nLab
formal smooth infinity-groupoid

Context

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A formal smooth \infty-groupoid is an ∞-groupoid equipped with a cohesive structure that subsumes that of smooth ∞-groupoids as well as of infinitesimal \infty-groupoids – ∞-Lie algebroids, hence equipped with “differential cohesion”.

In the cohesive (∞,1)-topos of formal smooth \infty-groupoids the canonical fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos Π(X)\mathbf{\Pi}(X) factors through a version relative to Smooth∞Grpd: the infinitesimal path ∞-functor Π inf(X)\mathbf{\Pi}_{inf}(X). In traditional terms this is the object modeled by the tangent Lie algebroid and the de Rham space of XX. The quasicoherent ∞-stacks on Π inf(X)\mathbf{\Pi}_{inf}(X) are D-modules on XX.

Definition

We consider (∞,1)-sheaves over a “twisted semidirect product” site or (∞,1)-site of

First in

we consider the 1-site, then in

we consider the (,1)(\infty,1)-site.

1-localic definition

Definition

Let T:=T := CartSp smooth{}_{smooth} be the syntactic category of the Lawvere theory of smooth algebras: the category of Cartesian spaces n\mathbb{R}^n and smooth functions between them.

Write

SmoothAlg:=TAlg SmoothAlg := T Alg

for its category of T-algebras – the smooth algebras (C C^\infty-rings).

Let

InfPointSmoothAlg op InfPoint \hookrightarrow SmoothAlg^{op}

be the full subcategory on the infinitesimally thickened points: this smooth algebras whose underlying abelian group is a vector space of the form V\mathbb{R} \oplus V with VV a finite-dimensional real vector space and nilpotent in the algebra structure.

Definition

Let

FormalCartSpSmoothLoc FormalCartSp \hookrightarrow SmoothLoc

be the full subcategory of the category of smooth loci on the objects of the form

U= n×D U = \mathbb{R}^n \times D

that are products of a Cartesian space n\mathbb{R}^n \in CartSp for nn \in \mathbb{N} and an infinitesimally thickened point DInfPointD \in InfPoint. (See at FormalCartSp.)

Equip this category with the coverage where a family of morphisms is covering precisely if it is of the form {U i×D(f i,Id D)U×D}\{U_i \times D \stackrel{(f_i, Id_D)}{\to} U \times D\} for {f i:U iU}\{f_i : U_i \to U\} a covering in CartSp smooth{}_{smooth} (a good open cover).

This appears as ([Kock 86, (5.1)]).

Note

The sheaf topos over FormalCartSp is (equivalent to) the topos known as the Cahiers topos, a smooth topos that constitutes a well adapted model for synthetic differential geometry. See at Cahiers topos for further references.

Definition

We say the (∞,1)-topos of formal smooth \infty-groupoids is the (∞,1)-category of (∞,1)-sheaves

FormalSmoothGrpd:=Sh (,1)(FormalCartSp) FormalSmooth \infty Grpd := Sh_{(\infty,1)}(FormalCartSp)

on FormalCartSpFormalCartSp.

\infty-localic

We now generalize the 1-category InfPointInfPoint of infinitesimally thickened points to the (∞,1)-category InfPoint InfPoint_\infty of “derived infinitesimally thickened points”, the formal dual of “small commutative \infty-algebras” from (Hinich, Lurie).

(…)

Properties

Proposition

FormalSmooth \infty Grpd$ is a cohesive (∞,1)-topos.

Proof

Because FormalCartSp is an ∞-cohesive site. See there for details.

Definition

Write FormalSmoothMfdSmoothAlg opFormalSmoothMfd \hookrightarrow SmoothAlg^{op} for the full subcategory of smooth loci on the formal smooth manifolds: those modeled on FormalCartSp equipped with the evident coverage.

Observation

FormalCartSpFormalCartSp is a dense sub-site of FormalSmoothMfdFormalSmoothMfd.

Proposition

There is an equivalence of (∞,1)-categories

FormalSmoothGrpdSh^ (,1)(FSmoothMfd) FormalSmooth\infty Grpd \simeq \hat Sh_{(\infty,1)}(FSmoothMfd)

with the hypercomplete (∞,1)-topos over FSmoothMfdFSmoothMfd.

Proof

With the above observation this is directly analogous to the corresponding proof at ETop∞Grpd.

Definition

Write i:CartSp smoothFormalCartSpi : CartSp_{smooth} \hookrightarrow FormalCartSp for the canonical embedding.

