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smooth infinity-groupoid -- structures

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

This is a sub-entry of Smooth∞Grpd. See there for background.


Contents

Structures

We discuss the general abstract structures in a cohesive (∞,1)-topos realized in Smooth∞Grpd.

Geometric homotopy and Galois theory

We discuss the intrinsic fundamental ∞-groupoid construction realized in SmoothGrpdSmooth \infty Grpd.

Proposition

If XSmoothGrpdX \in Smooth\infty Grpd is presented by X SmoothMfd Δ op[CartSp smooth op,sSet]X_\bullet \in SmoothMfd^{\Delta^{op}} \hookrightarrow [CartSp_{smooth}^{op}, sSet], then its image i !(X)i_!(X) \in ETop∞Grpd under the relative topological cohesion morphism is presented by the underlying simplicial topological space X TopMfd Δ op[CartSp top op,sSet]X_\bullet \in TopMfd^{\Delta^{op}} \hookrightarrow [CartSp_{top}^{op}, sSet].

Proof

Let first XSmoothMfdSmoothMfd Δ opX \in SmoothMfd \hookrightarrow SmoothMfd^{\Delta^{op}} be simplicially constant. Then there is a differentiably good open cover {U iX}\{U_i \to X\} such that the Cech nerve projection

( [k]ΔΔ[k] i 0,,i kU i 0× X× XU i k)X \left( \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} U_{i_0} \times_X \cdots \times_X U_{i_k} \right) \stackrel{\simeq}{\to} X

is a cofibrant resolution in [CartSp smooth op,sSet] proj,loc[CartSp_{smooth}^{op}, sSet]_{proj,loc} which is degreewise a coproduct of representables. That means that the left derived functor 𝕃Lan i\mathbb{L} Lan_i on XX is computed by the application of Lan iLan_i on this coend, which by the fact that this is defined to be the left Kan extension along ii is given degreewise by ii, and since ii preserves pullbacks along covers, this is

(𝕃Lan i)X Lan i( [k]ΔΔ[k] i 0,,i kU i 0× X× XU i k) = [k]ΔΔ[k] i 0,,i kLan i(U i 0× X× XU i k) [k]ΔΔ[k] i 0,,i ki(U i 0× X× XU i k) [k]ΔΔ[k] i 0,,i k(i(U i 0)× i(X)× i(X)i(U i k)) i(X), \begin{aligned} (\mathbb{L} Lan_i) X & \simeq Lan_i \left( \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} U_{i_0} \times_X \cdots \times_X U_{i_k} \right) \\ & = \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} Lan_i (U_{i_0} \times_X \cdots \times_X U_{i_k}) \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} i (U_{i_0} \times_X \cdots \times_X U_{i_k}) \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} (i(U_{i_0}) \times_{i(X)} \cdots \times_{i(X)} i(U_{i_k})) \\ & \simeq i(X) \end{aligned} \,,

The last step follows from observing that we have manifestly the Cech nerve as before, but now of the underlying topological spaces of the {U i}\{U_i\} and of XX.

The claim then follows for general simplicial spaces by observing that X = [k]ΔΔ[k]X k[CartSp smooth op,sSet] proj,locX_\bullet = \int^{[k] \in \Delta} \Delta[k] \cdot X_k \in [CartSp_{smooth}^{op}, sSet]_{proj,loc} presents the (∞,1)-colimit over X :Δ opSmoothMfdSmoothGrpdX_\bullet : \Delta^{op} \to SmoothMfd \hookrightarrow Smooth \infty Grpd and the left adjoint (∞,1)-functor i !i_! preserves these (∞,1)-colimits.

Corollary

If XSmoothGrpdX \in Smooth\infty Grpd is presented by X SmoothMfd Δ op[CartSp smooth op,sSet]X_\bullet \in SmoothMfd^{\Delta^{op}} \hookrightarrow [CartSp_{smooth}^{op}, sSet], then the image of XX under the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor

SmoothGrpdΠGrpdTop Smooth \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \underoverset{|-|}{\simeq}{\to} Top

is equivalent to the geometric realization of (a Reedy cofibrant replacement of) the underlying simplicial topological space

Π(X)QX . |\Pi(X)| \simeq |Q X_\bullet| \,.

In particular if XX is an ordinary smooth manifold then

Π(X)SingX \Pi(X) \simeq Sing X

is equivalent to the standard fundamental ∞-groupoid of XX.

Proof

By a proposition above the functor Π\Pi factors as ΠXΠ ETopi !X\Pi X \simeq \Pi_{ETop} i_! X. By the above proposition this is Π Etop\Pi_{Etop} applied to the underlying simplicial topological space. The claim then follows with the corresponding proposition discussed at ETop∞Grpd.

Corollary

The Π:SmoothGrpdGrpd\Pi : \mathrm{Smooth}\infty\mathrm{Grpd} \to \infty \mathrm{Grpd} preserves homotopy fibers of morphisms that are presented in [CartSp smooth op,sSet] proj[\mathrm{CartSp}_{\mathrm{smooth}}^{\mathrm{op}}, \mathrm{sSet}]_{\mathrm{proj}} by morphisms of the form XW¯GX \to \bar W G with XX fibrant and GG a simplicial group in SmoothMfd.

Proof

By the above the functor factors as

Π SmoothΠ ETopi !. \Pi_{\mathrm{Smooth}} \simeq \Pi_{\mathrm{ETop}} \circ i_! \,.

and i !i_! assigns the underlying topological spaces. If we can show that this preserves the homotopy fibers in question, then the claim follows with the analogous discussion at ETop∞Grpd.

We find this as in the proof of the latter proposition, by considering the pasting diagram of pullbacks of simplicial presheaves

P P WG QX X W¯G. \array{ P' &\to& P &\to& W G \\ \downarrow && \downarrow && \downarrow \\ Q X &\to& X &\to& \bar W G } \,.

Since the component maps of the right vertical morphisms are surjective, the degreewise pullbacks in SmoothMfd\mathrm{SmoothMfd} that define PP' are all along transversal maps, and thus the underlying objects in TopMfd are the pullbacks of the underlying topological manifolds. Therefore the degreewise forgetful functor SmoothMfdTopMfd\mathrm{SmoothMfd} \to \mathrm{TopMfd} presents i !i_! on the outer diagram and sends this homotopy pullback to a homotopy pullback.

Paths and geometric Postnikov towers

Paths and geometric Postnikov towers

We discuss the notion of geometric path ∞-groupoids realized in SmoothGrpdSmooth\infty Grpd.

Definition

For XX \in SmthMfd write

SingX[CartSp op,sSet] \mathbf{Sing} X \in [CartSp^{op}, sSet]

for the simplicial presheaf that sends a test space UU \in CartSp to the singular simplicial complex of smooth simplices smoothly parameterized over UU:

SingX:(U,k)Hom SmthMfd(U×Δ k,X). \mathbf{Sing} X : (U,k) \mapsto Hom_{SmthMfd}(U \times \Delta^k, X) \,.
Proposition

The object SingX[CartSp op,sSet]\mathbf{Sing} X \in [CartSp^{op}, sSet] presents the abstractly defined fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos Π(X)\mathbf{\Pi}(X).

Proof

Using the Steenrod-Wockel approximation theorem this comes down to the same argument as for Euclidean topological ∞-groupoids. See Presentation of the path ∞-groupoid there.

Concrete objects – diffeological \infty-groupoids

This section is at

Geometry and structure sheaves

(…)

Cohesive \infty-groups

We discuss cohesive ∞-groups in SmoothGrpdSmooth \infty Grpd: smooth \infty-groups.

Lie groups

Let GG be a Lie group. Under the embedding SmoothMfdSmoothGrpdSmoothMfd \hookrightarrow Smooth \infty Grpd this is canonically identifed as a 0-truncated ∞-group object in SmoothGrpdSmooth \infty Grpd. Write BGSmoothGrpd\mathbf{B}G \in Smooth \infty Grpd for the corresponding delooping object.

Proposition

A fibrant presentation of the delooping object BG\mathbf{B}G in the projective local model structure on simplicial presheaves [CartSp smooth op,sSet] proj,loc[CartSp^{op}_{smooth}, sSet]_{proj, loc} is given by the simplicial presheaf that is the nerve of the one-object Lie groupoid

BG c:=(G*) \mathbf{B}G_c := (G \stackrel{\to}{\to} * )

regarded as a simplicial manifold and canonically embedded into simplicial presheaves:

BG c:UN(C (U,G)*). \mathbf{B}G_c : U \mapsto N(C^\infty(U,G) \stackrel{\to}{\to} *) \,.
Proof

The presheaf is clearly objectwise a Kan complex, being objectwise the nerve of a groupoid. It satisfies descent along good open covers {U i n}\{U_i \to \mathbb{R}^n\} of Cartesian spaces, because the descent \infty-groupoid [CartSp op,sSet](C({U i}),BG)[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}G) is GBund( n)GTrivBund( n)\cdots \simeq G Bund(\mathbb{R}^n) \simeq G TrivBund(\mathbb{R}^n): an object is a Cech 1-cocycle with coefficients in GG, a morphism a Cech coboundary, which yields the groupoid of GG-principal bundles over UU, which for the Cartesian space UU is however equivalent to the groupoid of trivial GG-bundles over UU.

To show that BG\mathbf{B}G is indeed the delooping object of GG it is sufficient (by the discussion at model structure on simplicial presheaves – homotopy limits) to compute the (∞,1)-pullback G*× BG*G \simeq * \times_{\mathbf{B}G} * in the global model structure [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}.

This is accomplished by the ordinary pullback of the fibrant replacement diagram

G N(G×Gp 1p 1p 2G) p 2 * N(G*) \array{ G &\to& N(G\times G \stackrel{\overset{p_1 \cdot p_2}{\to}}{\underset{p_1}{\to}} G) \\ \downarrow && \downarrow^{\mathrlap{p_2}} \\ * &\to& N(G \stackrel{\to}{\to} *) }

as discussed at universal principal ∞-bundle.

Strict Lie 2-groups

Let now G=Ξ[G 2G 1]G = \Xi[G_2 \to G_1] be a strict Lie 2-group coming from a smooth crossed module G 2δG 1G_2 \stackrel{\delta}{\to} G_1 with action α:G 1Aut(G 2)\alpha : G_1 \to Aut(G_2).

Proposition

A fibrant representative of BG\mathbf{B}G in [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov} is given by the crossed complex

Ξ[G 2G 1*]. \Xi[G_2 \to G_1 \stackrel{\to}{\to} *] \,.
Proof

As above for Lie groups.

Circle Lie nn-groups

Definition

Write equivalently

U(1)=S 1=/ U(1) = S^1 = \mathbb{R}/\mathbb{Z}

for the abelian Lie group called the circle group or 1-dimensional unitary group , regarded as a 0-truncated ∞-group object in SmoothGrpdSmooth\infty Grpd under the embedding SmoothMfdSmoothGrpdSmoothMfd \hookrightarrow Smooth \infty Grpd.

For nn \in \mathbb{N} the nn-fold delooping B nU(1)SmoothGrpd\mathbf{B}^n U(1) \in Smooth \infty Grpd we call the circle Lie (n+1)-group.

Write

U(1)[n]:=[0C (,U(1))00][CartSp op,Ch +] U(1)[n] := [\cdots \to 0 \to C^\infty(-,U(1)) \to 0 \to \cdots \to 0] \in [CartSp^{op}, Ch^+_\bullet]

for the chain complex of sheaves concentrated in degree nn on U(1)U(1).

Recall the right Quillen functor Ξ:[CartSp op,Ch +][CartSp op,sSet]\Xi : [CartSp^{op}, Ch_\bullet^+] \to [CartSp^{op}, sSet] from above.

Proposition

The simplicial presheaf Ξ(U(1)[n])[CartSp op,sSet]\Xi(U(1)[n]) \in [CartSp^{op}, sSet] is a fibrant representative in [CartSp op,sSet] proj,loc[CartSp^{op},sSet]_{proj,loc} of the circle Lie (n+1)(n+1)-group B nU(1)\mathbf{B}^n U(1).

Proof

First notice that since U(1)[n]U(1)[n] is fibrant in [CartSp smooth op,Ch ] proj[CartSp_{smooth}^{op}, Ch_\bullet]_{proj} we have that ΞU(1)[n]\Xi U(1)[n] is fibrant in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}. We may compute the homotopy pullback that defines the loop space object in the global model structure (by the discussion at model structure on simplicial presheaves – homotopy (co)limits) we may check the statement about the delooping in the global model structure.

Consider the global fibration resolution of the point inclusion *Ξ(U(1)[n])* \to \Xi(U(1)[n]) given by

Ξ[C (,U(1))IdC (,U(1))00] Ξ[C (,U(1))000]. \array{ \Xi [C^\infty(-,U(1)) \stackrel{Id}{\to} C^\infty(-,U(1)) \to 0 \to \cdots \to 0] \\ \downarrow \\ \Xi [C^\infty(-,U(1)) \to 0 \to 0 \to \cdots \to 0] } \,.

The underlying morphism of chain complexes is clearly degreewise surjective, hence a projective fibration, hence its image under Ξ\Xi is a projective fibration. Therefore the homotopy pullback in question is given by the ordinary pullback

Ξ[0C (,U(1))00] Ξ[C (,U(1))IdC (,U(1))00] Ξ[0000] Ξ[C (,U(1))000], \array{ \Xi[0 \to C^\infty(-,U(1)) \to 0 \to \cdots \to 0] &\to& \Xi [C^\infty(-,U(1)) \stackrel{Id}{\to} C^\infty(-,U(1)) \to 0 \to \cdots \to 0] \\ \downarrow && \downarrow \\ \Xi [0 \to 0 \to 0 \to \cdots \to 0] & \to & \Xi [C^\infty(-,U(1)) \to 0 \to 0 \to \cdots \to 0] } \,,

computed in [CartSp op,Ch ][CartSp^{op}, Ch_\bullet] and then using that Ξ\Xi is the right adjoint and hence preserves pullbacks. This shows that the loop object ΩΞ(U(1)[n])\Omega \Xi(U(1)[n]) is indeed presented by Ξ(U(1)[n1])\Xi (U(1)[n-1]).

Now we discuss the fibrancy of U(1)[n]U(1)[n] in the local model structure [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc}. We need to check that for all good open covers {U iU}\{U_i \to U\} of a Cartesian space UU we have that the mophism

C (U,U(1))[n][CartSp op,sSet](C({U i}),ΞU(1)[n]) C^\infty(U,U(1))[n] \to [CartSp^{op}, sSet](C(\{U_i\}), \Xi U(1)[n])

is an equivalence of Kan complexes, where C({U i})C(\{U_i\}) is the Cech nerve of the cover. Observe that the Kan complex on the right is that whose vertices are cocycles in degree-nn Cech cohomology (see there for details) with coefficients in U(1)U(1) and whose morphisms are coboundaries between these.

We proceed by induction on nn. For n=0n = 0 the condition is just that C (,U(1))C^\infty(-,U(1)) is a sheaf, which clearly it is. For general nn we use that since C({U i})C(\{U_i\}) is cofibrant, the above is the derived hom-space functor which commutes with homotopy pullbacks and hence with forming loop space objects, so that

π 1[CartSp op,sSet](C({U i}),Ξ(U(1)[n]))π 0[CartSp op,sSet](C({U i}),Ξ(U(1)[n1])) \pi_1 [CartSp^{op}, sSet](C(\{U_i\}), \Xi (U(1)[n])) \simeq \pi_0 [CartSp^{op}, sSet](C(\{U_i\}), \Xi (U(1)[n-1]))

by the above result on delooping. So we find that for all 0kn0 \leq k \leq n that π k[CartSp op,sSet](C({U i}),Ξ(U(1)[n]))\pi_k [CartSp^{op}, sSet](C(\{U_i\}), \Xi(U(1)[n])) is the Cech cohomology of UU with coefficients in U(1)U(1) in degree nkn-k. By standard facts about Cech cohomology (using the short exact sequence of abelian groups U(1)\mathbb{Z} \to U(1)\to \mathbb{R} and the fact that the cohomology with coefficients in \mathbb{R} vanishes in positive degree, for instance by a partition of unity argument) we have that this is given by the integral cohomology groups

π 0[CartSp op,sSet](C({U i}),Ξ(U(1)[n]))H n+1(U,) \pi_0 [CartSp^{op}, sSet](C(\{U_i\}), \Xi (U(1)[n])) \simeq H^{n+1}(U, \mathbb{Z})

for n1n \geq 1. For the contractible Cartesian space all these cohomology groups vanish.

So we have that Ξ(U(1)[n])(U)\Xi(U(1)[n])(U) and [CartSp op,sSet](C({U i}),ΞU(1)[n])[CartSp^{op}, sSet](C(\{U_i\}), \Xi U(1)[n]) both have homotopy groups concentrated in degree nn on U(1)U(1). The above looping argument together with the fact that U(1)U(1) is a sheaf also shows that the morphism in question is an isomorphism on this degree-nn homotopy group, hence is indeed a weak homotopy equivalence.

Remark

In the equivalent presentation of SmoothGrpdSmooth\infty Grpd by simplicial presheaves on all of SmoothMfd the objects ΞU(1)[n]\Xi U(1)[n] are far from being locally fibrant. Instead, their local fibrant replacements are given by the nn-stacks of circle n-bundles.

Simplicial Lie groups

Every connected object XLieGrpdX \in \infty Lie Grpd is – by definition – the delooping X=BGX = \mathbf{B}G of a Lie ∞-group G=ΩXG = \Omega X, its loop space object formed in LieGrpd\infty LieGrpd. Since the discussion of group objects, loop space objects etc. involves only finite (∞,1)-limits and ∞-stackification preserves these, this may be discussed in the (∞,1)-category of (∞,1)-presheaves on CartSp. Since there (,1)(\infty,1)-limits are computed objectwise, an ∞-group object GG in LieGrpd\infty LieGrpd is modeled by a (∞,1)-presheaf with values in ∞-groups in ∞Grpd.

