two sides of this are the pasting composite
i !A× YX → i !X →ϕ X i *X ↓ i !p ↓ i !f ↓ i *f i !A → i !Y →ϕ Y i *Y \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
i !A× YX →ϕ A× YX i *A× YA → i *X ↓ i !p ↓ i *p ↓ i *f i !A →ϕ A i *A → i *Y. \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 !i_! preserves the given pullback;

  • the second square is a pullback, since ff is formally étale.

  • the total top rectangle is therefore a pullback, by the pasting law;

  • the fourth square is a pullback since i *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 pp is formally étale.

The statements about closure under composition and pullback appears as(KontsevichRosenberg, prop. 5.4, prop. 5.6). Notice that the extra assumption that i !i_! preserves the pullback is implicit in their setup, by remark \ref{Relation To RK?}.

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
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→Yf : X \to Y and g:Y→Zg : Y \to Z be two morphisms and consider the pasting diagram

i !X →i !f i !Y →i !g Z ↓ ↓ ↓ i *X →i *f i *Y →i *g i *Z. \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 ff and gg are formally étale then both small squares are pullback squares. Then the pasting law says that so is the outer rectangle and hence g∘fg \circ f is formally étale. Similarly, if gg and g∘fg \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 ff is formally étale.

For the fourth claim, let Id≃(g→f→g)Id \simeq (g \to f \to g) be a retract in the arrow (∞,1)-category H I\mathbf{H}^I. By applying the natural transformation ϕ:i !→I *\phi : i_! \to I_* we obtain a retract

Id≃((i !g→i *g)→(i !f→i *f)→(i !g→i *g)) Id \simeq ((i_! g \to i_*g) \to (i_! f \to i_*f) \to (i_! g \to i_*g))

in the category of squares H □\mathbf{H}^{\Box}. We claim that generally, if the middle piece in a retract in H □\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 CC presenting H\mathbf{H};

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

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

A× YX → X p↓ ↓ f A → Y \array{ A \times_Y X &\to& X \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{f}} \\ A &\to& Y }

where ff is formally étale.

Applying the natural transformation ϕ:i !→i *\phi : i_! \to i_* to this yields a square of squares. Two sides of this are the pasting composite

i !A× YX → i !X →ϕ X i *X ↓ i !p ↓ i !f ↓ i *f i !A → i !Y →ϕ Y i *Y \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

i !A× YX →ϕ A× YX i *A× YA → i *X ↓ i !p ↓ i *p ↓ i *f i !A →ϕ A i *A → i *Y. \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 !i_! preserves the given pullback;

  • the second square is a pullback, since ff is formally étale.

  • the total top rectangle is therefore a pullback, by the pasting law;

  • the fourth square is a pullback since i *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 pp 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 Red\mathbf{Red} is an idempotent projection of H th\mathbf{H}_{th} onto the image of H\mathbf{H}

RedRed≃Red. \mathbf{Red} \mathbf{Red} \simeq \mathbf{Red} \,.

Accordingly also

Π infΠ inf≃Π inf \mathbf{\Pi}_{inf} \mathbf{\Pi}_{inf} \simeq \mathbf{\Pi}_{inf}

and

♭ inf♭ inf≃♭ inf. \mathbf{\flat}_{inf} \mathbf{\flat}_{inf} \simeq \mathbf{\flat}_{inf} \,.
Proof

By definition of infinitesimal neighbourhood we have that i !i_! is a full and faithful (∞,1)-functor. It follows that i *i !≃Idi^* i_! \simeq Id and hence

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

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

Proof

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

Π inf(X)→Π infΠ infX \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∈H thX \in \mathbf{H}_{th} any object, write

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

for the full sub-(∞,1)-category of the slice (∞,1)-topos over XX on those maps into XX 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

(H th) /X th←Et↪←L(H th) /X fet. (\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:Y→Xf \colon Y \to X to X× Π inf(X)Π inf(Y)→YX \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y and the reflection unit is the left horizontal morphism in