Proposition

The functor i *i^* given by restriction along ii exhibits FormalSmoothGrpdFormalSmooth\infty Grpd as an infinitesimal cohesive neighbourhood of the (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids in that we have a quadruple of adjoint (∞,1)-functors

(i !i *i *i !):SmoothGrpdi !i *i *i !FormalSmoothGrpd, ( i_! \dashv i^* \dashv i_* \dashv i^! ) : Smooth \infty Grpd \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\to}}{\stackrel{i^!}{\leftarrow}}}} FormalSmooth \infty Grpd \,,

such that i !i_! is a full and faithful (∞,1)-functor.

Proof

Since i:CartSp smoothCartSp formalsmoothi : CartSp_{smooth} \hookrightarrow CartSp_{formalsmooth} is an infinitesimally ∞-cohesive site this follows with a proposition discussed at cohesive (infinity,1)-topos -- infinitesimal cohesion.

Structures

We discuss the realization of the general abstract structures in a cohesive (∞,1)-topos in FormalSmoothGrpdFormalSmooth \infty Grpd.

Since by the above discussion FormalSmoothGrpdFormalSmooth\infty Grpd is strongly \infty-connected relative to Smooth∞Grpd all of these structures that depend only on \infty-connectedness (and not on locality) acquire a relative version.

\infty-Lie algebroids and deformation theory

This subsection is at

Paths and geometric Postnikov towers

We discuss the intrinsic infinitesimal path adjunction realized in FormalSmoothGrpdFormalSmooth\infty Grpd.

(RedΠ inf inf):=(iΠ infDisc infΠ infDisc infΓ inf):FormalSmoothGrpdFormalSmoothGrpd. (\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf}) := (i \circ \Pi_{inf} \dashv Disc_{inf} \Pi_{inf} \dashv Disc_{inf} \circ \Gamma_{inf}) : FormalSmooth \infty Grpd \to FormalSmooth \infty Grpd \,.
Proposition

For U×DCartSp smoothInfinSmoothLoc=FormalCartSpFormalSmoothGrpdU \times D \in CartSp_{smooth} \ltimes InfinSmoothLoc = FormalCartSp \hookrightarrow FormalSmooth\infty Grpd we have that

Red(U×D)U \mathbf{Red}(U \times D) \simeq U

is the reduced smooth locus: the formal dual of the smooth algebra obtained by quotienting out all nilpotent elements in the smooth algebra C (K×D)C (K)C (D)C^\infty(K \times D) \simeq C^\infty(K) \otimes C^\infty(D).

Proof

By the model category presentation of Red𝕃Lan ii *\mathbf{Red} \simeq \mathbb{L} Lan_i \circ \mathbb{R}i^* of the above proof and using that every representable is cofibrant and fibrant in the local projective model structure on simplicial presheaves we have

Red(U×D) (𝕃Lan i)(i *)(U×D) (𝕃Lan i)i *(U×D) (𝕃Lan i)U Lan iU U \begin{aligned} \mathbf{Red}(U \times D) & \simeq (\mathbb{L}Lan_i) (\mathbb{R}i^*) (U \times D) \\ &\simeq (\mathbb{L}Lan_i) i^* (U \times D) \\ & \simeq (\mathbb{L} Lan_i) U \\ & \simeq Lan_i U \\ & \simeq U \end{aligned}

(using that ii is a full and faithful functor).

Proposition

For XSmoothAlg opFormalSmoothGrpdX \in SmoothAlg^{op} \to FormalSmooth \infty Grpd a smooth locus, we have that Π inf(X)\mathbf{\Pi}_{inf}(X) is the corresponding de Rham space, the object in which all infinitesimal neighbour points in XX are equivalent, characterized by

Π inf(X):U×DX(U). \mathbf{\Pi}_{inf}(X) : U \times D \mapsto X(U) \,.
Proof

By the (RedΠ inf)(\mathbf{Red} \dashv \mathbf{\Pi}_{inf})-adjunction relation

Π inf(X)(U×D) =FormalSmoothGrpd(U×D,Π inf(X)) FormalSmoothGrpd(Red(U×D),X) FormalSmoothGrpd(U,X). \begin{aligned} \mathbf{\Pi}_{inf}(X)(U \times D) & = FormalSmooth \infty Grpd(U \times D, \mathbf{\Pi}_{inf}(X)) \\ & \simeq FormalSmooth \infty Grpd( \mathbf{Red}(U \times D), X) \\ & \simeq FormalSmooth \infty Grpd( U, X ) \end{aligned} \,.