By standard results on Models for group objects in ∞Grpd the latter may equivalently be modeled by simplicial groups. A simplicial group is possibly weak ∞-groupoid equipped with a strict group object structure. While strict ∞-groupoids with weak group object structure do not model all ∞-groups, weak \infty-groupoids with strict group structure do.

There is a good supply of standard results for and constructions with simplicial groups which makes this model useful for applications.

Cohomology and principal \infty-bundles

We discuss the intrinsic cohomology and pricipal ∞-bundles in SmoothGrpdSmooth \infty Grpd.

Cohomology with constant coefficients

Definition

Let AA \in ∞Grpd be any discrete ∞-groupoid. Write A|A| \in Top for its geometric realization. For XX any topological space, the nonabelian cohomology of XX with coefficients in AA is the set of homotopy classes of maps XAX \to |A|

H Top(X,A):=π 0Top(X,A). H_{Top}(X,A) := \pi_0 Top(X,|A|) \,.

We say Top(X,A)Top(X,|A|) itself is the cocycle ∞-groupoid for AA-valued nonabelian cohomology on XX.

Similarly, for X,ASmoothGrpdX, \mathbf{A} \in Smooth \infty Grpd two smooth \infty-groupoids, write

H Smooth(X,A):=π 0SmoothGrpd(X,A) H_{Smooth}(X,\mathbf{A}) := \pi_0 Smooth\infty Grpd(X,\mathbf{A})

for the intrinsic cohomology of SmoothGrpdSmooth \infty Grpd on XX with coefficients in A\mathbf{A}.

Proposition

Let AA \in ∞Grpd, write DiscASmoothGrpdDisc A \in Smooth \infty Grpd for the corresponding discrete smooth ∞-groupoid. Let XSmoothMfdiSmoothGrpdX \in SmoothMfd \stackrel{i}{\hookrightarrow} Smooth \infty Grpd be a paracompact topological space regarded as a 0-truncated Euclidean-topological \infty-groupoid.

We have an isomorphism of cohomology sets

H Top(X,A)H Smooth(X,DiscA) H_{Top}(X,A) \simeq H_{Smooth}(X,Disc A)

and in fact an equivalence of cocycle ∞-groupoids

Top(X,A)SmoothGrpd(X,DiscA). Top(X,|A|) \simeq Smooth\infty Grpd(X, Disc A) \,.

More generally, for X SmoothMfd Δ opX_\bullet \in SmoothMfd^{\Delta^{op}} presenting an object in SmoothGrpdSmooth \infty Grpd we have

H Smooth(X ,DiscA)H Top(X,A). H_{Smooth}(X_\bullet, Disc A) \simeq H_{Top}(|X|, |A|) \,.
Proof

This follows from the (ΠDisc)(\Pi \dashv Disc)-adjunction and the above proposition asserting that Π(X )X |\Pi(X_\bullet)| \simeq |X_\bullet| is the ordinary geometric realization of simplicial topological spaces.

GG-Principal bundles for GG a Lie group

Let GG be a Lie group, regarded as a group object in SmoothGrpdSmooth \infty Grpd as in the above discussion.

The universal G-principal bundle is a replacement of the point inclusion *BG* \to \mathbf{B}G by a fibration EGBG\mathbf{E}G \to \mathbf{B}G.

For GG an ordinary group one model for this is given by the Lie groupoid

EG=(G×Gp 1G), \mathbf{E}G = (G\times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}} G) \,,

which is the action groupoid G//GG//G of GG acting on itself.

One noteworthy aspect of this object is that it is itself groupal, in fact itself a Lie strict 2-group in a way that is compatible with the canonical inclusion GEGG \to \mathbf{E}G: it is an example of a groupal model for universal principal ∞-bundles.

To emphasize this group structure, we also write INN(G)INN(G) for this groupoid, following SchrRob. The corresponding crossed module is

[INN(G)]=(GIdG). [INN(G)] = (G \stackrel{Id}{\to} G) \,.

Accordingly we write BEG\mathbf{B E}G or BINN(G)\mathbf{B}INN(G) for the 2-groupoid given by the Lie crossed complex

(GIdG*). (G \stackrel{Id}{\to}G \stackrel{\to}{\to} * ) \,.

The following proposition asserts that the general definition of principal ∞-bundles in an (∞,1)-topos H\mathbf{H} applied to the coefficient object BG\mathbf{B}G in H=LieGrpd\mathbf{H} = \infty LieGrpd for GG a Lie group does reprpduce the ordinary notion of GG-principal bundles.

Proposition

Let XX be a paracompact smooth manifold. The ordinary first nonabelian cohomology of XX with coefficients in GG coincided with the intrinsic cohomology of LieGrpd\infty Lie Grpd

H 1(X,G)π 0LieGrpd(X,BG) H^1(X,G) \simeq \pi_0 \infty LieGrpd(X, \mathbf{B}G)

and the GG-principal bundle PXP \to X corresponding to a cocycle XBGX \to \mathbf{B}G in LieGrpd\infty LieGrpd is indeed the homotopy fiber of that cocycle.

Proof

By the discussion at model structure on simplicial presheaves we have that a cofibrant resolution for XX in the model [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov} for LieGrpd\infty LieGrpd is civen by the Cech nerve C{U i}C\{U_i\} of a good open cover {U iX}\{U_i \to X\}. It follows that π 0LieGrpd(X,BG)\pi_0 \infty LieGrpd(X,\mathbf{B}G) is the Cech cohomology of XX with coefficients in GG (see there for details).

Concretely, a cocycle

g:C({U i})BG g : C(\{U_i\}) \to \mathbf{B}G

is canonically identified with a transition function

g: i,jU iU jG g : \coprod_{i,j} U_i \cap U_j \to G

satisfying on U iU jU kU_i \cap U_j \cap U_k the cocycle condition g ijg jk=g ikg_{i j} g_{j k} = g_{i k}.

From this we can compute the homotopy fiber of g:C({U i})BGg : C(\{U_i\}) \to \mathbf{B}G by forming the ordinary pullback of the fibrant replacement EGBG\mathbf{E}G \to \mathbf{B}G of the point inclusion *BG* \to \mathbf{B}G, where mathbEG=BG I× BG*\mathb{E}G = \mathbf{B}G^{I} \times_{\mathbf{B}G} * is the smooth groupoid

EG={ g 1 g 2 * h *g 1,g 2,hG,g 2=hg 1}. \mathbf{E}G = \left\{ \array{ && \bullet \\ & {}^{\mathllap{g_1}}\swarrow && \searrow^{\mathrlap{g_2}} \\ * &&\stackrel{h}{\to}&& * } \;\;| \;\; g_1,g_2,h \in G, g_2 = h g_1 \right\} \,.

From this we find the pullback P^\hat P in

P^ EG C({U i}) g BG \array{ \hat P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}G }

to be the smooth Lie groupoid

P^=( i,jU iU j×Gp 1×P 3p 2×g i,jp 3 iU i×G) \hat P = \left( \coprod_{i,j} U_i\cap U_j \times G \stackrel{\overset{p_2 \times g_{i,j} \cdot p_3}{\to}}{\underset{p_1 \times P_3}{\to}} \coprod_{i} U_i \times G \right)

i.e.

P^={ g 1 g 2 * g ij(x) * (x,i) (x,j)}. \hat P = \left\{ \array{ && \bullet \\ & {}^{\mathllap{g_1}}\swarrow && \searrow^{\mathrlap{g_2}} \\ * &&\stackrel{g_{i j}(x)}{\to}&& * \\ \\ (x,i) &&\to&& (x,j) } \right\} \,.

The evident projection P^P\hat P \to P is objectwise a surjective and full and faithful functor.

Cohomology of Lie groups

We consider the cohomology in H\mathbf{H} of smooth delooping groupoids BG\mathbf{B}G for GG an ordinary Lie group. This is a form of group cohomology for Lie groups.

H n(G,A):=π 0H(BG,B nA). H^n(G,A) := \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A) \,.

We discuss how this relates to other definitions of Lie group cohomology in the literature.

Definition

(naive Lie group cohomology)

For GG a Lie group and AA an abelian Lie group and nn \in \mathbb{N} define H rest n(G,A)H^n_{rest}(G,A) to be the group of equivalence classes of cocycles given by smooth functions G n ×AG^\times_n \to A by coboundatries given by smooth functions G × n1AG^{\times_{n-1}} \to A subject to the usual relations.

Observe that with BG=G × \mathbf{B}G = G^{\times_\bullet} regarded as an object of sPSh(CartSp)sPSh(CartSp), this is

H rest n(G,A)=π 0sPSh(BG,B nA). H^n_{rest}(G,A) = \pi_0 sPSh(\mathbf{B}G, \mathbf{B}^n A) \,.

Written this way it is evident that this definition misses to take into account any cofibrant replacement of BG\mathbf{B}G.

A more refined definition of cohomology of Lie groups has been given by (Segal), which was later rediscovered by (Brylinski), following (Blanc). A review is in section 4 of (Schommer-Pries).

Definition

(differential Lie group cohomology)

Let GG be a paracompact Lie group and AA an abelian Lie group.

For eack kk \in \mathbb{N} we can pick a good open cover {U i kG × kiI k}\{U^{k}_{i} \to G^{\times_k}| i \in I_k\} such that

  • the index sets arrange themselves into a simplicial set I:[k]I kI : [k] \mapsto I_k;

  • and for d j(U i k)d_j(U^k_i) and s j(U i k)s_j(U^k_i) the images of the face and degeneracy maps of G ×G^{\times\bullet} we have

    d j(U i k)U d j(i) k1 d_j(U^k_i) \subset U^{k-1}_{d_j(i)}

    and

    s j(U i k)U s j(i) k+1. s_j(U^k_i) \subset U^{k+1}_{s_j(i)} \,.

For instance start with a good open cover {U i 1G}\{U^1_i \to G\} and define a good open cover {U i 0i 1i 2 2}\{U^2_{i_0 i_1 i_2}\} of G×GG \times G by U i 0i 1i 2 2:=d 0 *U i 0 1d 1 *U i 1 1d 2 *U i 2 1U^2_{i_0 i_1 i_2} := d_0^* U^1_{i_0} \cap d_1^* U^1_{i_1} \cap d_2^* U^1_{i_2}. And so on.

Then the differentiable Lie group cohomology H diffr (G,A)H^\bullet_{diffr}(G,A) of GG with coefficients in AA is the cohomology of the total complex of the Cech double complex C (U i 0,,i ,A)C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A) whose differentials are the alternating sums of the face maps of G × G^{\times_\bullet} and of the Cech nerves, respectively:

H diff n(G,A):=H nTotC (U i 0,,i ,A) H^n_{diff}(G,A) := H^n Tot C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)

This is (Brylinski, definition 1.1).

As discussed there, this is equivalent to other definitions, notably to a definition given earlier in (Segal).

Observation

There is a natural map

H restr n(G,A)H diffr n(G,A) H^n_{restr}(G,A) \to H^n_{diffr}(G,A)

obtained by pulling back globally defined cocycles and coboundaries to good covers.

We can understand this differentiable Lie group cohomology in terms of maps out of a certain resolution of BG\mathbf{B}G in sPSh(CartSp) proj,covsPSh(CartSp)_{proj,cov}:

Proposition

For {U i }\{U^\bullet_{i_\bullet}\} a system of good open covers as above, we obtain a simplicial diagram of Cech nerves

C:[k]C({U i k}) C : [k] \mapsto C(\{U^k_{i}\})

which is degreewise a cofibrant resolution on sPSh(CartSp) proj,covsPSh(CartSp)_{proj,cov} of G × nG^{\times_n}. Its totalization coend is connected by a zig-zag of weak equivalences in sPSh(CartSp) proj,covsPSh(CartSp)_{proj,cov} to BG\mathbf{B}G

BG [k]Δ[k]C({U i k}) \mathbf{B}G \stackrel{\simeq}{\leftarrow} \stackrel{\simeq}{\to} \int^{[k]} \Delta[k] \cdot C(\{U^k_i\})

and we have

H diffr n(G,A)=π 0sPSh(CartSp)( [k]Δ[k]C({U i k}),B nA). H^n_{diffr}(G,A) = \pi_0 sPSh(CartSp)(\int^{[k]} \Delta[k] \cdot C(\{U^k_i\}), \mathbf{B}^n A) \,.

The proof of this will also show the following

Proposition

Write H (G,A):=π 0Sh (,1)(CartSp)(BG,B nA)H^\bullet(G,A) := \pi_0 Sh_{(\infty,1)}(CartSp)(\mathbf{B}G, \mathbf{B}^n A) for the intrinsic cohomology of BG\mathbf{B}G regarded as an object of the (,1)(\infty,1)-topos of \infty-Lie groupoids.

There is a natural morphism

H diffr n(G,A)H n(G,A). H^n_{diffr}(G,A) \to H^n(G,A) \,.
Proof

Since B nA\mathbf{B}^n A does satisfy descent with respect to good open covers of Cartesian spaces (every (n1)(n-1) AA-bundle gerbe over an n\mathbb{R}^n is trivializable), to compute the intrinsic cohomology we have to find a cofibrant replacement for BG\mathbf{B}G.

A cofibrant replacement of any paracompact manifold XX in [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov} is given by the Cech nerve C({U i})XC(\{U_i\}) \stackrel{\simeq}{\to} X of a good open cover {U iX}\{U_i \to X\}, because this is evidently a local epimorphism as described at model structure on simplicial presheaves - Cech localization.

Therefore from a choice of compatible families of open covers {U i kG ×k}\{U^k_i \to G^{\times k}\} as in the definition of differentiable group cohomology above, we obtain cofibrant replacements

C({U i k})G × k C(\{U^k_i\}) \stackrel{\simeq}{\to} G^{\times_k}

for each kk, and arranging themselves into a simplicial simplicial presheaf

C({U i }):Δ op[CartSp op,sSet], C(\{U^\bullet_i\}) : \Delta^{op} \to [CartSp^{op}, sSet] \,,

which is cofibrant in the injective model structure on functors [Δ op,[CartSp op,sSet] proj,cov] inj[\Delta^{op}, [CartSp^{op}, sSet]_{proj,cov}]_{inj}.

Observe that the fact that this is inded a simplicial diagram in sPSh(CartSp)sPSh(CartSp) is due to the extra compatibility condition in the above definition of differentiable Lie group cohomology.

Notice from the discussion at model structure on simplicial presheaves that we have canonically another cofibrant replacement Q(G × n)Q (G^{\times_n}) of G × nG^{\times_n} in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} which is functorial and has the special property that the diagonal

[k]ΔΔ[k]Q(G × k)BG \int^{[k] \in \Delta} \Delta[k] \cdot Q(G^{\times_k}) \stackrel{\simeq}{\to} \mathbf{B}G

is a cofibrant replacement of BG\mathbf{B}G. Notice that in [Δ op,[CartSp op,sSet] proj] inj[\Delta^{op}, [CartSp^{op}, sSet]_{proj}]_{inj} we therefore have a zig-zag of weak equivalences

C({U i })Q(G × ) C(\{U^\bullet_i\}) \stackrel{\simeq}{\to} \stackrel{\simeq}{\leftarrow} Q(G^{\times_\bullet})

between cofibrant replacements of G × G^{\times_\bullet}.

Write Δ :ΔsSet:[k]N(Δ/[k])\mathbf{\Delta}^\bullet : \Delta \to sSet : [k] \mapsto N(\Delta/[k]) for the standard Bousfield-Kan cofibrant replacement of the point, hence also of Δ\Delta, in [Δ,sSet Quillen] proj[\Delta, sSet_{Quillen}]_{proj} (more discussion of this is at homotopy colimit).

Then by the fact that the coend

()():[Δ,sSet Quillen] proj×[Δ op,[CartSp op,sSet Quilen] proj,cov] inj[CartSp op,sSet Quillen] proj,cov \int (-)\otimes (-) : [\Delta,sSet_{Quillen}]_{proj} \times [\Delta^{op}, [CartSp^{op}, sSet_{Quilen}]_{proj,cov}]_{inj} \to [CartSp^{op}, sSet_{Quillen}]_{proj,cov}

over the tensoring sSet Quillen×[CartSp op,sSet Quillen] proj,cov[CartSp op,sSet Quillen] proj,cov\otimes sSet_{Quillen} \times [CartSp^{op}, sSet_{Quillen}]_{proj,cov} \to [CartSp^{op}, sSet_{Quillen}]_{proj,cov} is a left Quillen bifunctor (as discussed there), we have weak equivalences in [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov}

BG= [k]ΔΔ[k]G × n [k]ΔΔ[k]Q(G × k)) [k]ΔΔ[k]Q(G × k)) [k]ΔΔ[k]C({U i k})) [k]ΔΔ[k]C({U i k})) \mathbf{B}G = \int^{[k] \in \Delta} \Delta[k] \cdot G^{\times_n} \stackrel{\simeq}{\leftarrow} \int^{[k] \in \Delta} \Delta[k] \cdot Q(G^{\times_k})) \stackrel{\simeq}{\leftarrow} \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot Q(G^{\times_k})) \stackrel{\simeq}{\to} \stackrel{\simeq}{\leftarrow} \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot C(\{U^k_i\})) \stackrel{\simeq}{\to} \int^{[k] \in \Delta} \Delta[k] \cdot C(\{U^k_i\}))

where

  • the first one is Dugger’s cofibrant replacement mentioned above, even a global weak equivalence;

  • the second one is objectwise the Bousfield-Kan map, hence also even a global weak equivalence;

  • the zig-zag is the image under the left Quillen functor [k]ΔΔ[k]()\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot (-) (using that mathfΔ\mathf{\Delta} is cofibrant) of the above zig-zag of weak equivalences between cofibrant objects;

  • the last one is again the global weak equivalence coming objectwise from the Bousfield-Kan map.

This shows the first claim, that BG kΔ[k]C({U i k})\mathbf{B}G \stackrel{\simeq}{\leftarrow}\stackrel{\simeq}{\rightarrow} \int^{k} \Delta[k] \cdot C(\{U^k_i\}).

