nLab
synthetic differential infinity-groupoid

Idea

A synthetic differential $\mathrm{\infty }$-groupoid is an ∞-groupoid equipped with a cohesive structure that subsumes that of smooth ∞-groupoids as well as of infinitesimal $\mathrm{\infty }$-groupoids: ∞-Lie algebroids.

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

Definition

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

• Cartesian spaces with smooth functions between them as for smooth ∞-groupoids,

• and a category or (∞,1)-category of infinitesimally thickened points.

First in

• 1-localic definition

we consider the 1-site, then in

• ∞-localic definition

we consider the $(\mathrm{\infty },1)$-site.

1-localic definition

Definition

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

Write

$$\mathrm{SmoothAlg}:=T\mathrm{Alg}$$SmoothAlg := T Alg

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

Let

$$\mathrm{InfPoint}\hookrightarrow {\mathrm{SmoothAlg}}^{\mathrm{op}}$$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 $ℝ\oplus V$ with $V$ a finite-dimensional real vector space and nilpotent in the algebra structure.

Definition

Let

$${\mathrm{CartSp}}_{\mathrm{synthdiff}}\hookrightarrow \mathrm{SmoothLoc}$$CartSp_{synthdiff} \hookrightarrow SmoothLoc

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

$$U={ℝ}^n\times D$$U = \mathbb{R}^n \times D

that are products of a Cartesian space ${ℝ}^n\in$ CartSp for $n\in \mathbb{N}$ and an infinitesimally thickened point $D\in \mathrm{InfPoint}$.

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

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

Note

The sheaf topos over CartSp${}_{\mathrm{synthdiff}}$ 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 synthetic differential $\mathrm{\infty }$-groupoids is the (∞,1)-category of (∞,1)-sheaves

$$\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}:={\mathrm{Sh}}_{(\mathrm{\infty },1)}({\mathrm{CartSp}}_{\mathrm{synthdiff}})$$SynthDiff \infty Grpd := Sh_{(\infty,1)}(CartSp_{synthdiff})

on ${\mathrm{CartSp}}_{\mathrm{synthdiff}}$.

$\mathrm{\infty }$-localic

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

(…)

Properties

Proposition

$\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ is a cohesive (∞,1)-topos.

Proof

Because CartSp${}_{\mathrm{synthdiff}}$ is an ∞-cohesive site. See there for details.

Definition

Write $\mathrm{FSmoothMfd}\hookrightarrow {\mathrm{SmoothAlg}}^{\mathrm{op}}$ for the full subcategory of smooth loci on the formal smooth manifolds: those modeled on CartSp${}_{\mathrm{synthdiff}}$ equipped with the evident coverage.

Observation

${\mathrm{CartSp}}_{\mathrm{synthdiff}}$ is a dense sub-site of $\mathrm{FSmoothMfd}$.

Proposition

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

$$\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}\simeq {\widehat{\mathrm{Sh}}}_{(\mathrm{\infty },1)}(\mathrm{FSmoothMfd})$$SynthDiff\infty Grpd \simeq \hat Sh_{(\infty,1)}(FSmoothMfd)

with the hypercomplete (∞,1)-topos over $\mathrm{FSmoothMfd}$.

Proof

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

Definition

Write $i:{\mathrm{CartSp}}_{\mathrm{smooth}}\hookrightarrow {\mathrm{CartSp}}_{\mathrm{synthdiff}}$ for the canonical embedding.

Proposition

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

$$(i_{!}⊣i^{*}⊣i_{*}⊣i^{!}):\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{i_{!}}{\hookrightarrow }}{\stackrel{\stackrel{i^{*}}{\leftarrow }}{\stackrel{\stackrel{i_{*}}{\to }}{\stackrel{i^{!}}{\leftarrow }}}}\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd},$$( 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}}}} SynthDiff \infty Grpd \,,

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

Proof

Since $i:{\mathrm{CartSp}}_{\mathrm{smooth}}\hookrightarrow {\mathrm{CartSp}}_{\mathrm{synthdiff}}$ 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 $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$.