Cohomology and principal \infty-bundles

We discuss the intrinsic cohomology in a cohesive (∞,1)-topos realized in FormalSmoothGrpdFormalSmooth\infty Grpd.

Cohomology localization

Proposition

The canonical line object of the Lawvere theory CartSp smooth{}_{smooth} is the real line, regarded as an object of the Cahiers topos, and hence of FormalSmoothGrpdFormalSmooth \infty Grpd

𝔸 CartSp smooth 1=. \mathbb{A}^1_{CartSp_{smooth}} = \mathbb{R} \,.

We shall write \mathbb{R} also for the underlying additive group

𝔾 a= \mathbb{G}_a = \mathbb{R}

regarded as an abelian ∞-group object in FormalSmoothGrpdFormalSmooth\infty Grpd. For nn \in \mathbb{N} write B nFormalSmoothGrpd\mathbf{B}^n \mathbb{R} \in FormalSmooth\infty Grpd for its nn-fold delooping.

For nn \in \mathbb{N} and XFormalSmoothGrpdX \in FormalSmooth\infty Grpd write

H synthdiff n(X):=π 0FormalSmoothGrpd(X,B n) H^n_{synthdiff}(X) := \pi_0 FormalSmooth\infty Grpd(X,\mathbf{B}^n \mathbb{R})

for the cohomology group of XX with coefficients in the canonical line object in degree nn.

Definition

Write

L sdiffFormalSmoothGrpd \mathbf{L}_{sdiff} \hookrightarrow FormalSmooth \infty Grpd

for the cohomology localization of FormalSmoothGrpdFormalSmooth\infty Grpd at \mathbb{R}-cohomology: the full sub-(∞,1)-category on the WW-local object with respect to the class WW of morphisms that induce isomorphisms in all \mathbb{R}-cohomology groups.

Proposition

Let SmoothAlg proj ΔSmoothAlg^{\Delta}_{proj} be the projective model structure on cosimplicial smooth algebras and let j:(SmoothAlg Δ) op[FormalCartSp,sSet]j : (SmoothAlg^{\Delta})^{op} \to [FormalCartSp, sSet] be the prolonged external Yoneda embedding.

  1. This constitutes the right adjoint of a Quillen adjunction

    (𝒪j):(SmoothAlg Δ) opj𝒪[FSmoothMfd op,sSet] proj,loc. (\mathcal{O} \dashv j) : (SmoothAlg^\Delta)^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}} [FSmoothMfd^{op}, sSet]_{proj,loc} \,.
  2. Restricted to simplicial formal smooth manifolds along

    FSmoothMfd Δ op(SmoothAlg Δ) op FSmoothMfd^{\Delta^{op}} \hookrightarrow (SmoothAlg^\Delta)^{op}

    the right derived functor of jj is a full and faithful (∞,1)-functor that factors through the cohomology localization and thus identifies a full reflective sub-(∞,1)-category

    (FSmoothMfd Δ op) L sdiffFormalSmoothGrpd (FSmoothMfd^{\Delta^{op}})^\circ \hookrightarrow \mathbf{L}_{sdiff} \hookrightarrow FormalSmooth\infty Grpd
  3. The intrinsic \mathbb{R}-cohomology of any object XFormalSmoothGrpdX \in FormalSmooth\infty Grpd is computed by the ordinary cochain cohomology of the Moore cochain complex underlying the cosimplicial abelian group of the image under the left derived functor(𝕃𝒪)(X)(\mathbb{L}\mathcal{O})(X) under the Dold-Kan correspondence:

    H fSmooth n(X)H cochain n(N (𝕃𝒪)(X)). H_{fSmooth}^n(X) \simeq H^n_{cochain}(N^\bullet(\mathbb{L}\mathcal{O})(X)) \,.
Proof

First a remark on the sites. By the above proposition FormalSmoothGrpdFormalSmooth\infty Grpd is equivalent to the hypercomplete (∞,1)-topos over formal smooth manifolds. This is presented by the left Bousfield localization of [FSmoothMfd op,sSet] proj,loc[FSmoothMfd^{op}, sSet]_{proj,loc} at the ∞-connected morphisms. But a fibrant object in [FSmoothMfd op,sSet] proj,loc[FSmoothMfd^{op}, sSet]_{proj,loc} that is also n-truncated for nn \in \mathbb{N} is also fibrant in the hyperlocalization (only for the untruncated object there is an additional condition). Therefore the right Quillen functor claimed above indeed lands in truncated objects in FormalSmoothinftyGrpdFormalSmooth \inftyGrpd.