Y → X× Π inf(Y)Π inf(Y) → Π inf(Y) ↘ ↓ ↓ Π inf(f) X → Π inf(X). \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 (H th) /X et(\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→(H th) /X etI \to (\mathbf{H}_{th})_{/X}^{et} out of a small (∞,1)-category. Since the inclusion of (H th) /X fet(\mathbf{H}_{th})_{/X}^{fet} is full, it is sufficient to show that the (∞,1)(\infty,1)-colimit over this diagram taken in (H th) /X(\mathbf{H}_{th})_{/X} lands again in (H th) /X et(\mathbf{H}_{th})_{/X}^{et} in order to have that (∞,1)(\infty,1)-colimits are preserved by the inclusion. Moreover, colimits in a slice of H th\mathbf{H}_{th} are computed in H th\mathbf{H}_{th} itself (this is discussed at slice category - Colimits).

Therefore we are reduced to showing that the square

lim→ iY i → Π inflim→ iY i ↓ ↓ X → Π inf(X) \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 Π inf\mathbf{\Pi}_{inf} is a left adjoint it commutes with the (∞,1)(\infty,1)-colimit on objects and hence this diagram is equivalent to

lim→ iY i → lim→ iΠ infY i ↓ ↓ X → Π inf(X). \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 H th\mathbf{H}_{th}, hence that on the left the component Y iY_i for each i∈Ii \in I is the (∞,1)-pullback of Π inf(Y i)→Π inf(X)\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X), by assumption that we are taking an (∞,1)(\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∈Grp(H th)G \in Grp(\mathbf{H}_{th}) be an ∞-group and let write ♭ dRBG∈H th\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th} for the corresponding de Rham coefficient object.

For X∈H thX \in \mathbf{H}_{th} any object, consider the projection X×♭ dRBG→XX \times \flat_{dR}\mathbf{B}G \to X as an object in the slice (∞,1)-topos (H th) /X(\mathbf{H}_{th})_{/X}. Then write

Et(X×♭ dRBG)∈(H th) /X fet 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 GG-valued cotangent bundle of XX.

Remark

For U∈H thU \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→XU \to X is like an open map/open embedding. Regarded as an object in (H th) /X fet(\mathbf{H}_{th})_{/X}^{fet} we may consider the sections over UU of the cotangent bundle as defined above, which in H th\mathbf{H}_{th} are diagrams

U → Et(X×♭ dRBG) ↘ ↙ X. \array{ U &&\to&& Et(X \times \flat_{dR} \mathbf{B}G) \\ & \searrow && \swarrow \\ && X } \,.

By the fact that Et(−)Et(-) is right adjoint, such diagrams are in bijection to diagrams

U → X×♭ dRBG ↘ ↙ X \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→X)(U \to X) along (H th) /X fet↪(H th) /X(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}.

In other words, the (flat) sections of the GG-valued cotangent bundle Et(X×♭ dRBG)→XEt(X \times \flat_{dR}\mathbf{B}G) \to X are just the sections of X×♭ dRBG→XX \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 XX.

But then by the very nature of ♭ dRBG\flat_{dR}\mathbf{B}G it follows that the flat sections of the GG-valued cotangent bundle of XX are indeed nothing but the flat GG-valued differential forms on XX.

Critical locus

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

Definition

For S:X→GS \colon X \to G a morphism in H th\mathbf{H}_{th}, hence GG-valued function, its derivative is the composite

dS:X→SG→d♭ dRBG. \mathbf{d}S \colon X \stackrel{S}{\to} G \stackrel{\mathbf{d}}{\to} \flat_{dR}\mathbf{B}G \,.

Since the identity on XX is formally étale, This we may regard as a section of the GG-valued cotangent bundle, def. \ref{Cotangent Bundle?},

X →(id,dS) Et(X×♭ dRBG) id↘ ↙ X. \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∈X|dS=0}\{x \in X | \mathbf{d}S = 0\} of SS is the homotopy fiber of this section in (H th) /X fet(\mathbf{H}_{th})_{/X}^{fet}, hence the (∞,1)(\infty,1)-pullback

{x∈X|dS=0} → X ↓ ↓ 0 X →dS Et(X×♭ dRBG). \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 03:18:21 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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