Moreover, again by the left Quillen bifunctor property of ()()\int(-)\cdot(-) we have that [k]ΔΔ[k]C({U i k}))\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot C(\{U^k_i\})) is cofibrant. Therefore the intrinsic cohomology H n(G,A)H^n(G,A) is

π 0H(BG,B nA)=[CartSp op,sSet] proj,cov( [k]ΔΔ[k]C({U i k})),B nA). \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A) = [CartSp^{op}, sSet]_{proj,cov} \left( \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot C(\{U^k_i\})) , \mathbf{B}^n A \right) \,.

By pullback along the Bousfield-Kan map Δ[k]Δ[k]\mathbf{\Delta[k]} \to \Delta[k] we have a natural morphism

π 0sPSh( [k]Δ[k]C({U i k}),B nA)H n(G,A). \pi_0 sPSh(\int^{[k]} \Delta[k]\cdot C(\{U^k_i\}), \mathbf{B}^n A) \to H^n(G,A) \,.

It remains to show that the set on the left is the differentiable Lie group cohomology.

For that observe that B nA\mathbf{B}^n A is in the image of the Dold-Kan correspondence Ξ:[CartSp op,Ch ][CartSp op,sAb]U[CartSp op,sSet]\Xi : [CartSp^{op}, Ch_\bullet] \to [CartSp^{op}, sAb] \stackrel{U}{\to} [CartSp^{op}, sSet] of the chain-complex valued presheaf A[n]=A [n]A[n] = A \otimes_\mathbb{Z} \mathbb{Z}[n] to reduce the computation of this simplicial hom-complex to that of a cochain complex:

sPSh(K ,ΞA[n]) UsSet(K (U),UΞA(U)[n]) UsAb(K (U),ΞA(U)[n]) UCh (C (K (U)),A(U) [n]) UCh ([n],C (Set(K (U),A(U)))) Ch ([n], UC (Set(K (U),A(U))) Ch ([n],C (PSh(K ,A)). \begin{aligned} sPSh(K_\bullet, \Xi A[n]) & \simeq \int_U sSet(K_\bullet(U) , U \Xi A(U)[n]) \\ & \simeq \int_U sAb(\mathbb{Z} K_\bullet(U), \Xi A(U)[n]) \\ &\simeq \int_U Ch_\bullet( C_\bullet(\mathbb{Z} K_\bullet(U)), A(U)\otimes_{\mathbb{Z}}\mathbb{Z}[n] ) \\ & \simeq \int_U Ch^\bullet( \mathbb{Z}[n], C^\bullet (Set(K_\bullet(U),A(U)))) \\ & \simeq Ch^\bullet (\mathbb{Z}[n], \int_U C^\bullet(Set(K_\bullet(U),A(U))) \\ & \simeq Ch^\bullet(\mathbb{Z}[n], C^\bullet( PSh(K_\bullet, A) ) \end{aligned} \,.

Setting here K = [k]Δ[k]C({U i k})K_\bullet = \int^{[k]} \Delta[k] \cdot C(\{U^k_i\}) and using the Eilenberg-Zilber theorem to identify the Moore complex of the diagonal to the total complex of the double complex of Moore complexes, the claim follows.

Proposition

For GG a Lie group and AA either

  1. a discrete abelian group;

  2. the additive Lie group of real numbers \mathbb{R};

the intrinsic cohomology of GG in SmoothGrpdSmooth\infty Grpd coincides with the refined Lie group cohomology of (Segal/Brylinski)

H SmoothGrpd n(BG,A)H Segal n(G,A). H^n_{Smooth\infty Grpd}(\mathbf{B}G, A) \simeq H^n_{Segal}(G,A) \,.

In particular we have in general

H SmoothGrpd n(BG,)H top n(BG,) H^n_{Smooth\infty Grpd}(\mathbf{B}G, \mathbb{Z}) \simeq H^{n}_{top}(B G, \mathbb{Z})

and for GG compact also

H SmoothGrpd n(BG,U(1))H top n+1(BG,). H^n_{Smooth\infty Grpd}(\mathbf{B}G, U(1)) \simeq H^{n+1}_{top}(B G, \mathbb{Z}) \,.
Proof

The first statement is a special case of the above proposition about cohomology with constant coefficients.

The second statement is a special case of the more general statement that is proven at SynthDiff∞Grpd.

The last statement follows then from the observation (Brylinski) that H diffr n(G,)H naive n(G,)H^n_{diffr}(G,\mathbb{R}) \simeq H^n_{naive}(G,\mathbb{R}) and the classical result (Blanc) that H naive n(G,)=0H_{naive}^n(G,\mathbb{R}) = 0 in positive degree, and using the fiber sequence induced from the short exact sequence of abelian groups /\mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z}.

Smooth principal 1-bundles and 2-bundles

Proposition

Let GG be a Lie group and BGH:=LieGrpd\mathbf{B}G \in \mathbf{H} := \infty LieGrpd its delooping as discussed above. Let XX be a paracompact smooth manifold. Then

H(X,BG)GBund(X). \mathbf{H}(X,\mathbf{B}G) \simeq G Bund(X) \,.

is equivalent to the groupoid of smooth GG-principal bundles on XX. In particular

H Smooth 1(X,G)π 0H(X,BG) H^1_{Smooth}(X,G) \simeq \pi_0 \mathbf{H}(X,\mathbf{B}G)

is the nonabelian smooth Cech cohomology of XX with coefficients in GG.

Analogously, let GG be a strict Lie 2-group such that π 0G\pi_0 G is a smooth manifold and G 0π 0GG_0 \to \pi_0 G is a submersion. Then we have that

H(X,BG)GBund(X) \mathbf{H}(X, \mathbf{B}G) \simeq G Bund(X)

is the 2-groupoid of GG-principal 2-bundles, whose connected components are given by first smooth nonabelian Cech cohomology H Smooth 1(X,G)H^1_{Smooth}(X,G).

Proof

The first case is a special case of the second, it is sufficient to consider that.

First we argue that BG\mathbf{B}G is fibrant in [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj, loc}, hence that for {U i n}\{U_i \to \mathbb{R}^n\} an open cover we have a weak equivalence

BC ( n,G)[CartSp op,sSet](C(U),BG), \mathbf{B}C^\infty(\mathbb{R}^n, G) \stackrel{\simeq}{\to} [CartSp^{op}, sSet](C(U), \mathbf{B}G) \,,

for C(U)C(U) the Cech nerve of the good cover. Since the site CartSp with good open cover coverage is a Verdier site, it follows by the statements discussed at hypercover that every hypercover has a refinement by a split hypercover, which is a cofibrant resolution in [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc}. But also C(U)XC(U) \to X is a cofibrant resolution. Hence by the existence of the global model structure and using that BG\mathbf{B}G is fibrant in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}, it follows that

π 0[CartSp op,sSet](C(U),BG)H 1( n,G) \pi_0[CartSp^{op}, sSet](C(U), \mathbf{B}G) \simeq H^1(\mathbb{R}^n, G)

is nonabelian degree 1 Cech cohomology on n\mathbb{R}^n with coefficients in the 2-group GG. By (NikolausWaldorf, prop. 4.1) this is the singleton set on the contractible n\mathbb{R}^n. This shows that the descent morphism in quesion is an isomorphism on π 0\pi_0.

Next, π 1\pi_1 on the right is similarly the set of equivalence classes of G 0//(G 0G 1)G_0//(G_0 \ltimes G_1)-groupoid principal bundles. The underlying G 0×G 1G_0 \times G_1-principal bundle on the contractible n\mathbb{R}^n is trivializable, hence π 1([CartSp op,sSet](C(U),B g))C ( n,G 0)\pi_1([CartSp^{op}, sSet](C(U), \mathbf{B}^g)) \simeq C^\infty(\mathbb{R}^n , G_0), and so we have also an isomorphism on π 1\pi_1. Finally the isomorphism on π 2\pi_2 is clear.

Now for XX an arbitrary paracompact smooth manifold and UXU \to X a good open cover, we have again that C(U)XC(U) \to X is a cofibrant resolution in the local model structure, hence hence by the above it follows that

SmoothGrpd(X,BG)[CartSp op,sSet](C(U),BG). Smooth \infty Grpd(X, \mathbf{B}G) \simeq [CartSp^{op}, sSet](C(U), \mathbf{B}G) \,.

By the same argument about hypercovers on Verdier sites as above, it follows that the connected components of this are

π 0SmoothGrpd(X,BG)H Smooth 1(X,G). \pi_0 Smooth \infty Grpd(X, \mathbf{B}G) \simeq H^1_{Smooth}(X, G) \,.

Twisted cohomology

We discuss implementation in SmoothGrpdSmooth \infty Grpd of the notion of twisted cohomology in a cohesive (∞,1)-topos.

Concordance

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van Kampen theorem

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Paths and geometric Postnikov towers

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Universal coverings and geometric Whitehead towers

Given a Lie group GG, we may regard it either as a topological group and form the classifying space G\mathcal{B}G \in Top \simeq ∞Grpd, or we may form its delooping Lie groupoid BGLieGrpd\mathbf{B}G \in \infty LieGrpd, as discussed in the section on Lie groups above. By the discussion in the section on geometric realization we have that under the canonical geometric realization functor

Π:LieGrpdGrpdTop \Pi : \infty LieGrpd \to \infty Grpd \simeq Top

we have

GΠ(BG). \mathcal{B}G \simeq \Pi(\mathbf{B}G) \,.

Analogous statements hold for \infty-Lie groups. For instance by the same result we have that the circle n-groupoid B nU(1)\mathbf{B}^n U(1) maps to the Eilenberg-MacLane space

Π(B nU(1))K(,n+1). \Pi(\mathbf{B}^n U(1)) \simeq K(\mathbb{Z}, n+1) \,.

This means that given a cocycle

c:GK(,n+1) c : \mathcal{B}G \to K(\mathbb{Z},n+1)

in integral cohomology (defining a characteristic class), it is of interest to ask of this morphism also lifts through Π\Pi to a morphism

c:BGB nU(1). \mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1) \,.

Classes of examples of such lifts have been described in the section on integration of ∞-Lie algebra cocycles.

By general reasoning, the extensions/principal ∞-bundles classified by such cocycles are their homotopy fiber

BG^ * BG c B nU(1) \array{ \mathbf{B}\hat G &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n U(1) }

in LieGrpd\infty LieGrpd and

G^ * G c K(,n) \array{ \mathcal{B}\hat G &\to& * \\ \downarrow && \downarrow \\ \mathcal{B}G &\stackrel{c}{\to}& K(\mathbb{Z}, n) }

in TopGrpdTop \simeq \infty Grpd. By the theorem discussed in the section on Geometric realization we have that these two homotopy fibers correspond to each other under Π:LieGrpdGrpd\Pi : \infty LieGrpd \to \infty Grpd:

Π(BG^)G^. \Pi(\mathbf{B}\hat G) \simeq \mathcal{B}\hat G \,.

So this means with a smooth lift of a cocycle, we automatically obtain the corresponding smooth lift of the extension that it classifies (assuming that the simplicial topological group GG is well-sectioned). This is notably useful for finding smooth lifts of Whitehead towers. Examples of this we discuss in the following.

The smooth Whitehead tower of the orthogonal group

Let OO denote the orthogonal group, regarded as a Lie group. We discuss steps of the Whitehead tower of OO refined to LieGrpd\infty LieGrpd.

First Stiefel-Whitney class
Proposition

A lift of the first Stiefel-Whitney class w 1:OK( 2,1)TopGrpdw_1 : \mathcal{B}O \to K(\mathbb{Z}_2 ,1) \in Top \simeq \infty Grpd to LieGrpd\infty LieGrpd is given by the morphism

w 1:BOB 2 \mathbf{w}_1 : \mathbf{B}O \to \mathbf{B}\mathbb{Z}_2

of Lie groupoids that is the delooping of the group homomorphism that sends gOg \in O to its signature σ(g) 2\sigma(g) \in \mathbb{Z}_2, where σ(g)=1\sigma(g) = 1 if gg is in the connected component of OO (is orientation preserving as an orthogonal map) and σ(g)=1\sigma(g) = -1 otherwise.

We may compute the principal bundle in LieGrpd\infty LieGrpd classified by w 1\mathbf{w}_1 as the ordinry pullback of E 2B 2\mathbf{E} \mathbb{Z}_2 \to \mathbf{B}\mathbb{Z}_2, where E 2\mathbf{E}\mathbb{Z}_2 is the groupoid 2× 2p 1 2ovserset\mathbb{Z}_2 \times \mathbb{Z}_2 \stackrel{\ovserset{\cdot}{\to}}{\underset{p_1}{\to} \mathbb{Z}_2}.

The resulting pullback groupoid

Q E 2 BO w 1 B 2 \array{ Q &\to& \mathbf{E}\mathbb{Z}_2 \\ \downarrow && \downarrow \\ \mathbf{B}O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B}\mathbb{Z}_2 }

has two objects, s=+1,1s = +1,-1 and morphisms of the form sgσ(g)ss \stackrel{g}{\to} \sigma(g)\cdot s for gOg \in O. The smooth structure is that induced from OO.

Evidently the canonical embedding BSOQ\mathbf{B} SO \to Q which sends a morphism h\bullet \stackrel{h}{\to} \bullet for hSOh \in SO an element of the special orthogonal group is an essentially surjective functor and a full and faithful functor and hence an equivalence of \infty-Lie groupoids.

This establishes the (∞,1)-pullback diagram

BSO * BO w 1 BZ 2 \array{ \mathbf{B} SO &\to& * \\ \downarrow &\swArrow& \downarrow \\ \mathbf{B} O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B}\mathbf{Z}_2 }

in LieGrpd\infty LieGrpd, covering under Π\Pi the pullback

SO * O w 1 Z 2. \array{ \mathcal{B} SO &\to& * \\ \downarrow &\swArrow& \downarrow \\ \mathcal{B} O &\stackrel{w_1}{\to}& \mathcal{B}\mathbf{Z}_2 } \,.
Second Stiefel-Whitney class

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BSpin * BSO w 2 B 2Z 2 \array{ \mathbf{B} Spin &\to& * \\ \downarrow &\swArrow& \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbf{Z}_2 }

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First fractional Pontryagin class

Let μ 3CE(𝔰𝔬(n))\mu_3 \in CE(\mathfrak{so}(n)) be the Lie algebra 3-cocycle ,[,]\langle -,[-,-]\rangle normalized such its left-invariant continuation to a differential 3-form on Spin(n)Spin(n) is the image in deRham cohomology of a generator of H 3(Spin,)H^3(Spin, \mathbb{Z}).

Proposition

The integration of this cocycle

12p 1:=exp(μ):BSpinB 3U(1) \frac{1}{2}\mathbf{p}_1 := \exp(\mu) : \mathbf{B}Spin \to \mathbf{B}^3 U(1)

is a smooth refinement of the first fractional Pontryagin class 12p 1:Spin 4\frac{1}{2} p_1 : \mathcal{B}Spin \to \mathcal{B}^4 \mathbb{Z}.

Proof

By the discussion at Chern-Simons circle 3-bundle.

Therefore the homotopy fiber

BString * BSpin 12p 1 B 3U(1) \array{ \mathbf{B} String &\to& * \\ \downarrow &\swArrow& \downarrow \\ \mathbf{B} Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) }

is a smooth model of the string group.

Proposition

Indeed, this homotopy fiber is given by the Lie string 2-group.

Proof

Compute the homotopy pullback as the ordinary limit over

EB 2U(1) cosk 3exp(𝔰𝔬) 12p 1 B 3U(1) BSpin. \array{ &\to& \mathbf{E} \mathbf{B}^2 U(1) \\ && \downarrow \\ \mathbf{cosk}_3 \exp(\mathfrak{so}) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}Spin } \,.

By inspection, this gives the Lie 2-group whose

  • objects are based paths in GG; the product is by concatenation of such paths;

  • morphisms are equivalence classes of based surfaces in GG labeled by some cU(1)c \in U(1); where two of these are equivalence if there is a ball cobounding them such that the integral of μ\mu over this ball accounts for the difference in the two labels.

Here me may equivalently take thin homotopy-classes of paths and surfaces. This is indeed one of the three incarnations of the string 2-group as a strict 2-group.

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Second fractional Pontryagin class
BFivebrane * BString 16p 2 B 7U(1) \array{ \mathbf{B} Fivebrane &\to& * \\ \downarrow &\swArrow& \downarrow \\ \mathbf{B} String &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 U(1) }

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Flat \infty-connections and local systems

We discuss the intrinsic flat cohomology in SmoothGrpdSmooth \infty Grpd.

Cohomology with constant coefficients

Theorem

Let XX be a paracompact smooth manifold regarded as a 0-truncated smooth \infty-groupoid under the embedding SmoothMfdSmoothGrpdSmoothMfd \hookrightarrow Smooth \infty Grpd. Let AA \in ∞Grpd be any ∞-groupoid and DiscASmoothGrpdDisc A \in Smooth \infty Grpd the coresponding discrete ∞-groupoid.

We have an equivalence of cocycle ∞-groupoids

SmoothGrpd(X,LConstA)Top(X,A) Smooth\infty Grpd(X, LConst A) \simeq Top(X, |A|)

and hence in particular an isomorphism on cohomology

H Top(X,A)π 0SmoothGrpd(X,DiscA). H_{Top}(X,A) \simeq \pi_0 Smooth \infty Grpd(X, Disc A) \,.

This also means that

H smooth,flat(X,A)H Top(X,ΓA). H_{smooth,flat}(X,A ) \simeq H_{Top}(X, \Gamma A) \,.
Proof

Same proof as of the analogous statement at ETop∞Grpd.

With coefficients in B nU(1){\mathbf{B}^n U(1)} or B n\mathbf{B}^n \mathbb{R}

Let B nU(1)\mathbf{B}^n U(1) be the circle (n+1)(n+1)-Lie group as discussed above. Recall the notation and model category presentations as discussed there.