The proof of the above statements is given in (Stel), following (Toën). Some details are spelled out at function algebras on ∞-stacks.

Cohomology of Lie groups

Proposition

Let GSmoothMfdSmoothGrpdFormalSmoothGrpdG \in SmoothMfd \hookrightarrow Smooth\infty Grpd \hookrightarrow FormalSmooth\infty Grpd be a Lie group.

Then the intrinsic group cohomology in Smooth∞Grpd and in FormalSmoothGrpdFormalSmooth\infty Grpd of GG with coefficients in \mathbb{R} coincides with Segal’s refined Lie group cohomology (Segal, Brylinski).

H smooth n(BG,)H fsmooth n(BG,)H Segal n(G,). H^n_{smooth}(\mathbf{B}G, \mathbb{R}) \simeq H^n_{fsmooth}(\mathbf{B}G, \mathbb{R}) \simeq H^n_{Segal}(G,\mathbb{R}) \,.

For the full proof please see here, section 3.4 for the moment.

Corollary

For GG a compact Lie group we have for all n1n \geq 1 that

H smooth n(G,U(1))H top n+1(BG,). H^n_{smooth}(G,U(1)) \simeq H_{top}^{n+1}(B G, \mathbb{Z}) \,.
Proof

This follows from applying the above result to the fiber sequence induced by the sequence /=U(1)\mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z} = U(1).

Note

This means that the intrinsic cohomology of compact Lie groups in Smooth∞Grpd and FormalSmoothGrpdFormalSmooth\infty Grpd coincides for these coefficients with the Segal-Blanc-Brylinski refined Lie group cohomology (Brylinski).

Cohomology of \infty-Lie algebroids

Proposition

Let 𝔞L Algd\mathfrak{a} \in L_\infty \mathrm{Algd} be an L-∞ algebroid. Then its intrinsic real cohomoloogy in FormalSmoothGrpd\mathrm{FormalSmooth}\infty \mathrm{Grpd}

H n(𝔞,):=π 0FormalSmoothGrpd(𝔞,B n) H^n(\mathfrak{a}, \mathbb{R}) := \pi_0 \mathrm{FormalSmooth}\infty \mathrm{Grpd}(\mathfrak{a}, \mathbf{B}^n \mathbb{R})

coincides with its ordinary L-∞ algebroid cohomology: the cochain cohomology of its Chevalley-Eilenberg algebra

H n(𝔞,)H n(CE(𝔞)). H^n(\mathfrak{a}, \mathbb{R}) \simeq H^n(\mathrm{CE}(\mathfrak{a})) \,.
Proof

By the above propoposition we have that

H n(𝔞,)H nN (𝕃𝒪)(i(𝔞)). H^n(\mathfrak{a}, \mathbb{R}) \simeq H^n N^\bullet(\mathbb{L}\mathcal{O})(i(\mathfrak{a})) \,.

By this lemma this is

H nN ( [k]ΔΔ[k]𝒪(i(𝔞) k)). \cdots \simeq H^n N^\bullet \left( \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \mathcal{O}(i(\mathfrak{a})_k) \right) \,.

Observe that 𝒪(𝔞) \mathcal{O}(\mathfrak{a})_\bullet is cofibrant in the Reedy model structure [Δ op,(SmoothAlg proj Δ) op] Reedy[\Delta^{\mathrm{op}}, (\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}})^{\mathrm{op}}]_{\mathrm{Reedy}} relative to the opposite of the projective model structure on cosimplicial algebras:
the map from the latching object in degree nn in SmoothAlg Δ) op\mathrm{SmoothAlg}^\Delta)^{\mathrm{op}} is dually in SmoothAlgSmoothAlg Δ\mathrm{SmoothAlg} \hookrightarrow \mathrm{SmoothAlg}^\Delta the projection

i=0 nCE(𝔞) i i n i=0 n1CE(𝔞) i i n \oplus_{i = 0}^n \mathrm{CE}(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n \to \oplus_{i = 0}^{n-1} \mathrm{CE}(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n

hence is a surjection, hence a fibration in SmoothAlg proj Δ\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}} and therefore indeed a cofibration in (SmoothAlg proj Δ) op(\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}})^{\mathrm{op}}.