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

$\mathrm{\infty }$-Lie algebroids and deformation theory

This subsection is at

• ∞-Lie algebroid.

Paths and geometric Postnikov towers

We discuss the intrinsic infinitesimal path adjunction realized in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$.

$$(\mathrm{Red}⊣{\Pi }_{\inf }⊣{♭}_{\inf }):=(i\circ {\Pi }_{\inf }⊣{\mathrm{Disc}}_{\inf }{\Pi }_{\inf }⊣{\mathrm{Disc}}_{\inf }\circ {\Gamma }_{\inf }):\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}\to \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}.$$(\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}) : SynthDiff \infty Grpd \to SynthDiff \infty Grpd \,.

Proposition

For $U\times D\in {\mathrm{CartSp}}_{\mathrm{smooth}}⋉\mathrm{InfinSmoothLoc}={\mathrm{CartSp}}_{\mathrm{synthdiff}}\hookrightarrow \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ we have that

$$\mathrm{Red}(U\times D)\simeq U$$\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 $C^{\mathrm{\infty }}(K\times D)\simeq C^{\mathrm{\infty }}(K)\otimes C^{\mathrm{\infty }}(D)$.

Proof

By the model category presentation of $\mathrm{Red}\simeq 𝕃{\mathrm{Lan}}_i\circ ℝi^{*}$ of the above proof and using that every representable is cofibrant and fibrant in the local projective model structure on simplicial presheaves we have

$$\begin{array}{rl}\mathrm{Red}(U\times D)& \simeq (𝕃{\mathrm{Lan}}_i)(ℝi^{*})(U\times D)\\ & \simeq (𝕃{\mathrm{Lan}}_i)i^{*}(U\times D)\\ & \simeq (𝕃{\mathrm{Lan}}_i)U\\ & \simeq {\mathrm{Lan}}_{i}U\\ & \simeq U\end{array}$$\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 $i$ is a full and faithful functor).

Proposition

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

$${\Pi }_{\inf }(X):U\times D\mapsto X(U).$$\mathbf{\Pi}_{inf}(X) : U \times D \mapsto X(U) \,.

Proof

By the $(\mathrm{Red}⊣{\Pi }_{\inf })$-adjunction relation

$$\begin{array}{rl}{\Pi }_{\inf }(X)(U\times D)& =\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}(U\times D,{\Pi }_{\inf }(X))\\ & \simeq \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}(\mathrm{Red}(U\times D),X)\\ & \simeq \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}(U,X)\end{array}.$$\begin{aligned} \mathbf{\Pi}_{inf}(X)(U \times D) & = SynthDiff \infty Grpd(U \times D, \mathbf{\Pi}_{inf}(X)) \\ & \simeq SynthDiff \infty Grpd( \mathbf{Red}(U \times D), X) \\ & \simeq SynthDiff \infty Grpd( U, X ) \end{aligned} \,.

Cohomology and principal $\mathrm{\infty }$-bundles

We discuss the intrinsic cohomology in a cohesive (∞,1)-topos realized in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$.

Cohomology localization

Proposition

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

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

We shall write $ℝ$ also for the underlying additive group

$${𝔾}_a=ℝ$$\mathbb{G}_a = \mathbb{R}

regarded as an abelian ∞-group object in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$. For $n\in \mathbb{N}$ write $B^nℝ\in \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ for its $n$-fold delooping.

For $n\in \mathbb{N}$ and $X\in \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ write

$$H_{\mathrm{synthdiff}}^n(X):={\pi }_0\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}(X,B^nℝ)$$H^n_{synthdiff}(X) := \pi_0 SynthDiff\infty Grpd(X,\mathbf{B}^n \mathbb{R})

for the cohomology group of $X$ with coefficients in the canonical line object in degree $n$.