Proposition

For n1n \geq 1 a fibration presentation in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} of the canonical morphism B nU(1)B nU(1)\mathbf{\flat} \mathbf{B}^n U(1) \to \mathbf{B}^n U(1) in SmoothGrpdSmooth \infty Grpd is given by the image under Ξ:[CartSp op,Ch +][CartSp op,sSet]\Xi : [CartSp^{op}, Ch_\bullet^+] \to [CartSp^{op}, sSet] of

C (,U(1)) d dR Ω 1() d dR d dR Ω cl n() C (,U(1)) 0 0, \array{ C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\stackrel{d_{dR}}{\to} & \cdots &\stackrel{d_{dR}}{\to}& \Omega^n_{cl}(-) \\ \downarrow && \downarrow && && \downarrow \\ C^\infty(-, U(1)) &\to& 0 &\to& \cdots &\to& 0 } \,,

where at the top we have the flat Deligne complex.

Proof

It is clear that the morphism of chain complexes is an objectwise surjection and hence maps to a projective fibration under Ξ\Xi. It remains to observe that the flat Deligne complex is a presentation of B nU(1)\mathbf{\flat} \mathbf{B}^n U(1):

By the discussion at ∞-cohesive site we have that =DiscΓ\mathbf{\flat} = Disc \Gamma is given on fibrant objects by first evaluating on the point and then extending back to a constant simplicial presheaf. By the above discussion we have that Ξ(U(1)[n])\Xi (U(1)[n]) is indeed fibrant, and so a fibrant presentation of B nU(1)\mathbf{\flat} \mathbf{B}^n U(1) is given by the constant presheaf U(1) const[n]:UΞ(U(1)[n])U(1)_{const} [n] : U \mapsto \Xi(U(1)[n]).

The inclusion U(1) const[n]U(1)[n]U(1)_{const}[n] \to U(1)[n] is not yet a fibration. But by a basic fact of abelian sheaf cohomology – using the Poincare lemma – we have a global weak equivalence U(1)[n] const[C (,U(1))d dRd dRΩ cl n()]U(1)[n]_{const} \stackrel{\simeq}{\to} [C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)] that factors this inclusion by the above fibration.

Flat coefficients of BG\mathbf{B}G for GG a Lie group

Let GG be a Lie group regarded as a 0-truncated ∞-group in SmoothGrpdSmooth \infty Grpd. Write 𝔤\mathfrak{g} for its Lie algebra. Write BGSmoothGrpd\mathbf{B}G \in Smooth \infty Grpd for its delooping. Recall the fibrant presentation BG c[CartSp smooth op,sSet] proj,loc\mathbf{B}G_c \in [CartSp_{smooth}^{op}, sSet]_{proj,loc} from above.

Proposition

The object BGSmoothGrpd\mathbf{\flat}\mathbf{B}G \in Smooth \infty Grpd has a fibrant presentation BG c[CartSp op,sSet] proj,loc\mathbf{\flat} \mathbf{B}G_{c} \in [CartSp^{op}, sSet]_{proj,loc} given by the groupoid of Lie-algebra valued forms

BG c=N(C (,G)×Ω flat 1(,𝔤)p 2Ad p 1(p 2)+p 1 1dp 1Ω flat 1(,𝔤)) \mathbf{\flat}\mathbf{B}G _{c} = N \left( C^\infty(-,G)\times \Omega^1_{flat}(-,\mathfrak{g}) \stackrel{ \overset{Ad_{p_1}(p_2)+ p_1^{-1} d p_1}{\to}}{\underset{p_2}{\to}} \Omega^1_{flat}(-,\mathfrak{g}) \right)

and this is such that the canonical morphism BGBG\mathbf{\flat} \mathbf{B}G \to \mathbf{B}G is presented by the canonical morphism of simplicial presheaves BG cBG c\mathbf{\flat}\mathbf{B}G_{c} \to \mathbf{B}G_{c} which is a fibration in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}.

Remark

This means that a UU-parameterized family of objects of BG c\mathbf{\flat}\mathbf{B}G_{c} is given by a Lie-algebra valued 1-form AΩ 1(U)𝔤A \in \Omega^1(U)\otimes \mathfrak{g} whose curvature 2-form F A=d dRA+[A,A]=0F_A = d_{dR} A + [A ,\wedge A] = 0 vanishes,
and a UU-parameterized family of morphisms g:AAg : A \to A' is given by a smooth function gC (U,G)g \in C^\infty(U,G) such that A=Ad gA+g 1dgA' = Ad_g A + g^{-1} d g, where Ad gA=g 1AgAd_g A = g^{-1} A g is the adjoint action of GG on its Lie algebra, and where g 1dg:=g *θg^{-1} d g := g^* \theta is the pullback of the Maurer-Cartan form on GG along gg.

Proof

By the discussion at ∞-cohesive site we have that BG\mathbf{\flat} \mathbf{B}G is presented by the simplicial presheaf that is constant on the nerve of the one-object groupoid

G disc*, G_{disc} \stackrel{\to}{\to} * \,,

for the discrete group underlying the Lie group GG. The canonical morphism of that into BG c\mathbf{B}G_c is however not a fibration.

We claim that the canonical inclusion N(G disc)BG cN(G_{disc}\stackrel{\to}{\to}) \to \mathbf{\flat} \mathbf{B}G_{c} factors the inclusion into BG c\mathbf{B}G_c by a weak equivalence followed by a global fibration.

To see the weak equivalence, notice that it is objectwise an equivalence of groupoids: it is essentially surjective since every flat 𝔤\mathfrak{g}-valued 1-form on the contractible n\mathbb{R}^n is of the form gdg 1g d g^{-1} for some function g: nGg : \mathbb{R}^n \to G (let g(x)=Pexp( 0 x)Ag(x) = P \exp(\int_{0}^x) A be the parallel transport of AA along any path from the origin to xx). Since the gauge transformation automorphism of the trivial 𝔤\mathfrak{g}-valued 1-form are precisely given by the constant GG-valued functions, this is also objectwise a full and faithful functor.

Similarly one sees that the map BG cBG\mathbf{\flat}\mathbf{B}G_c \to \mathbf{B}G is a fibration.

Finally we need to show that BG c\mathbf{\flat}\mathbf{B}G_c is fibrant in [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc}. This can be seen by observing that this sheaf is the coefficient object that in Cech cohomology computes GG-principal bundles with flat connection and then reasoning as above: every GG-principal bundle with flat connection on a Cartesian space is equivalent to a trivial GG-principal bundle whose connection is given by a globally defined 𝔤\mathfrak{g}-valued 1-form. Morphisms between these are precisely GG-valued functions that act on the 1-forms by gauge transformations as in the groupoid of Lie-algebra valued forms.

de Rham cohomology

We discuss the intrinsic de Rham cohomology in SmoothGrpdSmooth \infty Grpd.

With coefficients in B nU(1)\mathbf{B}^n U(1) or B n\mathbf{B}^n \mathbb{R}

Let B nU(1)\mathbf{B}^n U(1) be the circle Lie (n+1)(n+1)-group, as discussed above. Recall the notation and model category presentations from the discussion there.

Proposition

A fibrant representative in [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc} of the de Rham coefficent object dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) as well as dRB n\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R} is given by the truncated de Rham complex of sheaves of abelian groups of differential forms

dRB nU(1) chn:= dRB n chn:=Ξ[Ω 1()d dRΩ 2()d dRΩ n1()d dRΩ cl n()]. \mathbf{\flat}_{dR}\mathbf{B}^n U(1)_{chn} := \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{chn} := \Xi[\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to}\cdots \to \Omega^{n-1}(-) \stackrel{d_{dR}}{\to}\Omega^n_{cl}(-)] \,.
Proof

By definition and the fact that homotoppy pullbacks in the local structure may be computed in global structure (discussed here) the object dRB nU(1) chn\mathbf{\flat}_{dR}\mathbf{B}^n U(1)_{chn} is given, up to equivalence, by the homotopy pullback in [CartSp op,Ch ] proj[CartSp^{op}, Ch_\bullet]_{proj} of the inclusion U(1) const[n]U(1)[n]U(1)_{const}[n] \to U(1)[n] along the point inclusion *U(1)[n]* \to U(1)[n]. We may compute this as the ordinary pullback after passing to a resolution of this inclusion by a fibration. By the above discussion of flat cohomology with coefficients in B nU(1)\mathbf{B}^n U(1) such a fibration replacement is given by the map from the flat Deligne complex. Using this we find the ordinary pullback diagram

Ξ[0Ω 1()Ω cl n()] Ξ[C (,U(1))Ω 1()Ω cl n()] Ξ[000] Ξ[C (,U(1))00]. \array{ \Xi[0 \to \Omega^1(-) \to \cdots \to \Omega^n_{cl}(-)] &\to& \Xi[C^\infty(-,U(1)) \to \Omega^1(-) \to \cdots \to \Omega^n_{cl}(-)] \\ \downarrow && \downarrow \\ \Xi[0 \to 0 \to \cdots \to 0] &\to& \Xi[C^\infty(-,U(1)) \to 0 \to \cdots \to 0] } \,.
On smooth manifolds

Let XX be an orientable smooth manifold regarded under the embedding SmoothMfdSmoothGrpdSmoothMfd \hookrightarrow Smooth \infty Grpd.

Write H dR n(X)H^n_{dR}(X) for the ordinary de Rham cohomology of XX.

Proposition

For nn \in \mathbb{N} we have isomorphisms

π 0SmoothGrpd(X, dRB nU(1)){H dR n(X) n2 Ω cl 1(X) n=1 0 n=0 \pi_0 Smooth \infty Grpd(X, \mathbf{\flat}_{dR} \mathbf{B}^n U(1)) \simeq \left\{ \array{ H^n_{dR}(X) &| n \geq 2 \\ \Omega^1_{cl}(X) &| n = 1 \\ 0 &| n = 0 } \right.
Proof

Let {U iX}\{U_i \to X\} be a good open cover. By the disucssion at model structure on simplicial presheaves the Cech nerve C({U i})XC(\{U_i\}) \to X is a cofibrant resolution of XX in [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc}. Therefore we have for all nn

SmoothGrpd(X, dRB nU(1))[CartSp op,sSet](C({U i}),Ξ[Ω 1()d dRΩ cl n()]). Smooth \infty Grpd(X,\mathbf{\flat}_{dR} \mathbf{B}^n U(1)) \simeq [CartSp^{op}, sSet](C(\{U_i\}), \Xi[\Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)]) \,.

The right hand is the \infty-groupoid of cocylces in the Cech hypercohomology of the truncated complex of sheaves of differential forms. A cocycle is given by a collection

(C i,B ij,A ijk,,Z i 1,,i n) (C_i, B_{i j}, A_{i j k}, \cdots , Z_{i_1, \cdots, i_n})

of differential forms, with C iΩ cl n(U i)C_i \in \Omega^n_{cl}(U_i), B ijΩ n1(U iU j)B_{i j} \in \Omega^{n-1}(U_i \cap U_j), etc. , such that this collection is annihilated by the total differential D=d dR±δD = d_{dR} \pm \delta, where d dRd_{dR} is the de Rham differential and δ\delta the alternating sum of the pullbacks along the face maps of the Cech nerve.

It is a standard result of abelian sheaf cohomology that such cocycles represent classes in de Rham cohomology.

But for the record and since the details of this computation will show up again at some mildly subtle points in further discussion below, we spell this out in some detail.

For n=1n = 1 and n=0n= 0 our truncated de Rham complex degenerates to dRBU(1) chn=Ξ[Ω cl 1()]\mathbf{\flat}_{dR}\mathbf{B}U(1)_{chn} = \Xi[\Omega^1_{\mathrm{cl}}(-)] and dRU(1) chn=Ξ[0]\mathbf{\flat}_{dR}U(1)_{chn} = \Xi[0], respectively, which obviously has the cohomology as claimed above.

For n2n \geq 2 we can explicitly construct coboundaries connecting such a generic cocycle to one of the form

(F i,0,0,,0) (F_i, 0, 0, \cdots, 0)

by using a partition of unity (ρ iC (X))(\rho_i \in C^\infty(X)) subordinate to the cover {U iX}\{U_i \to X\}, i.e. xU iρ i(x)=0x \in U_i \Rightarrow \rho_i(x) = 0 and iρ i=1\sum_i \rho_i = 1.

For consider

(C i,B ij,A ijk,,Y i 1,,i n,Z i 1,,i n+1) + D(0,0,, i 0ρ i 0Z i 0,i 1,,i n,0) = (C i,B ij,A ijk,,Y i 1,,i n+d dR i 0ρ i 0Z i 0,i 1,,i n,0), \begin{aligned} & (C_i, B_{i j}, A_{i j k}, \cdots , Y_{i_1, \cdots, i_{n}}, Z_{i_1, \cdots, i_{n+1}}) \\ + & D (0, 0, \cdots, \sum_{i_0} \rho_{i_0} Z_{i_0, i_1, \cdots, i_{n}},0) \\ = & (C_i, B_{i j}, A_{i j k}, \cdots , Y_{i_1, \cdots, i_{n}} + d_{dR}\sum_{i_0} \rho_{i_0} Z_{i_0, i_1, \cdots, i_{n}}, 0) \end{aligned} \,,

where we use that from (δZ) i 1,,i n+2=0(\delta Z)_{i_1, \cdots, i_{n+2}} = 0 it follows that

(δρZ) i 1,,i n+1 = i 0ρ i 0 k=1 n+1(1) kZ i 0,i 1,i^ k,,i n+1 = i 0ρ i 0Z i 1,,i n+1 =Z i 1,,i n+1. \begin{aligned} (\delta \sum \rho Z)_{i_1, \cdots, i_{n+1}} &= \sum_{i_0} \rho_{i_0} \sum_{k = 1}^{n+1} (-1)^k Z_{i_0, i_1 \cdots, \hat i_k, \cdots, i_{n+1}} \\ & = \sum_{i_0} \rho_{i_0} Z_{i_1 ,\cdots, i_{n+1}} \\ & = Z_{i_1 ,\cdots, i_{n+1}} \end{aligned} \,.

where I am suppressing some evident signs

By recurseively adding coboundaries this way, we can annihilate all the higher Cech-components of the original cocycle and arrive at a cocycle of the form (F i,0,,0)(F_i, 0, \cdots, 0).

Such a cocycle being DD-closed says precisely that F i=F U iF_i = F|_{U_i} for FΩ cl n(X)F \in \Omega^n_{cl}(X) a globally defined closed differential form. Moreover, coboundaries between two cocycles both of this form

(F i,0,,0)=(F i,0,,0)+D(λ i,λ ij,) (F_i, 0, \cdots , 0) = (F'_i, 0, \cdots, 0) + D(\lambda_i, \lambda_{i j}, \cdots)

are necessarily themselves of the form (λ i,λ ij,)=(λ i,0,,0)(\lambda_i, \lambda_{i j}, \cdots) = (\lambda_i, 0 ,\cdots, 0) with λ i=λ U i\lambda_i = \lambda|_{U_i} for λΩ n1(X)\lambda \in \Omega^{n-1}(X) a globally defined differential nn-form and F=F+d dRλF = F' + d_{dR} \lambda.

On B nU(1)\mathbf{B}^n U(1)
Proposition

For n1n \geq 1 we have that the intrinsic de Rham cohomology of the circle n-groupoid B nU(1)\mathbf{B}^n U(1) in H=LieGrpd\mathbf{H} = \infty LieGrpd is concentrated in degree (n+1)(n+1), where it is \mathbb{R}:

H(B nU(1), dRB kU(1))={ fork=n+1 0 otherwise H(\mathbf{B}^n U(1), \mathbf{\flat}_{dR} \mathbf{B}^{k} U(1) ) = \left\{ \array{ \mathbb{R} & for \; k = n+1 \\ 0 & otherwise } \right.
Remark

By the discussion at circle n-group – differential coefficients above, we have that dRB kU(1) dRB k\mathbf{\flat}_{dR} \mathbf{B}^k U(1) \simeq \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}, reflecting the familiar fact that since Lie(U(1))Lie()Lie(U(1)) \simeq Lie(\mathbb{R}) we have that Lie(U(1))Lie(U(1))-valued forms are naturally identified with plain =Lie()\mathbb{R} = Lie(\mathbb{R})-valued forms. Therefore the above may equivalently be restated as

H dR k(B nU(1)):=H(B nU(1), dRB k)={ fork=n+1 0 otherwise. H^{k}_{dR}(\mathbf{B}^n U(1)) := H(\mathbf{B}^n U(1), \mathbf{\flat}_{dR} \mathbf{B}^{k} \mathbb{R} ) = \left\{ \array{ \mathbb{R} & for \; k = n+1 \\ 0 & otherwise } \right. \,.
Proof

Since both domain and codomain are abelian, it will be sufficient to demonstrate the statement for n=1n = 1, kk arbitrary, and for k=1k = 1, nn arbitrary. All other combinations of nn and kk are then implied by repeatedly using the delooping/looping (∞,1)-adjunction (ΣΩ)(\Sigma \dashv \Omega) on abelian objects in

H(B nU(1), dRB kU(1)) H(B nU(1),Ω dRB k+1U(1)) H(ΣB nU(1), dRB k+1U(1)) H(B n+1U(1), dRB k+1U(1)). \begin{aligned} \mathbf{H}(\mathbf{B}^n U(1), \mathbf{\flat}_{dR} \mathbf{B}^{k}U(1)) & \simeq \mathbf{H}(\mathbf{B}^n U(1), \Omega \mathbf{\flat}_{dR} \mathbf{B}^{k+1}U(1)) \\ & \simeq \mathbf{H}(\Sigma \mathbf{B}^n U(1), \mathbf{\flat}_{dR} \mathbf{B}^{k+1}U(1)) \\ & \simeq \mathbf{H}(\mathbf{B}^{n+1} U(1), \mathbf{\flat}_{dR} \mathbf{B}^{k+1}U(1)) \end{aligned} \,.

Here we use that since dR\mathbf{\flat}_{dR} is a right adjoint, it commutes with forming loop space objects.