Therefore using the Quillen bifunctor property of the coend over the tensoring in reverse to lemma \ref{CofibrantResolutionOfLinfinityAlgebroid} the above is equivalent to

H nN ( [k]ΔΔ[k]𝒪(i(𝔞) k)) \cdots \simeq H^n N^\bullet \left( \int^{[k] \in \Delta} \Delta[k] \cdot \mathcal{O}(i(\mathfrak{a})_k) \right)

with the fat simplex replaced again by the ordinary simplex. But in brackets this is now by definition the image under the monoidal Dold-Kan correspondence of the Chevalley-Eilenberg algebra

H n(N ΞCE(𝔞)). \cdots \simeq H^n( N^\bullet \Xi \mathrm{CE}(\mathfrak{a}) ) \,.

By the Dold-Kan correspondence we have hence

H n(CE(𝔞)). \cdots \simeq H^n(\mathrm{CE}(\mathfrak{a})) \,.

It follows that a degree-nn \mathbb{R}-cocycle on 𝔞\mathfrak{a} is presented by a morphism

μ:𝔞b n, \mu : \mathfrak{a} \to b^n \mathbb{R} \,,

where on the right we have the L L_\infty-algebroid whose CE\mathrm{CE}-algebra is concentrated in degree nn. Notice that if 𝔞=b𝔤\mathfrak{a} = b \mathfrak{g} is the delooping of an L L_\infty- algebra 𝔤\mathfrak{g} this is equivalently a morphism of L L_\infty-algebras

μ:𝔤b n1. \mu : \mathfrak{g} \to b^{n-1} \mathbb{R} \,.

de Rham theorem

under construction

We consider the de Rham theorem in FormalSmoothGrpdFormalSmooth \infty Grpd, with respect to the infinitesimal de Rham cohomology

H dR,inf n(X):=π 0FormalSmoothGrpd(X, infB n). H_{dR,inf}^n(X) := \pi_0 FormalSmooth \infty Grpd(X, \mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R}) \,.
Proposition

For all nn \in \mathbb{N} The canonical morphism

infB nB n \mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} \to \mathbf{\flat} \mathbf{B}^n \mathbb{R}

is an equivalence.

This means that for all XHX \in \mathbf{H} the infinitesimal de Rham cohomology coincides with the ordinary real cohomology of the geometric realization of XX

H dR,inf n(X)H n(|X|,). H^n_{dR, inf}(X) \simeq H^n(|X|, \mathbb{R}) \,.
Proof

Since all representables are formally smooth, we have a colimit

U× Π inf(U)UUΠ inf(U). U \times_{\mathbf{\Pi}_{inf}(U)} U \stackrel{\to}{\to} U \stackrel{}{\to} \mathbf{\Pi}_{inf}(U) \,.

In the presentation over the site we have that

X× Π inf(X)X:K×D{f,g:K×DU|KK×DU}. X \times_{\mathbf{\Pi}_{inf}(X)} X : K \times D \mapsto \{ f,g : K \times D \to U | K \to K \times D \stackrel{\to}{\to} U \} \,.

Therefore a morphism Π inf(U)\mathbf{\Pi}_{inf}(U) \to \mathbb{R} is equivalently a morphism ϕ:U\phi : U \to \mathbb{R} such that for all K×DUK \times D \to U that coincide on KK we have that all the composites

K×DUϕB n K \times D \to U \stackrel{\phi}{\to} \mathbf{B}^n \mathbb{R}

are equals. If UU is the point, then ϕ\phi is necessarily constant. If UU is not the point, there is one nontrivial tangent vector vv in UU. The composite produces the corresponding tangent vector ϕ *(v)\phi_*(v) in \mathbb{R}. But all these tangent vectors must be equal. Hence ϕ\phi must be constant.

This kind of argument is indicated in (Simpson-Teleman, prop. 3.4).

Proposition

Let XX \in SmoothMfd and write X Δ inf [CartSp formalsmooth op,sSet]X^{\Delta^\bullet_{inf}} \in [CartSp_{formalsmooth}^{op}, sSet] for the tangent Lie algebroid regarded as a simplicial object (see L-infinity algebroid for the details).

Then there is a morphism X Δ inf Π inf(X)X^{\Delta^\bullet_{inf}} \to \mathbf{\Pi}_{inf}(X) which is an equivalence in \mathbb{R}-cohomology.

(…)

Formally étale morphisms and cohesive étale \infty-groupoids

We discuss formally étale morphisms and étale objects with respect to the cohesive infinitesimal neighbourhood i:i : Smooth∞Grpd FormalSmoothGrpd\hookrightarrow FormalSmooth\infty Grpd.