Definition

Write

$$L_{\mathrm{sdiff}}\hookrightarrow \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$$\mathbf{L}_{sdiff} \hookrightarrow SynthDiff \infty Grpd

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

Proposition

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

1. This constitutes the right adjoint of a Quillen adjunction

   $$(𝒪⊣j):({\mathrm{SmoothAlg}}^{\Delta }{)}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{\leftarrow }}{\underset{j}{\to }}[{\mathrm{FSmoothMfd}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}.$$(\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

   $${\mathrm{FSmoothMfd}}^{{\Delta }^{\mathrm{op}}}\hookrightarrow ({\mathrm{SmoothAlg}}^{\Delta }{)}^{\mathrm{op}}$$FSmoothMfd^{\Delta^{op}} \hookrightarrow (SmoothAlg^\Delta)^{op}

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

   $$({\mathrm{FSmoothMfd}}^{{\Delta }^{\mathrm{op}}}{)}^{\circ }\hookrightarrow L_{\mathrm{sdiff}}\hookrightarrow \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$$(FSmoothMfd^{\Delta^{op}})^\circ \hookrightarrow \mathbf{L}_{sdiff} \hookrightarrow SynthDiff\infty Grpd

3. The intrinsic $ℝ$-cohomology of any object $X\in \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ 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$(𝕃𝒪)(X)$ under the Dold-Kan correspondence:

   $$H_{\mathrm{synthdiff}}^n(X)\simeq H_{\mathrm{cochain}}^n(N^{•}(𝕃𝒪)(X)).$$H_{synthdiff}^n(X) \simeq H^n_{cochain}(N^\bullet(\mathbb{L}\mathcal{O})(X)) \,.

Proof

First a remark on the sites. By the above proposition $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ is equivalent to the hypercomplete (∞,1)-topos over formal smooth manifolds. This is presented by the left Bousfield localization of $[{\mathrm{FSmoothMfd}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ at the ∞-connected morphisms. But a fibrant object in $[{\mathrm{FSmoothMfd}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ that is also n-truncated for $n\in \mathbb{N}$ 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 $\mathrm{SynthDiff}inftyGrpd$.

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 $G\in \mathrm{SmoothMfd}\hookrightarrow \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}\hookrightarrow \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ be a Lie group.

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

$$H_{\mathrm{smooth}}^n(BG,ℝ)\simeq H_{\mathrm{synthdiff}}^n(BG,ℝ)\simeq H_{\mathrm{Segal}}^n(G,ℝ).$$H^n_{smooth}(\mathbf{B}G, \mathbb{R}) \simeq H^n_{synthdiff}(\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 $G$ a compact Lie group we have for all $n\ge 1$ that

$$H_{\mathrm{smooth}}^n(G,U(1))\simeq H_{\mathrm{top}}^{n+1}(BG,\mathbb{Z}).$$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 $\mathbb{Z}\to ℝ\to ℝ/\mathbb{Z}=U(1)$.

Note

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

Cohomology of $\mathrm{\infty }$-Lie algebroids

Proposition

Let $𝔞\in L_{\mathrm{\infty }}\mathrm{Algd}$ be an L-∞ algebroid. Then its intrinsic real cohomoloogy in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$

$$H^n(𝔞,ℝ):={\pi }_0\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}(𝔞,B^nℝ)$$H^n(\mathfrak{a}, \mathbb{R}) := \pi_0 \mathrm{SynthDiff}\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

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

Proof

By the above propoposition we have that

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

By this lemma this is

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

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

$${\oplus }_{i=0}^n\mathrm{CE}(𝔞{)}_i\otimes {\wedge }^i{ℝ}^n\to {\oplus }_{i=0}^{n-1}\mathrm{CE}(𝔞{)}_i\otimes {\wedge }^i{ℝ}^n$$\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 ${\mathrm{SmoothAlg}}_{\mathrm{proj}}^{\Delta }$ and therefore indeed a cofibration in $({\mathrm{SmoothAlg}}_{\mathrm{proj}}^{\Delta }{)}^{\mathrm{op}}$.