Using this, the main work in proving the theorem is involved in establishing the statement for k=n+1=2k = n+1 = 2, which we now spell out. We note that this would be easy to prove if we did not have to pass to a cofibrant resolution of B nU(1)\mathbf{B}^n U(1) for computing the derived hom-space in the projectve model structure on simplicial presheaves [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov}. For then we would compute, using the above fibrant model for dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1)

[CartSp op,sSet](Ξ(C (U(1))[n]),Ξ[Ω 1()Ω cl n()]) [CartSp op,Ch ](NΞ(C (U(1))[n]),Ω 1()Ω cl n()) [CartSp op,Ch ](C (U(1))[n],Ω 1()Ω cl n()) [CartSp op,Ab](C (,U(1)),Ω 1()), \begin{aligned} & [CartSp^{op}, sSet](\;\;\Xi(C^\infty(-U(1))[n]), \;\;\Xi[\Omega^1(-) \to \cdots \to \Omega^n_{cl}(-)]\;\;) \\ & \simeq [CartSp^{op}, Ch_\bullet](\;\; N \Xi(C^\infty(-U(1))[n]), \;\;\Omega^1(-) \to \cdots \to \Omega^n_{cl}(-) \;\;) \\ & \simeq [CartSp^{op}, Ch_\bullet](\;\; C^\infty(-U(1))[n], \;\; \Omega^1(-) \to \cdots \to \Omega^n_{cl}(-) \;\; ) \\ &\simeq [CartSp^{op}, Ab](\;\; C^\infty(-,U(1)), \;\; \Omega^1(-) \;\;) \end{aligned} \,,

where Ξ:[CartSp op,Ch ][CartSp op,sSet]\Xi : [CartSp^{op}, Ch_\bullet] \to [CartSp^{op}, sSet] is the Dold-Kan correspondence map.

To see what this last set is, notice that if we forgot the abelian group structure, and looked at the last hom-set as one of sheaves, we find by the Yoneda lemma that is is the set Ω 1(U(1))\Omega^1(U(1)) of 1-forms. Among these the forms ωΩ 1(U(1))\omega \in \Omega^1(U(1)) that do respect the group structure are those such that for all UCartSpU \in CartSp and all f,g:UU(1)f,g : U \to U(1) we have

f *ω+g *ω=(fg) *ω. f^* \omega + g^* \omega = (f \cdot g)^* \omega \,.

A little reflection shows that this is satisfied precisely by the constant forms, i.e. those that are a linear multiple of the Maurer-Cartan form on U(1)U(1). Hence the above hom-complex is indeed just

. \cdots \simeq \mathbb{R} \,.

But B nU(1)\mathbf{B}^n U(1) is not cofibrant in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}. We will pass now to a cofibrant resolution and discuss that computing the homs out of that does nevertheless reduce to the above computation.

By Dugger’s construction of cofibrant replacements in the projective model structure we have that a cofibrant replacement for a simplciial presheaf AA is given by the diagonal of the bisimplicial presheaf which in degree (n,k)(n,k) has

U nU n1U 0A kU n, \coprod_{U_n \to U_{n-1} \to \cdots \to U_0 \to A_k} U_n \,,

where the coproduct ranges over all sequences of morphisms of representables U iCartSpU_i \in CartSp into A kA_k. And face and degeneracy maps are the evident ones.

So we prove now that the computation of H(BU(1), dRB 2U(1))\mathbf{H}(\mathbf{B}U(1), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)) does reduce to the above computation of morphisms out of BU(1)\mathbf{B}U(1). Write Q(B 1U(1))B nU(1)Q(\mathbf{B}^1 U(1)) \stackrel{\simeq}{\to} \mathbf{B}^n U(1) for the above cofibrant replacement.

Then a morphism Q(BU(1) dRB 2U(1))Q(\mathbf{B} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)) is a collection of horizontal morphisms in the diagram

Δ[2] U 0U 1U:2U(1)Ω cl 2(U 0)×Ω 1(U 0)×Ω 1(U 0) Δ[1] U 0U 1U(1)Ω cl 2(U 0)×Ω 1(U 0) Δ[0] U 0CartSpΩ cl 2(U 0) \array{ \Delta[2] &\to& \coprod_{U_0 \to U_1 \to U:2 \to U(1)} \Omega^2_{cl}(U_0) \times \Omega^1(U_0)\times \Omega^1(U_0) \\ \downarrow \downarrow \downarrow && \downarrow \downarrow \downarrow \\ \Delta[1] &\to& \coprod_{U_0 \to U_1 \to U(1)} \Omega^2_{cl}(U_0) \times \Omega^1(U_0) \\ \downarrow \downarrow && \downarrow \downarrow \\ \Delta[0] &\to& \coprod_{U_0 \in CartSp} \Omega^2_{cl}(U_0) }

Explicitly, such a cocycle is

  • a collection {λ UΩ cl 2(U)UCartSp}\{\lambda_{U} \in \Omega^2_{cl}(U) | U \in CartSp\};

  • a collection {ω U 0U 1U(1)Ω 1(U 9)}\{\omega_{U_0 \to U_1 \to U(1)} \in \Omega^1(U_9)\}

such that

  • for all f:U 0U 1f : U_0 \to U_1 we have f *λλ U 1=λ U 0+ω U 0fU 1U(1)f^* \lambda \lambda_{U_1} = \lambda_{U_0} + \omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)};

  • for all U 0fU 1U 2U(1)×U(1)U_0 \stackrel{f}{\to} U_1 \to U_2 \to U(1)\times U(1) we have

    ω U 0U 1U(1)×U(1)p 1U(1)+f *ω U 1U 2U(1)×U(1)p 2U(1)=ω U 0U 2U(1)×U(1)U(1) \omega_{U_0 \to U_1 \to U(1)\times U(1) \stackrel{p_1}{\to} U(1)} + f^* \omega_{U_1 \to U_2 \to U(1) \times U(1) \stackrel{p_2}{\to} U(1)} = \omega_{U_0 \to U_2 \to U(1) \times U(1) \stackrel{\cdot}{\to} U(1)}
  • for all UU we have ω UIdUeU(1)=0\omega_{U \stackrel{Id}{\to} U \to \stackrel{e}{\to} U(1)} = 0.

A coboundary between two such cocycles is

  • for all UCartSpU \in CartSp a κ UΩ 1(U)\kappa_U \in \Omega^1(U)

such that

  • λ U=λ U+dκ U\lambda'_U = \lambda_U + d \kappa_U;

  • ω U 0fU 1U(1)=ω U 0fU 1U(1)+f *κ U 1κ U 0\omega'_{U_0 \stackrel{f}{\to} U_1 \to U(1)} = \omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)} + f^* \kappa_{U_1} - \kappa_{U_0}.

We claim that any such cocycle (λ,ω)(\lambda,\omega) is coboundant to one such that

  • for all UU we have λ U=0\lambda_U = 0;

  • for all U 0fU(1)U_0 \stackrel{f}{\to} U(1) we have ω U 0fU 1eU(1)=0\omega_{U_0 \stackrel{f}{\to} U_1 \stackrel{e}{\to} U(1)} = 0.

The coboundary establishing this is given by setting

κ U:=ω U*eU(1). \kappa_U := \omega_{U \to * \stackrel{e}{\to} U(1)} \,.

This follows from the cocycle law for for maps of the form

U 1 f U 0 * const e,e const e,e U(1)×U(1) \array{ && U_1 \\ & {}^{\mathrlap{f}}\nearrow && \searrow \\ U_0 &&\to&& * \\ & {}_{\mathllap{const_{e,e}}}\searrow && \swarrow_{\mathrlap{const_{e,e}}} \\ && U(1) \times U(1) }

which asserts that

ω U 0fU 1eU(1)=ω U 0*eU(1)f *ω U 1*eU(1). \omega_{U_0 \stackrel{f}{\to} U_1 \stackrel{e}{\to} U(1)} = \omega_{U_0 \to * \stackrel{e}{\to} U(1)} - f^* \omega_{U_1 \to * \stackrel{e}{\to} U(1)} \,.

Moreover we claim that such cocycles with λ U=0\lambda_U = 0 for all UU and ω U 0U 1eU(1)\omega_{U_0 \to U_1 \stackrel{e}{\to} U(1)} for all U 0fU 1U_0 \stackrel{f}{\to} U_1 have no nontrivial morphisms between them, which means that these do constitute unique representatives of their cohomology classes. This is because such a morphism would be given by a collection κ UΩ cl 1(U)\kappa_U \in \Omega^1_{cl}(U) of closed 1-forms for each UCartSpU \in CartSp, such that in particular for all U 0fU 1U_0 \stackrel{f}{\to} U_1 we’d have κ U 0=f *κ U 1\kappa_{U_0} = f^* \kappa_{U_1}. Clearly for any choice of κ U\kappa_Us, one can find ff such that this is not satisfied. For instance simply take f:U*f : U \to *, which imposes the relation κ U=0\kappa_U = 0 explicitly.

This shows that the cohomology in question is the set of cocycles as above, satisfying the two extra constraints. Now using the cocycle law for the diagram

U 0 Id f U 0 U 1 (h,const e) (g,const e) U(1)×U(1) \array{ && U_0 \\ & {}^{\mathllap{Id}}\nearrow && \searrow^{\mathrlap{f}} \\ U_0 &&\to&& U_1 \\ & {}_{(h,const_e)}\searrow && \swarrow_{\mathrlap{(g,const_e)}} \\ && U(1) \times U(1) }

we find

ω U 0fU 1U(1)=ω U 0IdU 0U(1)+ω U 0fU 1eU(1) \omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)} = \omega_{U_0 \stackrel{Id}{\to} U_0 \to U(1)} + \omega_{U_0 \stackrel{f}{\to} U_1 \stackrel{e}{\to} U(1)}

and using it for

U 1 f Id U 0 U 1 (const e,h) (const e,g) U(1)×U(1) \array{ && U_1 \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{Id}} \\ U_0 &&\to&& U_1 \\ & {}_{(const_e,h)}\searrow && \swarrow_{\mathrlap{(const_e,g)}} \\ && U(1) \times U(1) }

we find

ω U 0fU 1U(1)=ω U 0fU 1eU(1)+f *ω U 1IdU 1U(1). \omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)} = \omega_{U_0 \stackrel{f}{\to} U_1 \stackrel{e}{\to} U(1)} + f^* \omega_{U_1 \stackrel{Id}{\to} U_1 \to U(1)} \,.

In view of the above gauge condition this is equivalently

ω U 0fU 1U(1)=ω U 0IdU 0U(1)=f *ω U 1IdU 1U(1). \omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)} = \omega_{U_0 \stackrel{Id}{\to} U_0 \to U(1)} = f^* \omega_{U_1 \stackrel{Id}{\to} U_1 \to U(1)} \,.

This says that ω U 0fU 1U(1)=:ω U 0U(1)\omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)} =: \omega_{U_0 \to U(1)} in fact only depends on the domain of ff, hence only on the map h:U 0U(1)h : U_0 \to U(1) and that as such these forms constitute the components of a morphism of sheaves

C (,U(1))Ω 1(). C^\infty(-,U(1)) \to \Omega^1(-) \,.

But this means that the cocycle Q(BU(1)) dRB 2U(1)Q(\mathbf{B}U(1)) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1) indeed factors through Q(BU(1))BU(1)Q(\mathbf{B}U(1)) \to \mathbf{B}U(1) and therefore the naive computation at the beginning indeed applies and shows that these cocycle are in bijection with multiples of the standard Maurer-Cartan form on U(1)U(1), hence with \mathbb{R}, so that

H(BU(1), dRB 2U(1))=. H(\mathbf{B}U(1), \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)) = \mathbb{R} \,.

The same kind of argument as above shows that for k2k \geq 2 cocycles Q(BU(1)) dRB kU(1)Q(\mathbf{B}U(1)) \to \mathbf{\flat}_{dR}\mathbf{B}^k U(1) have underlying them sheaf morphisms C (,U(1))Ω k1()C^\infty(-,U(1)) \to \Omega^{k-1}(-). But all these are necessarily trivial, due to the fact that U(1)U(1) is 1-dimensional. This establishes the theorem for n=1n=1 and arbitrary kk.

Finally the same argument that above showed that no nontrivial automorphisms of certain cocycles exist shows the theorem for k=1k=1 and arbitrary n1n \geq 1, namely that no nontrivial morphisms Q(B nU(1)) dRBU(1)Q(\mathbf{B}^n U(1)) \to \mathbf{\flat}_{dR} \mathbf{B}U(1) exist: such are given by a collection of 1-forms lamba UΩ 1(U)\lamba_U \in \Omega^1(U) for all UU, satisfyine for all possible maps f:U 0U 1f : U_0 \to U_1 the relation λ U 0=f *λ U 1\lambda_{U_0} = f^* \lambda_{U_1}. If λ U\lambda_{U} does not vanish for all UU, one can always find some ff for which this is not satisfied.

With coefficients in BG\mathbf{B}G for a Lie group GG

Let GG be a Lie group. Write 𝔤\mathfrak{g} for its Lie algebra.

Proposition

The object dRBGSmoothGrpd\mathbf{\flat}_{dR}\mathbf{B}G \in Smooth \infty Grpd has a fibrant presentation in [CartSp smooth op,sSet] proj,loc[CartSp_{smooth}^{op}, sSet]_{proj,loc} by the sheaf BG c=Ω flat 1(,𝔤)\mathbf{\flat}\mathbf{B}G_c = \Omega^1_{flat}(-, \mathfrak{g}) of flat Lie-algebra valued forms

dRBG c:UΩ flat 1(U,𝔤). \mathbf{\flat}_{dR}\mathbf{B}G_c : U \mapsto \Omega^1_{flat}(U,\mathfrak{g}) \,.
Proof

By a proposition above we have a fibration BG cBG c\mathbf{\flat}\mathbf{B}G_c \to \mathbf{B}G_c in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj} modeling the canonical inclusion BGBG\mathbf{\flat}\mathbf{B}G \to \mathbf{B}G. Therefore we may get a presentation for the defining (∞,1)-pullback

dRBG:=*× BGBG \mathbf{\flat}_{dR}\mathbf{B}G := * \times_{\mathbf{B}G} \mathbf{\flat} \mathbf{B}G

in SmoothGrpdSmooth \infty Grpd by the ordinary pullback

dRBG c*× BG cBG c \mathbf{\flat}_{dR}\mathbf{B}G_c \simeq * \times_{\mathbf{B}G_c} \mathbf{\flat} \mathbf{B}G_c

in [CartSp op,sSet][CartSp^{op}, sSet].

The resulting simplicial presheaf is fibrant in [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc} because it is a sheaf.

Remark

Writing TUT U for the tangent Lie algebroid of UU we may equivalently write

dRBG :UHom(TU,𝔤), \mathbf{\flat}_{dR} \mathbf{B}G_ : U \mapsto Hom(T U, \mathfrak{g}) \,,

where on the right we have the set of morphisms of Lie algebroids. Equivalently in terms of Chevalley-Eilenberg algebras this is

dRBG c:UHom dgAlg(CE(𝔤),(Ω (U),d dR)), \mathbf{\flat}_{dR} \mathbf{B}G_c : U \mapsto Hom_{dgAlg}(CE(\mathfrak{g}),(\Omega^\bullet(U), d_{dR})) \,,
Corollary

For XX \in SmoothMfd SmoothGrpd\hookrightarrow Smooth \infty Grpd we find the intrinsic de Rham cohomology set

H dR,Smooth 1(X,G):=π 0SmoothGrpd(X, dRBG)Ω flat 1(X,𝔤) H^1_{dR, Smooth}(X, G) := \pi_0 Smooth \infty Grpd(X, \mathbf{\flat}_{dR} \mathbf{B}G) \simeq \Omega^1_{flat}(X, \mathfrak{g})

is the set of smooth flat 𝔤\mathfrak{g}-valued differential forms on XX.

With coefficients in BG\mathbf{B}G for GG a strict 2-group

We work out, following the general definition the coefficient object for Lie 2-algabra valued forms dRB[G 2G 1]\mathbf{\flat}_{dR} \mathbf{B}[G_2\to G_1] for (G 2G 1)(G_2 \to G_1) a Lie crossed module.

Let Ξ:CrsdCplxKanCplx\Xi : CrsdCplx \to KanCplx now denote the inclusion of crossed complexes into all Kan complexes/∞-groupoids.

Write [𝔤 2δ *𝔤 1][\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1] for the corresponding differential crossed module with action α *:𝔤 1der(𝔤 2)\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2) corresponding to the Lie strict 2-group crossed module (G 2δG 1)(G_2 \stackrel{\delta}{\to} G_1) with action α:G 1Aut(G 2)\alpha : G_1 \to Aut(G_2).

Proposition

The Lie 2-groupoid B[G 2δG 1]\mathbf{\flat} \mathbf{B}[G_2 \stackrel{\delta}{\to} G_1] is represented in [CartSp op,sSet][CartSp^{op}, sSet] by the Lie 2-groupoid which on UCartSpU \in CartSp s the following 2-groupoid:

  • An object is a pair

    AΩ 1(U,𝔤 1),BΩ 2(U,𝔤 2) A \in \Omega^1(U,\mathfrak{g}_1)\,, \;\;\; B \in \Omega^2(U,\mathfrak{g}_2)

    such that

    δ *BdA+[AA]=0 \delta_* B - d A + [A \wedge A] = 0

    and

    dB+[AB]=0. d B + [A \wedge B] = 0 \,.
  • A 1-morphism (g,a):(A,B)(A,B)(g,a) : (A,B) \to (A',B') is a pair

    gC (U,G 1),aΩ 1(U,𝔤 2) g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)

    such that

    A=g 1Ag+g 1dg+g 1δ *ag A' = g^{-1} A g + g^{-1} d g + g^{-1} \delta_* a g

    and

    B=α g 1(B+da+[aa]+α *(Aa)). B' = \alpha_{g^{-1}}( B + d a + [a \wedge a] + \alpha_*(A \wedge a) ) \,.