Proposition

Let X X_\bullet be a degreewise smooth paracompact simplicial manifold, canonically regarded as an object of Smooth∞Grpd.

Then j !X j_! X_\bullet in FormalSmoothGrpdFormalSmooth \infty Grpd is presented by the same simplicial manifold.

Proof

First consider an ordinary smooth paracompact manifold XX. It admits a good open cover {U iX}\{U_i \to X\} and the corresponding Cech nerve C({U i})in[CartSp smooth op,sSet] projC(\{U_i\}) in [CartSp_{smooth}^{op}, sSet]_{proj} is a cofibrant resolution of XX. Therefore the \infty-functor j !j_! is computed on XX by evaluating the corresponding simplicial functor (of which it is the derived functor) on C({U i})C(\{U_i\}).

Since the simplicial functor

j !:[CartSp smooth op,sSet] proj,loc[CartSp formalsmooth op,sSet] proj,loc j_! : [CartSp_{smooth}^{op}, sSet]_{proj, loc} \to [CartSp_{formalsmooth}^{op}, sSet]_{proj, loc}

is a left adjoint (indeed a left Quillen functor) it preserves the coproducts and coend that the Cech nerve is built from:

j !C({U i}) =j ! [n]ΔΔ[n] i 0,,i nU i 0,,i n = [n]ΔΔ[n] i 0,,i nj !(U i 0,,i n) = [n]ΔΔ[n] i 0,,i nU i 0,,i n. \begin{aligned} j_! C(\{U_i\}) & = j_! \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n} \\ & = \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_0, \cdots, i_n} j_! (U_{i_0, \cdots, i_n}) \\ & = \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_0, \cdots, i_n} U_{i_0, \cdots, i_n} \end{aligned} \,.

Here we used that, by assumption on a good open cover, all the U i 0,,i nU_{i_0, \cdots, i_n} are Cartesian spaces, and that j !j_! coincides on representables with the inclusion CartSp smoothCartSp formalsmoothCartSp_{smooth} \hookrightarrow CartSp_{formalsmooth}.

Let now X X_\bullet be a general simplicial manifold. Assume that in each degree there is a good open cover {U p,iX p}\{U_{p,i} \to X_p\} such that these fit into a simplicial system giving a bisimplicial Cech nerve such that there is a morphism of bisimplicial presheaves

C(𝒰) ,X C(\mathcal{U})_{\bullet, \bullet} \to X_{\bullet}

with X X_\bullet regarded as simplicially constant in one direction. Each C(𝒰) p,X pC(\mathcal{U})_{p, \bullet} \to X_p is a cofibrant resolution.

We claim that the coend

[n]ΔC(𝒰) n,Δ[n]X \int^{[n] \in \Delta} C(\mathcal{U})_{n, \bullet} \cdot \mathbf{\Delta}[n] \;\;\; \to \;\;\; X

is a cofibrant resolution of XX, where Δ\mathbf{\Delta} is the fat simplex. From this the proposition follows by again applying the left Quillen functor j !j_! degreewise and pulling it through all the colimits.

This remaining claim follows from the same argument as used above in prop. 9.

Proposition

A morphism in FormalSmoothGrpdFormalSmooth\infty Grpd, is a formally étale morphism with respect to the infinitesimal cohesion i:SmoothGrpdFormalSmoothGrpdi \colon Smooth \infty Grpd \hookrightarrow FormalSmooth\infty Grpd precisely if for all infinitesimally thickened points DD the diagram

X D f D Y D Y f Y \array{ X^D &\stackrel{f^D}{\to}& Y^D \\ \downarrow && \downarrow \\ Y &\stackrel{f}{\to}& Y }

is an \infty-pullback under i *i^*.

Remark

Since i *i^* preserves \infty-limits, this is the case in particular if the diagram is an \infty-pullback already in FormalSmoothGrodFormalSmooth\infty Grod. In this form, restricted to 0-truncated objects, hence to the Cahiers topos, this characterization of formally étale morphisms appears axiomatized around p. 82 of (Kock 81, p. 82).

In particular, a smooth function f:XYf : X \to Y in SmoothMfd \hookrightarrow Smooth∞Grpd between smooth manifolds is a formally étale morphism with respect to the infinitesimal cohesion SmoothGrpdFormalSmoothGrpdSmooth \infty Grpd \hookrightarrow FormalSmooth\infty Grpd precisely if it is a local diffeomorphism in the traditional sense.