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

$$\cdots \simeq H^n{N}^{•}({\int }^{[k]\in \Delta }\Delta [k]\cdot 𝒪(i(𝔞{)}_k))$$\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

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

By the Dold-Kan correspondence we have hence

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

It follows that a degree-$n$ $ℝ$-cocycle on $𝔞$ is presented by a morphism

$$\mu :𝔞\to b^nℝ,$$\mu : \mathfrak{a} \to b^n \mathbb{R} \,,

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

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

de Rham theorem

under construction

We consider the de Rham theorem in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$, with respect to the infinitesimal de Rham cohomology

$$H_{\mathrm{dR},\inf }^n(X):={\pi }_0\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}(X,{♭}_{\inf }{B}^nℝ).$$H_{dR,inf}^n(X) := \pi_0 SynthDiff \infty Grpd(X, \mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R}) \,.

Proposition

For all $n\in \mathbb{N}$ The canonical morphism

$${♭}_{\inf }{B}^nℝ\to ♭B^nℝ$$\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} \to \mathbf{\flat} \mathbf{B}^n \mathbb{R}

is an equivalence.

This means that for all $X\in H$ the infinitesimal de Rham cohomology coincides with the ordinary real cohomology of the geometric realization of $X$

$$H_{\mathrm{dR},\inf }^n(X)\simeq H^n(\mid X\mid ,ℝ).$$H^n_{dR, inf}(X) \simeq H^n(|X|, \mathbb{R}) \,.

Proof

Since all representables are formally smooth, we have a colimit

$$U{\times }_{{\Pi }_{\inf }(U)}U\overrightarrow{\to }U\stackrel{}{\to }{\Pi }_{\inf }(U).$$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

$$X{\times }_{{\Pi }_{\inf }(X)}X:K\times D\mapsto \{f,g:K\times D\to U\mid K\to K\times D\overrightarrow{\to }U\}.$$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 ${\Pi }_{\inf }(U)\to ℝ$ is equivalently a morphism $\varphi :U\to ℝ$ such that for all $K\times D\to U$ that coincide on $K$ we have that all the composites

$$K\times D\to U\stackrel{\varphi }{\to }{B}^nℝ$$K \times D \to U \stackrel{\phi}{\to} \mathbf{B}^n \mathbb{R}

are equals. If $U$ is the point, then $\varphi$ is necessarily constant. If $U$ is not the point, there is one nontrivial tangent vector $v$ in $U$. The composite produces the corresponding tangent vector ${\varphi }_{*}(v)$ in $ℝ$. But all these tangent vectors must be equal. Hence $\varphi$ must be constant.

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

Proposition

Let $X\in$ SmoothMfd and write $X^{{\Delta }_{\inf }^{•}}\in [{\mathrm{CartSp}}_{\mathrm{synthdiff}}^{\mathrm{op}},\mathrm{sSet}]$ for the tangent Lie algebroid regarded as a simplicial object (see L-infinity algebroid for the details).

Then there is a morphism $X^{{\Delta }_{\inf }^{•}}\to {\Pi }_{\inf }(X)$ which is an equivalence in $ℝ$-cohomology.

(…)

Formally étale morphisms and cohesive étale $\mathrm{\infty }$-groupoids

We discuss formally étale morphisms and étale objects with respect to the cohesive infinitesimal neighbourhood $i:$ Smooth∞Grpd $\hookrightarrow \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$.

Proposition

Let $X_{•}$ be a degreewise smooth paracompact simplicial manifold, canonically regarded as an object of Smooth∞Grpd.

Then $j_{!}{X}_{•}$ in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ is presented by the same simplicial manifold.

Proof

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

Since the simplicial functor

$$j_{!}:[{\mathrm{CartSp}}_{\mathrm{smooth}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}\to [{\mathrm{CartSp}}_{\mathrm{synthdiff}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$$j_! : [CartSp_{smooth}^{op}, sSet]_{proj, loc} \to [CartSp_{synthdiff}^{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:

$$\begin{array}{rl}{j}_{!}C(\{U_i\})& =j_{!}{\int }^{[n]\in \Delta }\Delta [n]\cdot \coprod _{i_0,\cdots ,i_n}{U}_{i_0,\cdots ,i_n}\\ & ={\int }^{[n]\in \Delta }\Delta [n]\cdot \coprod _{i_0,\cdots ,i_n}{j}_{!}(U_{i_0,\cdots ,i_n})\\ & ={\int }^{[n]\in \Delta }\Delta [n]\cdot \coprod _{i_0,\cdots ,i_n}{U}_{i_0,\cdots ,i_n}\end{array}.$$\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 $U_{i_0,\cdots ,i_n}$ are Cartesian spaces, and that $j_{!}$ coincides on representables with the inclusion ${\mathrm{CartSp}}_{\mathrm{smooth}}\hookrightarrow {\mathrm{CartSp}}_{\mathrm{synthdiff}}$.