    The composite of two 1-morphisms

    (A,B)(g 1,a 1)(A,B)(g 2,a 2)(A,B) (A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to} (A'', B'')

    is given by the pair

    (g 1g 2,a 1+(α g 2) *a 2). (g_1 g_2, a_1 + (\alpha_{g_2})_* a_2) \,.
  • a 2-morphism f:(λ,a)(λ,a)f : (\lambda,a) \to (\lambda', a') is a function

    fC (U,G 2) f \in C^\infty(U,G_2)

    such that

    g=δ(f) 1g g' = \delta(f)^{-1} \cdot g

    and

    a=f 1df+f 1af+f 1(r f 1α f) *(a)f a' = f^{-1} d f + f^{-1} a f + f^{-1}(r_f^{-1} \circ \alpha_f)_*(a)f

and composition is defined as follows

(…)

This is the 2-groupoid of Lie 2-algebra valued forms as described in definition 2.11 of SchrWalII. There are many possible conventions. The above is supposed to describe the bidual opposite 2-category of the 2-groupoid as defined in that article, with the direction of 1- and 2-morphisms reversed.

Corollary

The 2-groupoid dRB[G 2G 1]\mathbf{\flat}_{dR} \mathbf{B}[G_2 \to G_1] is as the one above, discarding the piece C (,G 1)C^\infty(-,G_1) in the 1-morphisms and the piece fC (,G 2)C^\infty(-,G_2) in the 2-morphismms.

Proof

Form the defining pullback as before. (…)

Exponentiated \infty-Lie algebras

We discuss here the intrinsic exponentiated ∞-Lie algebras in SmoothGrpdSmooth \infty Grpd.

Lie integration

Definition

Write dgAlg for the category of dg-algebras over the real numbers \mathbb{R}.

Write

L CEdgAlg op L_\infty \stackrel{CE}{\hookrightarrow} dgAlg^{op}

for the full subcategory of the opposite category on the graded-commutative semi-free dgas of finite type on generators in positive degree. This is the category of L-∞ algebras identified by their dual Chevalley-Eilenberg algebras.

We describe a presentation of the exponentiation an L-∞ algebra to a smooth \infty-group. The following somewhat technical definition serves to control the smooth structure on these exponentiated objects.

Definition

For kk \in \mathbb{N} regard the kk-simplex Δ k\Delta^k as a smooth manifold with corners in the standard way. We think of this embedded into the Cartesian space k\mathbb{R}^k in the standard way with maximal rotation symmetry about the center of the simplex, and equip Δ k\Delta^k with the metric space structure induced this way.

A smooth differential form ω\omega on Δ k\Delta^k is said to have sitting instants along the boundary if, for every (r<k)(r \lt k)-face FF of Δ k\Delta^k there is an open neighbourhood U FU_F of FF in Δ k\Delta^k such that ω\omega restricted to UU is constant in the directions perpendicular to the rr-face on its value restricted to that face.

More generally, for any UU \in CartSp a smooth differential form ω\omega on U×Δ kU \times\Delta^k is said to have sitting instants if there is 0<ϵ0 \lt \epsilon \in \mathbb{R} such that for all points u:*Uu : * \to U the pullback along (u,Id):Δ kU×Δ k(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k is a form with sitting instants on ϵ\epsilon-neighbourhoods of faces.

Smooth forms with sitting instants form a sub-dg-algebra of all smooth forms. We write Ω si (U×Δ k)\Omega^\bullet_{si}(U \times \Delta^k) for this sub-dg-algebra.

We write Ω si,vert (U×Δ k)\Omega_{si,vert}^\bullet(U \times \Delta^k) for the further sub-dg-algebra of vertical differential forms with respect to the projection p:U×Δ kUp : U \times \Delta^k \to U, hence the coequalizer

Ω 1(U)0p *Ω si (U×Δ k)Ω si,vert (U×Δ k). \Omega^{\bullet \geq 1}(U) \stackrel{\stackrel{p^*}{\to}}{\underset{0}{\to}} \Omega^\bullet_{si}(U \times \Delta^k) \to \Omega^\bullet_{si, vert}(U \times \Delta^k) \,.
Definition

For 𝔤L \mathfrak{g} \in L_\infty write exp(𝔤)[CartSp smooth op,sSet]\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet] for the simplicial presheaf defined over UU \in CartSp and nn \in \mathbb{N} by

exp(𝔤):(U,[n])Hom dgAlg(CE(𝔤),Ω si,vert (U×Δ n),) \exp(\mathfrak{g}) : (U, [n]) \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega_{si,vert}^\bullet(U \times \Delta^n), )

with the evident structure maps given by pullback of differential forms.

For references related to this definition see Lie integration .

Proposition

The objects exp(𝔤)[CartSp smooth op,sSet]\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet] are

Proof

That exp(𝔤) 0=*\exp(\mathfrak{g})_0 = * follows from degree-counting: Ω si,vert (U×Δ 0)=C (U)\Omega^\bullet_{si,vert}(U \times \Delta^0) = C^\infty(U) is entirely in degree 0 and CE(𝔤)CE(\mathfrak{g}) is in degree 0 the ground field \mathbb{R}.

To see that exp(𝔤)\exp(\mathfrak{g}) has all horn-fillers over each UCartSpU \in CartSp observe that the standard continuous horn retracts f:Δ kΛ i kf : \Delta^k \to \Lambda^k_i are smooth away from the preimages of the (r<k)(r \lt k)-faces of Λ[k] i\Lambda[k]^i.

For ωΩ si,vert (U×Λ[k] i)\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i) a differential form with sitting instants on ϵ\epsilon-neighbourhoods, let therefore KΔ kK \subset \partial \Delta^k be the set of points of distance ϵ\leq \epsilon from any subface. Then we have a smooth function

f:Δ kKΛ i kK. f : \Delta^k \setminus K \to \Lambda^k_i \setminus K \,.

The pullback f *ωΩ (Δ kK)f^* \omega \in \Omega^\bullet(\Delta^k \setminus K) may be extended constantly back to a form with sitting instants on all of Δ k\Delta^k.

The resulting assignment

(CE(𝔤)AΩ si,vert (U×Λ i k))(CE(𝔤)AΩ si,vert (U×Λ i k)f *Ω si,vert (U×Δ n)) (CE(\mathfrak{g}) \stackrel{A}{\to} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i)) \mapsto (CE(\mathfrak{g}) \stackrel{A}{\to} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i) \stackrel{f^*}{\to} \Omega^\bullet_{si,vert}(U \times \Delta^n))

provides fillers for all horns over all UU \in CartSp.

Definition

We say that the loop space object Ωexp(𝔤)\Omega \exp(\mathfrak{g}) is the smooth \infty-group exponentiating 𝔤\mathfrak{g}.

Proposition

The objects exp(𝔤)SmoothGrpd\exp(\mathfrak{g}) \in Smooth \infty Grpd are geometrically contractible:

Πexp(𝔤)*. \Pi \exp(\mathfrak{g}) \simeq * \,.
Proof

Observe that every simplicial presheaf XX is the homotopy colimit over its component presheaves X n[CartSp smooth op,Set][CartSp smooth op,sSet] locX_n \in [CartSp_{smooth}^{op}, Set] \hookrightarrow [CartSp_{smooth}^{op}, sSet]_{loc}

X𝕃lim nX n. X \simeq \mathbb{L} {\lim_{\to}}_n X_n \,.

(Use for instance the injective model structure for which X X_\bullet is cofibrant in the Reedy model structure [Δ op,[CartSp smooth op,sSet] inj,loc] Reedy[\Delta^{op},[CartSp_{smooth}^{op}, sSet]_{inj,loc}]_{Reedy} ).

Therefore it is sufficient to show that in each degree nn the 0-truncated object exp(𝔤) n\exp(\mathfrak{g})_{n} is geometrically contractible. To exhibit a geometric contraction, choose for each nn \in \mathbb{N}, a smooth retraction

η n:Δ n×[0,1]Δ n \eta_n : \Delta^n \times [0,1] \to \Delta^n

of the nn-simplex: a smooth map such that η n(,1)=Id\eta_n(-,1) = Id and η n(,0)\eta_n(-,0) factors through the point.

We claim that this induces a diagram of presheaves

exp(𝔤) n (id,1) id exp(𝔤) n×[0,1] η n * exp(𝔤) n (id,0) exp(𝔤) n *, \array{ \exp(\mathfrak{g})_n \\ {}^{\mathllap{(id,1)}}\downarrow & \searrow^{\mathrm{id}} \\ \exp(\mathfrak{g})_n \times [0,1] &\stackrel{\eta_n^*}{\to}& \exp(\mathfrak{g})_n \\ {}^{\mathllap{(id,0)}}\uparrow & & \uparrow \\ \exp(\mathfrak{g})_n &\to& * } \,,

where over UCartSpU \in CartSp the middle morphism is given by

η n *:(α,f)(f,η n) *α, \eta_n^* : (\alpha, f) \mapsto (f,\eta_n)^* \alpha \,,

where

  • α:CE(𝔤)Ω vert (U×Δ n)\alpha : CE(\mathfrak{g}) \to \Omega^\bullet_{vert}(U \times \Delta^n) is an element of the set exp(𝔤) n(U)\exp(\mathfrak{g})_n(U),

  • ff is an element of [0,1](U)[0,1](U);

  • (f,η n)(f,\eta_n) is the composite morphism

    U×Δ n(Id,f)×idU×[0,1]×Δ n(Id,η n)U×Δ n U \times \Delta^n \stackrel{(Id,f)\times id}{\to} U \times [0,1] \times \Delta^n \stackrel{(Id,\eta_n)}{\to} U \times \Delta^n \;
  • (f,η) *α(f,\eta)^* \alpha is the postcomposition of α\alpha with the image of (f,η n)(f,\eta_n) under Ω vert ()\Omega^\bullet_{vert}(-).

Here the last item is well defined given the coequalizer definition of Ω vert \Omega^\bullet_{vert} because (f,η n)(f,\eta_n) is a morphism of bundles over UU

U×Δ n (id,f)×id U×[0,1]×Δ n id×η n U×Δ n U id U id U. \array{ U \times \Delta^n &\stackrel{(id,f) \times id}{\to}& U \times [0,1] \times \Delta^n &\stackrel{id \times \eta_n}{\to}& U \times \Delta^n \\ \downarrow && \downarrow && \downarrow \\ U &\stackrel{id}{\to}& U &\stackrel{id}{\to}& U } \,.

Similarly, for h:KUh : K \to U any morphism in CartSp smooth{}_{smooth} the naturality condition for a morphism of presheaves follows from the fact that the composites of bundle morphisms

K×Δ n h×id U×Δ n (Id,f)×id U×[0,1]×Δ n (Id,η n) U×Δ n K h U id U id U \array{ K \times \Delta^n &\stackrel{h \times id}{\to}& U \times \Delta^n &\stackrel{(Id,f)\times id}{\to}& U \times [0,1] \times \Delta^n &\stackrel{(Id,\eta_n)}{\to}& U \times \Delta^n \\ \downarrow && \downarrow && \downarrow && \downarrow \\ K &\stackrel{h}{\to}& U &\stackrel{id}{\to}& U &\stackrel{id}{\to}& U }

and

K×Δ n ((id,fh)×id K×[0,1]×Δ n id×η n K×Δ n h×id U×Δ n K id K id K h U \array{ K \times \Delta^n &\stackrel{((id,f \circ h) \times id }{\to}& K \times [0,1] \times \Delta^n &\stackrel{id \times \eta_n}{\to}& K \times \Delta^n &\stackrel{h \times id}{\to}& U \times \Delta^n \\ \downarrow && \downarrow && \downarrow && \downarrow \\ K &\stackrel{id}{\to}& K &\stackrel{id}{\to}& K &\stackrel{h}{\to}& U }

coincide.

Moreover, notice that the lower morphism in our diagram of presheaves indeed factors through the point as indicated, becase for an L-∞ algebra 𝔤\mathfrak{g} we have that the Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) is in degree 0 the ground field algebra algebra \mathbb{R}, so that there is a unique morphism CE(𝔤)Ω vert (U×Δ 0)C (U)CE(\mathfrak{g}) \to \Omega^\bullet_{vert}(U \times \Delta^0) \simeq C^\infty(U) in dgAlgdgAlg.

Finally, since [0,1][0,1] is a contractible paracompact manifold, we have that Π([0,1])*\Pi([0,1]) \simeq * by the above proposition. Therefore the above diagram of presheaves presents a geometric homotopy in SmoothGrpdSmooth \infty Grpd from the identity map to a map that factors through the point.

By the propositon about geometric homotopy in a cohesive (∞,1)-topos it follows that Π(exp(𝔤) n)*\Pi(\exp(\mathfrak{g})_n) \simeq * for all nn \in \mathbb{N}. And since Π\Pi preserves the homotopy colimit exp(𝔤)𝕃lim nexp(𝔤) n\exp(\mathfrak{g}) \simeq \mathbb{L} {\lim_\to}_n \exp(\mathfrak{g})_n we have that Π(exp(𝔤))*\Pi(\exp(\mathfrak{g})) \simeq *, too.

Remark

We may think of Ωexp(𝔤)\Omega \exp(\mathfrak{g}) as the smooth ∞-simply connected Lie integration of 𝔤\mathfrak{g}.

Notice however that Ωexp(𝔤)Smooth\Omega \exp(\mathfrak{g}) \in Smooth \infty in general has nontrivial and interesting categorical homotopy groups. The above statement says that its geometric homotopy groups vanish .

For a Lie algebra

Let 𝔤L \mathfrak{g} \in L_\infty be an ordinary (finite dimensional) Lie algebra. Standard Lie theory (see Lie's three theorems) provides a simply connected Lie group GG integrating 𝔤\mathfrak{g}.

With GG regarded as a smooth ∞-group write BGSmoothGrpd\mathbf{B}G \in Smooth\infty Grpd for its delooping. Recall from above the standard presentation of this by a simplicial presheaf BG c[CartSp smooth op,sSet]\mathbf{B}G_c \in [CartSp_{smooth}^{op}, sSet].

Proposition

The operation of parallel transport Pexp():Ω 1([0,1],𝔤)GP \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G yields a weak equivalence (in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj})

Pexp():cosk 3exp(𝔤)cosk 2exp(𝔤)BG c. P \exp(\int - ) : \mathbf{cosk}_3 \exp(\mathfrak{g}) \simeq \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_c \,.

The proof is spelled out at Lie integration. In the section Integrating Lie algebras to Lie groups.

For a line Lie nn-algebra

Let nn \in \mathbb{N}, n1n \geq 1.

Definition

Write

b n1L b^{n-1} \mathbb{R} \in L_\infty

for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree nn and vanishing differential. We call this the line Lie n-algebra

Observation

The discrete ∞-groupoid underlying exp(b n1)\exp(b^{n-1} \mathbb{R}) is given by the Kan complex that in degree kk has the set of closed differential nn-forms (with sitting instants) on the kk-simplex

Γ(exp(b n1)):[k]Ω si,cl n(Δ k) \Gamma(\exp(b^{n-1} \mathbb{R})) : [k] \mapsto \Omega^n_{si,cl}(\Delta^k)
Definition

We write equivalently

B n smp:=exp(b n1)[CartSp smooth op,sSet]. \mathbf{B}^n \mathbb{R}_{smp} := \exp(b^{n-1}\mathbb{R}) \in [CartSp_{smooth}^{op}, sSet] \,.
Proposition

We have that B n smp\mathbf{B}^n \mathbb{R}_{smp} is a presentation of the smooth line n-group B n\mathbf{B}^{n} \mathbb{R}.

Concretely, with B n chn[CartSp smooth op,sSet]\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet] the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree nn on C (,)C^\infty(-, \mathbb{R}) the equivalence is induced by the fiber integration of differential nn-forms over the nn-simplex:

Δ :B n smpB n chn. \int_{\Delta^\bullet} : \mathbf{B}^n \mathbb{R}_{smp} \stackrel{\simeq}{\to} \mathbf{B}^{n} \mathbb{R}_{chn} \,.

The proof of this is spelled out at Lie integration in the section Integration to line n-groups.

Flat coefficients

We consider now the flat coefficient objects exp(𝔤)\mathbf{\flat} \exp(\mathfrak{g}) of exponentiated ∞-Lie algebras exp(𝔤)\exp(\mathfrak{g}).

Definition

Write exp(𝔤) smp\mathbf{\flat}\exp(\mathfrak{g})_{smp} for the simplicial presheaf given by

exp(𝔤) smp:(U,[n])Hom dgAlg(CE(𝔤),Ω si (U×Δ n)). \mathbf{\flat}\exp(\mathfrak{g})_{smp} : (U,[n]) \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega_{si}^\bullet(U \times \Delta^n)) \,.
Proposition

The canonical morphism B nB n\mathbf{\flat} \mathbf{B}^n \mathbb{R} \to \mathbf{B}^n \mathbb{R} in SmoothGrpdSmooth \infty Grpd is presented in [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet] by the composite

constΓexp(b n1)exp(b n1) smpexp(b n1), const \Gamma \exp(b^{n-1} \mathbb{R}) \to \mathbf{\flat} \exp(b^{n-1} \mathbb{R})_{smp} \to \exp(b^{n-1} \mathbb{R}) \,,

where the first morphism is a weak equivalence and the second a fibration in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}.

We discuss the two morphisms in the composite separately in two lemmas.

Lemma

The canonical inclusion

constΓ(exp(𝔤))exp(𝔤) smp const \Gamma(\exp(\mathfrak{g})) \to \mathbf{\flat}\exp(\mathfrak{g})_{smp}

is a weak equivalence in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}.

Proof

The morphism in question is on each object UCartSpU \in CartSp the morphism of simplicial sets

Hom dgAlg(CE(𝔤),Ω si (Δ k))Hom dgAlg(CE(𝔤),Ω si (U×Δ k)), Hom_{dgAlg}(CE(\mathfrak{g}), \Omega_{si}^\bullet(\Delta^k)) \to Hom_{dgAlg}(CE(\mathfrak{g}), \Omega_{si}^\bullet(U \times \Delta^k)) \,,

which is given by pullback of differential forms along the projection U×Δ kΔ kU \times \Delta^k \to \Delta^k.