Proof

We spell out the case for smooth manifolds. Here we need to to show that

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 a pullback in Sh(CartSp formalsmooth)Sh(CartSp_{formalsmooth}) precisely if ff is a local diffeomorphism. This is a pullback precisely if for all U×DCartSp smoothInfPointCartSp formalsmoothU \times D \in CartSp_{smooth} \ltimes InfPoint \simeq CartSp_{formalsmooth} the diagram of sets of plots

Hom(U×D,i !X) i !f Hom(U×D,i !Y) Hom(U×D,i *X) i *f Hom(U×D,i *Y) \array{ Hom(U \times D, i_! X) &\stackrel{i_! f}{\to}& Hom(U \times D, i_! Y) \\ \downarrow && \downarrow \\ Hom(U \times D, i_* X) &\stackrel{i_* f}{\to}& Hom(U \times D, i_* Y) }

is a pullback. Using, by the discussion at ∞-cohesive site, that i !i_! preserves colimits and restricts to ii on representables, and using that i *(U×D)=Ui^* (U \times D ) = U, this is equivalently the diagram

Hom(U×D,X) f * Hom(U×D,Y) Hom(U,X) f * Hom(U,Y), \array{ Hom(U \times D, X) &\stackrel{f_*}{\to}& Hom(U \times D, Y) \\ \downarrow && \downarrow \\ Hom(U , X) &\stackrel{f_*}{\to}& Hom(U , Y) } \,,

where the vertical morphisms are given by restriction along the inclusion (id U,*):UU×D(id_U, *) : U \to U \times D.

For one direction of the claim it is sufficient to consider this situation for U=*U = * the point and DD the first order infinitesimal interval. Then Hom(*,X)Hom(*,X) is the underlying set of points of the manifold XX and Hom(D,X)Hom(D,X) is the set of tangent vectors, the set of points of the tangent bundle TXT X. The pullback

Hom(*,X)× Hom(*,Y)Hom(D,Y) Hom(*,X) \times_{Hom(*,Y)} Hom(D,Y)

is therefore the set of pairs consisting of a point xXx \in X and a tangent vector vT f(x)Yv \in T_{f(x)} Y. This set is in fiberwise bijection with Hom(D,X)=TXHom(D, X) = T X precisely if for each xXx \in X there is a bijection T xXT f(x)YT_x X \simeq T_{f(x)}Y , hence precisely if ff is a local diffeomorphism. Therefore ff being a local diffeo is necessary for ff being formally étale with respect to the given notion of infinitesimal cohesion.

To see that this is also sufficient notice that this is evident for the case that ff is in fact a monomorphism, and that since smooth functions are characterized locally, we can reduce the general case to that case.

Proposition

A Lie groupoid 𝒢\mathcal{G} is an étale groupoid in the traditional sense, precisely if regarded as an object in i:i : Smooth∞Grpd \hookrightarrow FormalSmooth∞Grpd it is an cohesive étale ∞-groupoid.

Proof

Let 𝒢 0𝒢\mathcal{G}_0 \to \mathcal{G} be the inclusion of the smooth manifold of objects. This is an effective epimorphism. It remains to show that this is formally étale with respect to the given cohesive neighbourhood.

By the discussion at (∞,1)-pullback we may compute the characteristic (,1)(\infty,1)-pullback by an ordinary pullback of a fibration of simplicial presheaves that presents 𝒢 0𝒢\mathcal{G}_0 \to \mathcal{G}.

By the factorization lemma such is given by

𝒢 I× 𝒢𝒢 0𝒢. \mathcal{G}^I \times_{\mathcal{G}} \mathcal{G}_0 \to \mathcal{G} \,.

By inspection one see that this morphis is

  • in degree 0 the target-map t:Mor(𝒢)𝒢 0t : Mor(\mathcal{G}) \to \mathcal{G}_0;

  • in degree 1 the projection Mor(𝒢) t× sMor(𝒢)Mor(𝒢)Mor(\mathcal{G}) {}_t\times_{s} Mor(\mathcal{G}) \to Mor(\mathcal{G}).

By prop. 14 both of these need to be étale maps in the ordinary sense. By definition, this is the case precisely if 𝒢\mathcal{G} is an étale groupoid.