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

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

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

We claim that the coend

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

is a cofibrant resolution of $X$, where $\Delta$ is the fat simplex. From this the proposition follows by again applying the left Quillen functor $j_{!}$ 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 $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$, is a formally étale morphism with respect to the infinitesimal cohesion $i:\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}\hookrightarrow \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ precisely if for all infinitesimally thickened points $D$ the diagram

$$\begin{array}{ccc}{X}^D& \stackrel{f^D}{\to }& Y^D\\ \downarrow & & \downarrow \\ Y& \stackrel{f}{\to }& Y\end{array}$$\array{ X^D &\stackrel{f^D}{\to}& Y^D \\ \downarrow && \downarrow \\ Y &\stackrel{f}{\to}& Y }

is an $\mathrm{\infty }$-pullback under $i^{*}$.

Remark

Since $i^{*}$ preserves $\mathrm{\infty }$-limits, this is the case in particular if the diagram is an $\mathrm{\infty }$-pullback already in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grod}$. 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 $f:X\to Y$ in SmoothMfd $\hookrightarrow$ Smooth∞Grpd between smooth manifolds is a formally étale morphism with respect to the infinitesimal cohesion $\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}\hookrightarrow \mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ 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

$$\begin{array}{ccc}{i}_{!}X& \stackrel{i_{!}f}{\to }& i_{!}Y\\ \downarrow & & \downarrow \\ i_{*}X& \stackrel{i_{*}f}{\to }& i_{*}Y\end{array}$$\array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }

is a pullback in $\mathrm{Sh}({\mathrm{CartSp}}_{\mathrm{synthdiff}})$ precisely if $f$ is a local diffeomorphism. This is a pullback precisely if for all $U\times D\in {\mathrm{CartSp}}_{\mathrm{smooth}}⋉\mathrm{InfPoint}\simeq {\mathrm{CartSp}}_{\mathrm{synthdiff}}$ the diagram of sets of plots

$$\begin{array}{ccc}\mathrm{Hom}(U\times D,i_{!}X)& \stackrel{i_{!}f}{\to }& \mathrm{Hom}(U\times D,i_{!}Y)\\ \downarrow & & \downarrow \\ \mathrm{Hom}(U\times D,i_{*}X)& \stackrel{i_{*}f}{\to }& \mathrm{Hom}(U\times D,i_{*}Y)\end{array}$$\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 $i_{!}$ preserves colimits and restricts to $i$ on representables, and using that $i^{*}(U\times D)=U$, this is equivalently the diagram

$$\begin{array}{ccc}\mathrm{Hom}(U\times D,X)& \stackrel{f_{*}}{\to }& \mathrm{Hom}(U\times D,Y)\\ \downarrow & & \downarrow \\ \mathrm{Hom}(U,X)& \stackrel{f_{*}}{\to }& \mathrm{Hom}(U,Y)\end{array},$$\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 $({\mathrm{id}}_U,*):U\to U\times D$.

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

$$\mathrm{Hom}(*,X){\times }_{\mathrm{Hom}(*,Y)}\mathrm{Hom}(D,Y)$$Hom(*,X) \times_{Hom(*,Y)} Hom(D,Y)

is therefore the set of pairs consisting of a point $x\in X$ and a tangent vector $v\in T_{f(x)}Y$. This set is in fiberwise bijection with $\mathrm{Hom}(D,X)=TX$ precisely if for each $x\in X$ there is a bijection $T_{x}X\simeq T_{f(x)}Y$, hence precisely if $f$ is a local diffeomorphism. Therefore $f$ being a local diffeo is necessary for $f$ 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 $f$ 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 $𝒢$ is an étale groupoid in the traditional sense, precisely if regarded as an object in $i:$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd it is an cohesive étale ∞-groupoid.