To show that for fixed UU this is a weak equivalence in the standard model structure on simplicial sets we produce objectwise a left inverse

F U:Hom dgAlg(CE(𝔤),Ω si (U×Δ ))Hom dgAlg(CE(𝔤),Ω si (Δ )) F_U : Hom_{dgAlg}(CE(\mathfrak{g}), \Omega_{si}^\bullet(U \times \Delta^\bullet)) \to Hom_{dgAlg}(CE(\mathfrak{g}), \Omega_{si}^\bullet(\Delta^\bullet))

and show that this is an acyclic fibration of simplicial sets. The statement then follows by the 2-out-of-3-property of weak equivalences.

We take F UF_U to be given by evaluation at 0:*U0: * \to U, i.e. by postcomposition with the morphisms

Ω (U×Δ k)Id×0 *Ω (*×Δ k)=Ω (Δ k). \Omega^\bullet(U \times \Delta^k ) \stackrel{Id \times 0^*}{\to} \Omega^\bullet(* \times \Delta^k ) = \Omega^\bullet(\Delta^k) \,.

(This of course is not natural in UU and hence does not extend to a morphism of simplicial presheaves. But for our argument here it need not.)

The morphism F UF_U is an acyclic Kan fibration precisely if all diagrams of the form

Δ[n] Hom(CE(𝔤),Ω si (U×Δ )) F U Δ[n] Hom(CE(𝔤),Ω si (Δ )) \array{ \partial \Delta[n] &\to& Hom(CE(\mathfrak{g}), \Omega_{si}^\bullet(U \times \Delta^\bullet )) \\ {}^{\mathllap{}}\downarrow && \downarrow^\mathrlap{F_U} \\ \Delta[n] &\stackrel{}{\to}& Hom(CE(\mathfrak{g}), \Omega_{si}^\bullet(\Delta^\bullet)) }

have a lift. Using the Yoneda lemma over the simplex category and since the differential forms on the simplices have sitting instants, we may, as above, equivalently reformulate this in terms of spheres as follows:

for every morphism CE(𝔤)Ω si (D n)CE(\mathfrak{g}) \to \Omega^\bullet_{si}(D^n) and morphism CE(𝔤)Ω si (U×S n1)CE(\mathfrak{g}) \to \Omega^\bullet_{si}(U \times S^{n-1}) such that the diagram

CE(𝔤) Ω (U×S n1) Ω si (D n) Ω (S n1) \array{ CE(\mathfrak{g}) &\to& \Omega^\bullet(U \times S^{n-1}) \\ \downarrow && \downarrow \\ \Omega_{si}^\bullet(D^n) &\to& \Omega^\bullet(S^{n-1}) }

commutes, this may be factored as

CE(𝔤) Ω si (U×D n) Ω (U×S n1) Ω (D n) Ω (S n1). \array{ CE(\mathfrak{g}) \\ & \searrow \\ &&\Omega_{si}^\bullet(U \times D^n) &\to& \Omega^\bullet(U \times S^{n-1}) \\ &&\downarrow && \downarrow \\ &&\Omega^\bullet(D^n) &\to& \Omega^\bullet(S^{n-1}) } \,.

(Here the subscript ” si{}_{si}” denotes differential forms on the disk that are radially constant in a neighbourhood of the boundary.)

This factorization we now construct.

Let first f:[0,1][0,1]f : [0,1] \to [0,1] be any smoothing function, i.e. a smooth function which is surjective, non-decreasing, and constant in a neighbourhood of the boundary. Define a smooth map

U×[0,1]U U \times [0,1] \to U

by

(u,σ)uf(1σ), (u,\sigma) \mapsto u \cdot f(1-\sigma) \,,

where we use the multiplicative structure on the Cartesian space UU. This function is the identity at σ=0\sigma = 0 and is the constant map to the origin at σ=1\sigma = 1. It exhibits a smooth contraction of UU.

Pullback of differential forms along this map produces a morphism

Ω (U×S n1)Ω (U×S n1×[0,1]) \Omega^\bullet(U \times S^{n-1}) \to \Omega^\bullet(U \times S^{n-1} \times [0,1])

which is such that a form ω\omega is sent to a form which in a neighbourhood (1ϵ,1](1-\epsilon,1] of 1[0,1]1 \in [0,1] is constant along (1ϵ,1]×U(1-\epsilon,1] \times U on the value (0,Id S n1) *ω(0 , Id_{S^{n-1}})^* \omega.

(Notice that this step does not respect vertical forms. This is the crucial difference between {Ω si (U×Δ k)CE(𝔤)}\{\Omega^\bullet_{si}(U \times \Delta ^k) \leftarrow CE(\mathfrak{g})\} and {Ω si,vert (U×Δ k)CE(𝔤)}\{\Omega^\bullet_{si,vert}(U \times \Delta ^k) \leftarrow CE(\mathfrak{g})\}).

Let now 0<ϵ0 \lt \epsilon \in \mathbb{R} some value such that the given forms CE(𝔤)Ω si (D k)CE(\mathfrak{g}) \to \Omega^\bullet_{si}(D^k) are constant a distance dϵd \leq \epsilon from the boundary of the disk. Let q:[0,ϵ/2][0,1]q : [0,\epsilon/2] \to [0,1] be given by multiplication by 1/(ϵ/2)1/(\epsilon/2) and h:D 1ϵ/2 kD 1 nh : D_{1-\epsilon/2}^k \to D_1^n the injection of the nn-disk of radius 1ϵ/21-\epsilon/2 into the unit nn-disk.

We can then glue to the morphism

CE(𝔤)Ω (U×S n1)Ω (U×[0,1]×S n1) id×q *×idΩ (U×[0,ϵ/2]×S n1) CE(\mathfrak{g}) \to \Omega^\bullet(U \times S^{n-1}) \to \Omega^\bullet(U \times [0,1] \times S^{n-1}) \stackrel{id \times q^* \times id}{\to_\simeq} \Omega^\bullet(U \times [0,\epsilon/2] \times S^{n-1})

to the morphism

CE(𝔤)Ω (D n)Ω (U×{1}×D n) h *Ω (U×{1}×D 1ϵ/2 n) CE(\mathfrak{g}) \to \Omega^\bullet(D^n) \to \Omega^\bullet(U \times \{1\} \times D^n) \stackrel{h^*}{\to_\simeq} \Omega^\bullet(U \times \{1\} \times D^n_{1-\epsilon/2})

by smoothly identifying the union [0,ϵ/2]×S n1 S n1D 1ϵ/2 n[0,\epsilon/2] \times S^{n-1} \coprod_{S^{n-1}} D^n_{1-\epsilon/2} with D nD^n (we glue a disk into an annulus to obtain a new disk) to obtain in total a morphism

CE(𝔤)Ω (U×D n) CE(\mathfrak{g}) \to \Omega^\bullet(U \times D^n)

with the desired properties: at u=0u = 0 the homotopy that we constructed is constant and the above construction hence restricts the forms to radius 1ϵ/2\leq 1-\epsilon/2 and then extends back to radius 1\leq 1 by the constant value that they had before. Away from 0 the homotopy in the rmaining ϵ/2\epsilon/2 bit smoothly interpolates to the boundary value.

Lemma

The canonical morphism

exp(𝔤) smpexp(𝔤) \mathbf{\flat}\exp(\mathfrak{g})_{smp} \to \exp(\mathfrak{g})

is a fibration in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}.

Proof

Over each UCartSpU \in CartSp the morphism is induced from the morphism of dg-algebras

Ω (U)C (U) \Omega^\bullet(U) \to C^\infty(U)

that discards all differential forms of non-vanishing degree.

It is sufficient to show that for

CE(𝔤)Ω si,vert (U×(D n×[0,1])) CE(\mathfrak{g}) \to \Omega_{si,vert}^\bullet( U \times (D^n \times [0,1]) )

a morphism and

CE(𝔤)Ω si (U×D n) CE(\mathfrak{g}) \to \Omega^\bullet_{si}(U \times D^n )

a lift of its restriction to σ=0[0,1]\sigma = 0 \in [0,1] we have an extension to a lift

CE(𝔤)Ω si,vert (U×(D n×[0,1])). CE(\mathfrak{g}) \to \Omega_{si,vert}^\bullet(U \times (D^n \times [0,1])) \,.

From these lifts all the required lifts are obtained by precomposition with some evident smooth retractions.

The idea of the proof is that the lifts in question are obtained from solving differential equations with boundary conditions, and exist due to the existence of solutions of first order systems of partial differential equations and the Bianchi identities for flat ∞-Lie algebroid valued differential forms.

1st case: 𝔤=b n1\mathfrak{g} = b^{n-1} \mathbb{R}

We condsider the case 𝔤=b n1\mathfrak{g} = b^{n-1} \mathbb{R}.

A morphism CE(b n1)Ω si,vert (U×(D k×[0,1]))CE(b^{n-1} \mathbb{R}) \to \Omega_{si,vert}^\bullet( U \times (D^k \times [0,1])) is a UU-parameterized family of nn-forms A=A D k+(dσ)λA = A_{D^k} + (d \sigma) \wedge \lambda on D k×[0,1]D^k \times [0,1] satisfying

d D kA D k=0 d_{D^k} A_{D^k} = 0

and

σA D k=d D kλ. \partial_\sigma A_{D^k} = d_{D^k} \lambda \,.

Consider a lift of the restriction to σ=0\sigma = 0 by a term A D k+A U×D kA_{D^k} + A_{U \times D^k} being a morphism CE(b n1)Ω si (U×D k)CE(b^{n-1}\mathbb{R}) \to \Omega^\bullet_{si}(U \times D_k ).

We can extend this to a morphism CE(b n1)Ω si (U×D k×[0,1])CE(b^{n-1}\mathbb{R}) \to \Omega^\bullet_{si}(U \times D_k \times [0,1]) by lifting λ\lambda without adding any terms to it and solving the differential equation

σ(A D k+A U×D k)=(d U+d D k)λ, \partial_\sigma (A_{D^k} + A_{U \times D^k} ) = (d_U + d_{D^k}) \lambda \,,

equivalently for the new term

σA U×D k=d Uλ, \partial_\sigma A_{U \times D^k} = d_U \lambda \,,

with boundary condition the given value at σ=0\sigma = 0.

For this unique solution to be a lift we need in addition that (d U+d D k)(A D k+A U×D k)=0(d_U + d_{D^k}) (A_{D^k} + A_{U \times D^k}) = 0. Since this is the case at σ=0\sigma = 0 by assumption of the lift at that end, we need to check that

σ(d U+d D k)(A D k+A U×D k)=0. \partial_\sigma (d_U + d_{D^k}) (A_{D^k} + A_{U \times D^k}) = 0 \,.

But by the above this follows from the Bianchi identity, which for the special abelian case that we are considering is just the nilpotency of the de Rham differential

=(d U+d D k)(d U+d D k)(A D k+A U×D k)=0. \cdots = (d_U + d_{D^k}) (d_U + d_{D^k}) (A_{D^k} + A_{U \times D^k}) = 0 \,.

2nd case: 𝔤anarbitraryLiealgebra\mathfrak{g} an arbitrary Lie algebra

Now let 𝔤\mathfrak{g} be an ordinary Lie algebra. Choose a dual basis {t a}\{t^a\} and structue constants C a bcC^a{}_{b c}. We get a discussion analogous to the above with structure constant terms thrown in:

the original element is a collection of 1-forms A v adv+A σ adσA^a_v d v + A^a_\sigma d \sigma satisfying

σA v a= vA σC a bcA v bA σ c. \partial_\sigma A_v^a = \partial_v A_\sigma - C^a{}_{b c} A_v^b A_\sigma^c \,.

We lift by adding a term A u aduA_u^a d u that is uniquely fixed by the condition that it solves the differential equation

σA u= uA σC a bcA σ bA u c \partial_\sigma A_u = \partial_u A_\sigma - C^a{}_{b c} A_\sigma^b A_u^c

for given boundary value at σ=0\sigma = 0.

We need to show that the lift found this way also satisfies the equation

F vu a:= vA u a uA v a+C a bcA v bA u c=0. F_{v u}^a := \partial_v A^a_u - \partial_u A^a_v + C^a{}_{b c} A_v^b A_u^c = 0 \,.

By assumption, this is true at σ=0\sigma = 0. We now show that the σ\sigma-derivative of this expression satisfies the Binachi-type equation

σF vu a=C a bcA σ bF vu c. \partial_\sigma F^a_{v u} = C^a{}_{b c} A^b_\sigma F^c_{v u} \,.

A solution to this differential equation with initial value 0 is F vu a=0F^a_{v u} = 0. Since this solution is guaranteed to be unique, we will have shown our claim.

Now compute:

σF uv a := σ uA v a σ vA u a+ σC a bcA u bA v c = u σA v a v σA u a+ σC a bcA u bA v c = u( vA σ aC a bcA σ bA v c) v( uA σ aC a bcA σ bA u c)+ σC a bcA u bA v c = uC a bcA σ bA v c vC a bcA σ bA u c+ σC a bcA u bA v c =C a bcA σ b( uA v c vA u c)+C a bc( σA u bC b deA u dA σ e)A v cC a bc( σA ν bC b deA v dA σ e)A u b+ σC a bcA v bA u c =C a bcA σ cF uv b. \begin{aligned} \partial_\sigma F^a_{u v} &:= \partial_\sigma \partial_u A^a_v - \partial_\sigma \partial_v A^a_u + \partial_\sigma C^a{}_{b c} A_u^b A^c_v \\ & = \partial_u \partial_\sigma A^a_v - \partial_v \partial_\sigma A^a_u + \partial_\sigma C^a{}_{b c} A_u^b \wedge A^c_v \\ &= \partial_u (\partial_v A^a_\sigma - C^a{}_{b c} A^b_\sigma A^c_v) - \partial_v (\partial_u A^a_\sigma - C^a{}_{b c} A^b_\sigma A^c_u) + \partial_\sigma C^a{}_{b c} A_u^b A^c_v \\ & = \partial_u C^a{}_{b c} A^b_\sigma A^c_v - \partial_v C^a{}_{b c} A^b_\sigma A^c_u + \partial_\sigma C^a{}_{b c} A_u^b A^c_v \\ & = C^a{}_{b c} A^b_\sigma (\partial_u A^c_v - \partial_v A^c_u) + C^a{}_{b c} (\partial_\sigma A^b_u - C^b{}_{d e} A^d_u A^e_\sigma) A^c_v - C^a{}_{b c} (\partial_\sigma A^b_\nu - C^b{}_{d e} A^d_v A^e_\sigma) A^b_u + \partial_\sigma C^a{}_{b c} A^b_v A^c_u \\ & = C^a{}_{b c} A^c_\sigma F^b_{u v} \end{aligned} \,.

Here in the last step we use the Jacobi identity

C a bcC b de+C a bdC b ec+C a beC b cd=0. C^a{}_{b c} C^b{}_{d e} + C^a{}_{b d} C^b{}_{e c} + C^a{}_{b e} C^b{}_{c d} = 0 \,.

general case

For 𝔤\mathfrak{g} a general L L_\infty-algebra, the computation is essentially as above for the Lie algebra case only that all indices become multi-indices in a suitable sense.

For instance the structure constants now have components of arbitrary arity. But for the discussion of the lift it is still always just the components with two legs along the uu-, vv-, σ\sigma- direction that matter, all other indices just run along.

I’ll try to think of a convenient notation to express this.

For a line Lie nn-algebra

We have discussed now two different presentations for the flat coefficient object B n\mathbf{\flat}\mathbf{B}^n \mathbb{R}:

  1. B n chn\mathbf{\flat} \mathbf{B}^n \mathbb{R}_{chn} – discussed here;

  2. B n smp\mathbf{\flat} \mathbf{B}^n \mathbb{R}_{smp} – discusse here;

There is an evident degreewise map

(1) +1 Δ :B n smpB n chn (-1)^{\bullet+1} \int_{\Delta^\bullet} : \mathbf{\flat} \mathbf{B}^n \mathbb{R}_{smp} \to \mathbf{\flat} \mathbf{B}^n \mathbb{R}_{chn}

that sends a closed nn-form ωΩ cl n(U×Δ k)\omega \in \Omega^n_{cl}(U \times \Delta^k) to (1) k+1(-1)^{k+1} times its fiber integration Δ kω\int_{\Delta^k} \omega.

Proposition

This map yields a morphism of simplicial presheaves

:B n smpB n chn \int : \mathbf{\flat} \mathbf{B}^n \mathbb{R}_{smp} \to \mathbf{\flat} \mathbf{B}^n \mathbb{R}_{chn}

which is a weak equivalence in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}.

Proof

First we check that we have a morphism of simplicial sets over each UCartSpU \in CartSp. Since both objects are abelian simplicial groups we may, by the Dold-Kan correspondence, check the statement for sheaves of normalized chain complexes.

Notice that the chain complex differential on the forms ωΩ cl n(U×Δ k)\omega \in \Omega^n_{cl}(U \times \Delta^k) on simplices sends a form to the alternating sum of its restriction to the faces of the simplex. Postcomposed with the integration map this is the operation ω Δ kω\omega \mapsto \int_{\partial \Delta^k} \omega of integration over the boundary.

Conversely, first integrating over the simplex and then applying the de Rham differential on UU yields

ω(1) k+1d U Δ kω = Δ kd Uω = Δ kd Δ kω = Δ kω, \begin{aligned} \omega \mapsto (-1)^{k+1} d_U \int_{\Delta^k} \omega &= - \int_{\Delta^k} d_U \omega \\ & = \int_{\Delta^k} d_{\Delta^k} \omega \\ & = \int_{\partial \Delta^k} \omega \end{aligned} \,,

where we first used that ω\omega is closed, so that d dRω=(d U+d Δ k)ω=0d_{dR} \omega = (d_U + d_{\Delta^k}) \omega = 0, and then used Stokes' theorem. Therefore we have indeed objectwise a chain map.

By the discussion of the two objects we already know that both present the homotopy type of B n\mathbf{\flat} \mathbf{B}^n \mathbb{R}. Therefore it suffices to show that the integration map is over each UCartSpU \in CartSp an isomorphism on the simplicial homotopy group in degree nn.