Formally smooth / formally unramified morphisms

As a direct consequence of prop. 14 we have the following

Proposition

A smooth function f:XYf : X \to Y between smooth manifolds, is a submersion or immersion, respectively, precisely if, when canonically regarded as a morphism in FormalSmoothGrpdFormalSmooth \infty Grpd, it is a formally smooth morphism or formally unramified morphism, respectively.

Proof

As in the proof of prop. 14 we find that the pullback i *X× i *Yi !Yi_* X \times_{i_* Y} i_! Y is over the infinitesimal interval isomorphic to

X× YTY X \times_Y T Y

and the canonical morphism from i !Xi_! X into this pullback is

TXX× YTY. T X \to X \times_Y T Y \,.

Lie differentiation

We sketch how to formalize Lie differentiation in the context of formal smooth \infty-groupoids.

Let

inf:InfPoint H */ inf : InfPoint_\infty \hookrightarrow \mathbf{H}^{*/}

be the canonical inclusion. By (Lurie) we have the full inclusion

Lie Sh (InfPoint ) Lie_\infty \hookrightarrow Sh_\infty(InfPoint_\infty)

on those objects whose space of global sections is contractible and which are infinitesimally cohesive (for a somewhat different notion of “infinitesimal cohesion”, beware the terminology). Consider then the \infty-functor

Grp(H)H 1 */yonedaPSh (H */)inf *PSh (InfPoint ) Grp(\mathbf{H}) \simeq \mathbf{H}^{*/}_{\geq 1} \stackrel{yoneda}{\to} PSh_\infty( \mathbf{H}^{*/}) \stackrel{inf^*}{\to} PSh_\infty(InfPoint_\infty)

which sends a pointed connected formal smooth \infty-groupoid BG\mathbf{B}G to the (,1)(\infty,1)-presheaf of pointed morphisms

ptBG \mathbf{pt} \to \mathbf{B}G

for ptInfPoint \mathbf{pt} \in InfPoint_\infty.

By assumption that BG\mathbf{B}G is connected (and we need to assume that it is geometric, which will gives infinitesimal cohesion by the Artin-Lurie representability theorem) this factors as

H 1 */LieLie Sh (InfPoint ). \mathbf{H}^{*/}_{\geq 1} \stackrel{Lie}{\to} Lie_\infty \hookrightarrow Sh_\infty(InfPoint_\infty) \,.

The resulting \infty-functor

Lie:Grp(H)H 1 */Lie Lie : Grp(\mathbf{H}) \simeq \mathbf{H}^{*/}_{\geq 1} \to Lie_\infty

is Lie differentiation.

For differentiation of smooth groupoids with atlas UXU \to X to L-infinity algebroids this happens under UU

H U/ \mathbf{H}^{U/}

(…)

References

The site FormalCartSp is discussed in section 5 of

  • Anders Kock, Convenient vector spaces embed into the Cahiers topos , Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 (numdam).

For more on this see at Cahiers topos.

The notion of formally étale maps as obtained here from the general abstract definition in differential cohesion coincided on 0-truncated objects with that defined on p. 82 of

The infinitesimal path ∞-groupoid adjunction (RedΠ inf inf)(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf}) and the de Rham theorem for \infty-stacks is discussed at the level of homotopy categories in section 3 of

The (,1)(\infty,1)-topos SynthDiffGrpdSynthDiff\infty Grpd is discussed in section 4.4 of

The cohomology localization of SynthDiffGrpdSynthDiff\infty Grpd and the infinitesimal singular simplicial complex as a presentation for infinitesimal paths objects in SynthDiffGrpdSynthDiff\infty Grpd is discussed in

  • Herman Stel, \infty-Stacks and their function algebras – with applications to \infty-Lie theory , master thesis (2010) (web)

The discussion of the cohomology localization of SynthDiffGrpdSynthDiff\infty Grpd follows that in another context in

The construction of the infinitesimal path object has been amplified and discussed by Anders Kock under the name combinatorial differential forms, for instance in

The discussion that the normalized cosimplicial algebra of functions on the infinitesimal singular simplicial complex is the de Rham complex is further discussed in

The results on differentiable Lie group cohomology used above is in

  • P. Blanc, Cohomologie différentiable et changement de groupes Astérisque, vol. 124-125 (1985), pp. 113-130.

recalled in

which parallels

  • Graeme Segal, Cohomology of topological groups , Symposia Mathematica, Vol IV (1970) (1986?) p. 377

The (,1)(\infty,1)-site of derived infinitesimal points is discussed in

following

Revised on November 4, 2014 08:03:59 by Urs Schreiber (185.26.182.34)