Proof

Let ${𝒢}_0\to 𝒢$ 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 $(\mathrm{\infty },1)$-pullback by an ordinary pullback of a fibration of simplicial presheaves that presents ${𝒢}_0\to 𝒢$.

By the factorization lemma such is given by

$${𝒢}^I{\times }_{𝒢}{𝒢}_0\to 𝒢.$$\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 $t:\mathrm{Mor}(𝒢)\to {𝒢}_0$;

• in degree 1 the projection $\mathrm{Mor}(𝒢){}_t{\times }_s\mathrm{Mor}(𝒢)\to \mathrm{Mor}(𝒢)$.

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

Formally smooth / formally unramified morphisms

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

Proposition

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

Proof

As in the proof of prop. 14 we find that the pullback $i_{*}X{\times }_{i_{*}Y}{i}_{!}Y$ is over the infinitesimal interval? isomorphic to

$$X{\times }_{Y}TY$$X \times_Y T Y

and the canonical morphism from $i_{!}X$ into this pullback is

$$TX\to X{\times }_{Y}TY.$$T X \to X \times_Y T Y \,.

Lie differentiation

We sketch how to formalize Lie differentiation in the context of synthetic differential $\mathrm{\infty }$-groupoids.

Let

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

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

$${\mathrm{Lie}}_{\mathrm{\infty }}\hookrightarrow {\mathrm{Sh}}_{\mathrm{\infty }}({\mathrm{InfPoint}}_{\mathrm{\infty }})$$Lie_\infty \hookrightarrow Sh_\infty(InfPoint_\infty)

on those objects whose space of global sections is contractible. Consider then the $\mathrm{\infty }$-functor

$$\mathrm{Grp}(H)\simeq H_{\ge 1}^{*/}\stackrel{\mathrm{yoneda}}{\to }{\mathrm{PSh}}_{\mathrm{\infty }}(H^{*/})\stackrel{{\inf }^{*}}{\to }{\mathrm{PSh}}_{\mathrm{\infty }}({\mathrm{InfPoint}}_{\mathrm{\infty }})$$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 synthetic differential $\mathrm{\infty }$-groupoid $BG$ to the $(\mathrm{\infty },1)$-presheaf of pointed morphisms

$$\mathrm{pt}\to BG$$\mathbf{pt} \to \mathbf{B}G

for $\mathrm{pt}\in {\mathrm{InfPoint}}_{\mathrm{\infty }}$.

By assumption that $BG$ is connected this factors as

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

The resulting $\mathrm{\infty }$-functor

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

is Lie differentiation.

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

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

(…)

Related concepts

• cohesive (∞,1)-topos

  • discrete ∞-groupoid

  • Euclidean-topological ∞-groupoid

  • smooth ∞-groupoid

  • synthetic differential ∞-groupoid

  • super ∞-groupoid

  • smooth super ∞-groupoid

  • synthetic differential super ∞-groupoid

References

The site CartSp${}_{\mathrm{synthdiff}}$ 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

• Anders Kock, Synthetic Differential Geometry (1981) (pdf)

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

• Carlos Simpson, Constantin Teleman, deRham theorem for $\mathrm{\infty }$-stacks (pdf)

The $(\mathrm{\infty },1)$-topos $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ is discussed in section 3.4 of

• Urs Schreiber, differential cohomology in a cohesive topos

The cohomology localization of $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ and the infinitesimal singular simplicial complex as a presentation for infinitesimal paths objects in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ is discussed in

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

The discussion of the cohomology localization of $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$ follows that in another context in

• Bertrand Toën, Champs affine (arXiv:math/0012219)

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

• Anders Kock, Synthetic Geometry of Manifolds (pdf)

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

• Jean-Luc Brylinski, Differentiable Cohomology of Gauge Groups (arXiv)

which parallels

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

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

• Jacob Lurie, Formal moduli problems

following

• Vladimir Hinich, DG coalgebras as formal stacks, J. Pure Appl. Algebra, 162 (2001), 209-250 (arXiv:math/9812034)