nLab formally étale morphism

A morphisms of spaces $X \overset{p}{\longrightarrow} Y$ is called formally étale if it has a lifting property as étalé spaces do locally, but for infinitesmal extensions: If for $Z \overset{f}{\to} Y$ any morphism and $\Re(Z) \to X$ a lift of its restriction along its reduction $\Re(Z) \to Z$, there is a unique extension to a complete lift.

(If there exists at least one such infinitesimal extension, it is called a formally smooth morphism. If there exists at most one such extension, it is called a formally unramified morphism. The formally étale morphisms are precisely those that are both formally smooth and formally unramified.)

Traditionally this has been considered in the context of geometry over formal duals of rings and associative algebras. This we discuss in the section (Concrete notion). But generally the notion makes sense in any context of differential cohesion. This we discuss in the section General abstract notion.

General abstract notion

Definition

Let

$$\mathbf{H} \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}} \mathbf{H}_{th}$$

be an adjoint triple of functor with $u^*$ a full and faithful functor that preserves the terminal object.

We may think of this as exhibiting differential cohesion (see there for details, but notice that in the notation used there we have $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$).

We think of the objects of $\mathbf{H}$ as cohesive spaces and of the objects of $\mathbf{H}_{th}$ as such cohesive spaces possibly equipped with infinitesimal extension.

As a class of examples that is useful to keep in mind consider a Q-category

$$(cod \dashv \epsilon \dashv dom) : \bar A \to A$$

of infinitesimal thickening of rings and let

$$((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) := ([dom,Set] \dashv [\epsilon, Set] \dashv [cod,Set] : [\bar A, Set] \to [A,Set])$$

be the corresponding Q-category of copresheaves.

For any such setup there is a canonical natural transformation

$$\phi : u^* \to u^! \,.$$

Details of this are in the section Adjoint quadruples at cohesive topos.

From this we get for every morphism $f : X \to Y$ in $\mathbf{H}$ a canonical morphism

(1)$$u^* X \to u^* Y \prod_{u^! Y} u^! X \,.$$

Definition

A morphism $f : X \to Y$ in $\mathbf{H}$ is called formally étale if (1) is an isomorphism.

This appears as (KontsevichRosenberg, def. 5.1, prop. 5.3.1.1).

In other words, $f$ is formally étale if the $f$-component naturality square

$$\array{ u^* X &\stackrel{u^* f}{\to}& u^* Y \\ {}^{\mathllap{\phi_X}}\downarrow && \downarrow^{\mathrlap{\phi_Y}} \\ u^! X &\stackrel{u^! f}{\to}& u^! Y }$$

of the natural transformation $\phi$ is a pullback diagram.

Remark

The partial notions of this condition are: if the above morphism is a monomorphism then $f$ is a formally unramified morphism, if it is an epimorphism then $f$ is a formally smooth morphism.

Definition

An object $X \in \mathbf{H}$ is called formally étale if the morphism $X \to *$ to the terminal object is formally étale.

Proposition

The object $X$ is formally étale precisely if

$$u^* X \to u^! X$$

is an isomorphism.

This appears as (KontsevichRosenberg, def. 5.3.2).

Properties

Proposition

Formally étale morphisms are closed under composition.

This appears as (KontsevichRosenberg, prop. 5.4).

Proof

This follows by the pasting law for pullbacks: let $f : X \to Y$ and $g : Y \to Z$ be two formally étale morphisms. Then by definition both of the small squares in

$$\array{ u^* X &\stackrel{u^* f }{\to}& u^* Y &\stackrel{u^* g}{\to}& u^* Z \\ \downarrow && \downarrow && \downarrow \\ u^! X &\stackrel{u^! f }{\to}& u^! Y &\stackrel{u^! g}{\to}& u^! Z }$$

are pullback squares. Hence so is the total outer square.

Using also the other case of the pasting law, the above proof shows more:

Proposition

If

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

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

Proposition

Formally étale morphisms are closed under retracts.

This means that if $f : X \to Y$ is formally étale and

$$\array{ A &\to & X &\to& A \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{p}} \\ B &\to& Y &\to& B }$$

is a commuting diagram such that the two horizontal composites are identities, then also $p$ is formally étale.

Proof

By applying the natural transformation $\phi : u^* \to u^!$ to this diagram we obtain a retract diagram in the category of squares, given by the naturality squares of $\phi$ on $f$ and $p$, where the middle square is a pullback square. By this proposition at retract this implies that also the retracting square is a pullback, which means that $p$ is formally étale.

Proposition

If $u^*$ preserves pullbacks, then formally étale morphisms are stable under pullback.

Proof

Consider a 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 $\phi : u^* \to u^!$ to this yields a square of squares. Two sides of this are the pasting composite

$$\array{ u^* A \times_Y X &\to& u^* X &\stackrel{\phi_X}{\to}& u^! X \\ \downarrow && \downarrow^{\mathrlap{u^* f}} && \downarrow^{\mathrlap{u^! f}} \\ u^* A &\to& u^* Y &\stackrel{\phi_Y}{\to}& u^! Y }$$

and the other two sides are the pasting composite

$$\array{ u^* A \times_Y X &\stackrel{\phi_{A \times_Y X}}{\to}& u^! A \times_Y A &\stackrel{}{\to}& u^! X \\ \downarrow^{} && \downarrow && \downarrow^{\mathrlap{u^! f}} \\ u^* A &\stackrel{\phi_A}{\to}& u^! A &\to& u^! Y } \,.$$

Counting left to right and top to bottom, we have that

• the first square is a pullback by assumption on $u^*$;

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

• the fourth square is a pullback since $u^!$ is right adjoint and so also preserves pullbacks;

• also the total bottom rectangle is a pullback, since it is equal to the bottom total rectangle;

• therefore finally the third square is a pullback, by the pasting law, hence also $p$ is formally étale.

Concrete notions

We discuss realizations of the above general abstract definition in concrete models of the axioms.

See also the concrete notions of formally smooth morphism and formally unramified morphism.

In differential geometry

The category SmoothMfd of smooth manifolds may naturally be thought of as sitting inside the more general context of the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids. This is canonically equipped with a notion of infinitesimal cohesion exhibited by its inclusion into SynthDiff∞Grpd. This implies that there is an intrinsic notion of formally étale morphisms of smooth $\infty$-groupoids in general and of smooth manifolds in particular

Proposition

A smooth function is a formally étale morphism in this sense precisely if it is a local diffeomorphism in the traditional sense.

See this section for more details.

In algebraic geometry

• formally étale morphism of schemes

In noncommutative geometry

See (RosenbergKontsevich, section 5.8)

Related concepts

formally smooth morphism and formally unramified morphism $\Rightarrow$ formally étale morphism

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

References

The idea of defining étale morphisms $f$ as those that get send to a pullback square by a natural transformation goes back to lectures by André Joyal in the 1970s.

See the introduction and see section 4 of

• Eduardo Dubuc, Axiomatic etal maps and a theory of spectrum, Journal of pure and applied algebra, 149 (2000)

The identification of the natural transformation in question with that induced by an adjoint triple (“Q-categories”) and the relation to formal étaleness is observed (apparently independently?) in

• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (pdf, ps, dvi)

Formalization and discussion in the context of cohesive (∞,1)-toposes is in section 2.5.3 (and defn 5.3.19) of

• Urs Schreiber, differential cohomology in a cohesive topos