Clearly the morphism

Δ n:Ω si,cl (U×Δ n)C (U,) \int_{\Delta^n} : \Omega^\bullet_{si,cl}(U \times \Delta^n) \to C^\infty(U, \mathbb{R})

is surjective on degree nn homotopy groups: for f:U*f : U \to * \to \mathbb{R} constant, a preimage is fvol Δ nf \cdot vol_{\Delta^n}, the normalized volume form of the nn-simplex times ff.

Moreover, these preimages clearly span the whole homotopy group π n(B n)mathbR disc\pi_n (\mathbf{\flat} \mathbf{B}^n \mathbb{R}) \simeq \mathb{R}_{disc} (they are in fact the images of the weak equivalence constΓexp(b n1)B n smpconst \Gamma \exp(b^{n-1}\mathbb{R}) \to \mathbf{\flat} \mathbf{B}^n \mathbb{R}_{smp} ) and the integration map is injective on them. Therefore it is an isomorphism on the homotopy groups in degree nn.

de Rham coefficients

We now consider the de Rham coefficient object dRexp(𝔤)\mathbf{\flat}_{dR} \exp(\mathfrak{g}) of exponentiated L-∞ algebras exp(𝔤)\exp(\mathfrak{g}).

Corollary

For 𝔤L \mathfrak{g} \in L_\infty a representive in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} of the object de Rham coefficient object dRexp(𝔤)\mathbf{\flat}_{dR} \exp(\mathfrak{g}) is the presheaf

dRB n smp:(U,[n])Hom dgAlg(CE(𝔤),Ω si 1,(U×Δ n)), \mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{smp} : (U,[n]) \mapsto Hom_{dgAlg}( CE(\mathfrak{g}), \Omega_{si}^{\bullet\geq 1,\bullet}(U \times \Delta^n) ) \,,

where the notation on the right denotes the dg-algebra of differential forms on U×Δ nU \times\Delta^n that (apart from having sitting instants on the faces of Δ n\Delta^n) are along UU of non-vanishing degree.

Proof

By the above proposition we may present the defining (∞,1)-pullback dRB n:=*× B nB n\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R} := * \times_{\mathbf{B}^n \mathbb{R}} \mathbf{\flat} \mathbf{B}^n \mathbb{R} in SmoothGrpdSmooth \infty Grpd by the ordinary pullback

dRB n smp B n smp * B n \array{ \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{smp} &\to& \mathbf{\flat}\mathbf{B}^n \mathbb{R}_{smp} \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}^n \mathbb{R} }

in [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet].

For a line Lie nn-algebra

We have discussed now two different presentations for the de Rham coefficient object dRB n\mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}:

  1. dRB n chn\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{chn} – discussed here;

  2. dRB n smp\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{smp} – discussed here;

There is an evident degreewise map

(1) +1 Δ : dRB n smp dRB n chn (-1)^{\bullet+1} \int_{\Delta^\bullet} : \mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{smp} \to \mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{chn}

that sends a closed nn-form ωΩ cl n(U×Δ k)\omega \in \Omega^n_{cl}(U \times \Delta^k) to (1) k+1(-1)^{k+1} times its fiber integration Δ kω\int_{\Delta^k} \omega.

Proposition

This map yields a morphism of simplicial presheaves

: dRB n smp dRB n chn \int : \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{smp} \to \mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{chn}

which is a weak equivalence in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}.

Proof

By the Dold-Kan correspondence we may check the statement for sheaves of (normalized) chain complexes.

Notice that the chain complex differential on the forms ωΩ cl n(U×Δ k)\omega \in \Omega^n_{cl}(U \times \Delta^k) on simplices sends a form to the alternating sum of its restriction to the faces of the simplex. Postcomposed with the integration map this is the operation ω Δ kω\omega \mapsto \int_{\partial \Delta^k} \omega.

Conversely, first integrating over the simplex and then applying the de Rham differential on UU yields

ω(1) k+1d U Δ kω = Δ kd Uω = Δ kd Δ kω = Δ kω, \begin{aligned} \omega \mapsto (-1)^{k+1} d_U \int_{\Delta^k} \omega &= - \int_{\Delta^k} d_U \omega \\ & = \int_{\Delta^k} d_{\Delta^k} \omega \\ & = \int_{\partial \Delta^k} \omega \end{aligned} \,,

where we first used that ω\omega is closed, so that d dRω=(d U+d Δ k)ω=0d_{dR} \omega = (d_U + d_{\Delta^k}) \omega = 0, and then Stokes' theorem.

Therefore we have indeed objectwise a chain map.

To see that it gives a weak equivalence, notice that this morphism is the morphism on pullbacks induced from the weak equivalence of diagrams

* exp(b n1) B n smp = ± Δ ± Δ * B n chn B n chn. \array{ * &\to& \exp(b^{n-1}\mathbb{R}) &\leftarrow& \mathbf{\flat}\mathbf{B}^n \mathbb{R}_{smp} \\ \downarrow^{\mathrlap{=}} && {}^{\mathllap{\pm \int_{\Delta^\bullet}}} \downarrow^{\mathrlap{\simeq}} && {}^{\mathllap{\pm \int_{\Delta^\bullet}}} \downarrow^{\mathrlap{\simeq}} \\ * &\to& \mathbf{B}^n \mathbb{R}_{chn} & \leftarrow& \mathbf{\flat}\mathbf{B}^n \mathbb{R}_{chn} } \,.

Since both of these pullbacks are homotopy pullbacks by the above discussion, the induced morphism between the pullbacks is also a weak equivalence.

Maurer-Cartan forms and curvature characteristic forms

We discuss the intrinsic Maurer-Cartan and curvature characteristic forms defined in any cohesive (,1)(\infty,1)-topos realized in SmoothGrpdSmooth \infty Grpd.

The canonical form on a Lie group

Let GG be a Lie group. Write 𝔤\mathfrak{g} for its Lie algebra.

Proposition

Under the identification

SmoothGrpd(X, dRBG)Ω flat 1(X,𝔤) Smooth \infty Grpd(X, \mathbf{\flat}_{dR}\mathbf{B}G) \simeq \Omega^1_{flat}(X,\mathfrak{g})

from the above proposition, for XX \in SmoothMfd, we have that the canonical morphism

θ:G dRBG \theta : G \to \mathbf{\flat}_{dR} \mathbf{B}G

in SmoothGrpdSmooth \infty Grpd corresponds to the ordinary Maurer-Cartan form on GG.

Proof

We compute the defining double (∞,1)-pullback

G * θ dRBG BG * BG \array{ G &\to& * \\ {}^{\mathllap{\theta}}\downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat} \mathbf{B}G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G }

in SmoothGrpdSmooth \infty Grpd as a homotopy pullback in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}

In the above discussion of differential coefficients we already modeled the lower (,1)(\infty,1)-pullback square by the ordinary pullback

dRBG c BG c * BG c \array{ \mathbf{\flat}_{dR}\mathbf{B}G_c &\to& \mathbf{\flat}\mathbf{B}G_c \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G_c }

A standard fibration replacement of the point inclusion *BG* \to \mathbf{\flat}\mathbf{B}G is (as discussed at universal principal ∞-bundle) given by replacing the point by the presheaf that assigns groupoids of the form

Q:U{ A 0=0 g 1 g 2 A 1 h A 2}, Q : U \mapsto \left\{ \array{ && A_0 = 0 \\ & {}^{\mathllap{g_1}}\swarrow && \searrow^{\mathrlap{g_2}} \\ A_1 &&\stackrel{h}{\to}&& A_2 } \right\} \,,

where on the right the commuting triangle is in ( dRBG c)(U)(\mathbf{\flat}_{dR}\mathbf{B}G_c)(U) and here regarded as a morphism from (g 1,A 1)(g_1,A_1) to (g 2,A 2)(g_2,A_2). And the fibration QBG cQ \to \mathbf{\flat}\mathbf{B}G_c is given by projecting out the base of these triangles.

The pullback of this along dRBG cBG c\mathbf{\flat}_{dR}\mathbf{B}G_c \to \mathbf{\flat}\mathbf{B}G_c is over each UU the restriction of the groupoid Q(U)Q(U) to its set of objects, hence is the sheaf

U{A 0=0 g g *θ}C (U,G)=G(U), U \mapsto \left\{ \array{ A_0 = 0 \\ \downarrow^{\mathrlap{g}} \\ g^* \theta } \right\} \simeq C^\infty(U,G) = G(U) \,,

equipped with the projection

t U:G dRBG c t_U : G \to \mathbf{\flat}_{dR} \mathbf{B}G_c

given by

t U:(g:UG)g *θ. t_U : (g : U \to G) \mapsto g^* \theta \,.

Under the Yoneda lemma (over SmoothMfd) this identifies the morphism tt with the Maurer-Cartan form θΩ flat 1(G,𝔤)\theta \in \Omega^1_{flat}(G,\mathfrak{g}).

The universal curvature characteristic on B nU(1)\mathbf{B}^n U(1)

We discuss presentations of universal curvature characteristics B nU(1)mthbf dRB n+1U(1)\mathbf{B}^n U(1)\to \mthbf{\flat}_{dR}\mathbf{B}^{n+1} U(1) and B nmthbf dRB n+1\mathbf{B}^n \mathbb{R}\to \mthbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R} in SmoothGrpdSmooth \infty Grpd by constructions in [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet].

Recall the discussion of B nU(1)\mathbf{B}^n U(1) and of dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) from above.

Definition

For nn \in \mathbb{N} define the simplicial presheaf

B nU(1) diff,chn:=Ξ(0C (,U(1))Ω 1()d dRIdΩ 1()Ω 2()d dR+Idd dR±IdΩ n()). \mathbf{B}^n U(1)_{diff,chn} := \array{ \Xi\left( 0\stackrel{}{\to} {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)} \stackrel{d_{dR} - Id}{\to} {\Omega^1(-) \atop \oplus \Omega^2(-)} \stackrel{d_{dR} + Id}{\to} \cdots \stackrel{d_{dR} \pm Id}{\to} \Omega^n(-) \right) } \,.
Proposition

The evident projection

B nU(1) diff,chnB nU(1) chn \mathbf{B}^n U(1)_{diff,chn} \to \mathbf{B}^n U(1)_{chn}

is a weak equivalence in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}. Moreover, the universal curvature characteristic

curv:B nU(1) dRB n+1U(1) curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)

in SmoothGrpdSmooth \infty Grpd is presented in [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc} by a span

B nU(1) diff,chn curv chn dRB n+1U(1) chn B nU(1) \array{ \mathbf{B}^n U(1)_{diff,chn} &\stackrel{curv_{chn}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{chn} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n U(1) }

where the horizontal morphism is the evident projection

Ξ(0C (,U(1))Ω 1()d dRIdΩ 1()Ω 2()d dR+Idd dR±IdΩ n()) Ξ(0Ω 1()d dRΩ 2()d dRd dRΩ cl n+1())). \array{ \Xi\left(\; 0\stackrel{}{\to} {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)} \stackrel{d_{dR} - Id}{\to} {\Omega^1(-) \atop \oplus \Omega^2(-)} \stackrel{d_{dR} + Id}{\to} \cdots \stackrel{d_{dR} \pm Id}{\to} \Omega^n(-) \right) \\ \downarrow \\ \Xi\left( \; 0 \stackrel{}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega_{cl}^{n+1}(-)) \right) } \,.
Proof

We need to present the defining (∞,1)-pullback

B nU(1) * curv dRB n+1U(1) B n+1U(1) * B n+1U(1) \array{ \mathbf{B}^n U(1) &\to& * \\ {}^{\mathllap{curv}}\downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) &\to& \mathbf{\flat} \mathbf{B}^{n+1} U(1) \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}^{n+1} U(1) }

by a homotopy pullback in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} (since (∞,1)-sheafification preserves finite (∞,1)-pullbacks it is sufficient to present the (,1)(\infty,1)-pullback in (∞,1)-presheaves).

We claim that we have a commuting diagram

Ξ(0C (,U(1))Ω 1()d dRIdΩ 1()Ω 2()d dR+Idd dR±IdΩ n()) Ξ(C (,U(1))Id+d dRC (,U(1))Ω 1()d dRIdΩ 1()Ω 2()d dR+IdΩ n1()Ω n()d dR±IdΩ n()) (Id,p 2,p 2,,p 2,d dR) Ξ(0Ω 1()d dRΩ 2()d dRd dRΩ cl n+1()) (C (,U(1))d dRΩ 1()d dRΩ 2()d dRd dRΩ cl n+1()) Ξ(0000) Ξ(C (,U(1))000) \array{ \Xi(\; 0\stackrel{}{\to} {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)} \stackrel{d_{dR} - Id}{\to} {\Omega^1(-) \atop \oplus \Omega^2(-)} \stackrel{d_{dR} + Id}{\to} \cdots \stackrel{d_{dR} \pm Id}{\to} \Omega^n(-) \;) &\to& \Xi(\; C^\infty(-,U(1)) \stackrel{Id + d_{dR}}{\to} {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)} \stackrel{d_{dR} - Id}{\to} {\Omega^1(-) \atop \oplus \Omega^2(-)} \stackrel{d_{dR} + Id}{\to} \cdots { \Omega^{n-1}(-) \atop \oplus \Omega^n(-)} \stackrel{d_{dR} \pm Id}{\to} \Omega^n(-) \;) \\ \downarrow && \downarrow^{\mathrlap{(Id, p_2, p_2, \cdots, p_2,d_{dR})}} \\ \Xi( \; 0 \stackrel{}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega_{cl}^{n+1}(-)) &\to& (C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega_{cl}^{n+1}(-) \;) \\ \downarrow && \downarrow \\ \Xi( \; 0 \to 0 \to 0 \to \cdots \to 0\;) &\to& \Xi( \; C^\infty(-,U(1)) \to 0 \to 0 \to \cdots \to 0 \;) }

in [CartSp op,sSet] proj[CartSp^{op},sSet]_{proj} where

  • the objects are fibrant models for the corresponding objects in the above (,1)(\infty,1)-pullback diagram;

  • the two right vertical morphisms are fibrations;

  • the two squares are pullback squares.

Therefore this is a homotopy pullback in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} that realizes the (∞,1)-pullback in question.

For the lower square we had discussed this already above. For the upper square the same type of reasoning applies. The main point is to find the chain complex in the top right such that it is a resolution of the point and maps by a fibration onto our model for B nU(1)\mathbf{\flat}\mathbf{B}^n U(1). The top right complex is

C (,U(1)) d dR Ω 1() d dR Ω 2() d dR d dR Ω n() id id id id id C (,U(1)) d dR Ω 1() d dR Ω 2() d dR d dR Ω n() \array{ && C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\stackrel{d_{dR}}{\to}& \Omega^2(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) \\ &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \cdots & {}^{\mathrlap{id}}\nearrow& \\ C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\stackrel{d_{dR}}{\to}& \Omega^2(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) }

and the vertical map out of it into C (,U(1))d dRΩ 1()d dRΩ n()d dRΩ cl n+1()C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-) \stackrel{d_{dR}}{\to} \Omega^{n+1}_{cl}(-) is in positive degree the projection onto the lower row and in degree 0 the de Rham differential. This is manifestly surjective (by the Poincare lemma applied to each object UU \in CartSp) hence this is a fibration.

The pullback object in the top left is in this notation

B nU(1) diff,chn:=Ξ(C (,U(1)) d dR Ω 1() d dR d dR Ω n() id id id Ω 1() d dR d dR Ω n()) \mathbf{B}^n U(1)_{diff,chn} := \Xi \left( \array{ C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) \\ \oplus &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \cdots & {}^{\mathrlap{id}}\nearrow& \\ \Omega^1(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) } \right)

and in turn the top left vertical morphism curv:B diff nU(1) dRB n+1U(1)curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) is in positive degree the projection on the lower row and in degree 0 the de Rham differential.

Notice that the evident forgetful morphism B nU(1)B diff nU(1)\mathbf{B}^n U(1) \stackrel{}{\leftarrow} \mathbf{B}^n_{diff} U(1) is indeed a weak equivalence.

In the section on de Rham coefficients for exponentiated Lie algebras we had discussed an equivalent presentation of most of the objects above. We now formulate the curvature characteristic in this alternative form.

Observation

We may write the simplicial presheaf dRB n+1 smp\mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}_{smp} from above equivalently as follows

dRB n+1 smp:(U,[k]){Ω si,vert (U×Δ k) 0 0 Ω si (U×Δ k) A CE(b n)}, \mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}_{smp} : (U, [k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\stackrel{0}{\leftarrow}& 0 \\ \uparrow && \uparrow \\ \Omega_{si}^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& CE(b^{n}\mathbb{R}) } \right\} \,,

where on the right we have the set of commuting diagrams in dgAlg of the given form, with the vertical morphisms being the canonical projections.

Definition

Write W(b n1)W(b^{n-1}\mathbb{R}) \in dgAlg for the Weil algebra of the line Lie n-algebra, defined to be the free commutative dg-algebra on a single generator in degree nn, hence the graded commutative algebra on a generator in degree nn and a generator in degree (n+1)(n+1) equipped with the differential that takes the former to the latter.

Observation

We have the following properties of W(b n1)\mathrm{W}(b^{n-1}\mathbb{R})

  • There is a canonical natural isomorphism

    Hom dgAlg(W(b n1),Ω (U))Ω n(U) Hom_{dgAlg}(W(b^{n-1} \mathbb{R}), \Omega^\bullet(U)) \simeq \Omega^n(U)

    between dg-algebra homomorphisms A:W(b n1)Ω (X)A : W(b^{n-1}\mathbb{R}) \to \Omega^\bullet(X) from the Weil algebra of b n1b^{n-1}\mathbb{R} to the de Rham complex and degree-nn differential forms, not necessarily closed.

  • There is a canonical dg-algebra homomorphism W(b n1)CE(b n1)W(b^{n-1}\mathbb{R}) \to CE(b^{n-1}\mathbb{R}) and the differential nn-form corresponding to AA factors through this morphism precisely if the curvature d dRAd_{dR} A of AA vanishes.

  • The image under exp()\exp(-)

    exp(inn(b n1))exp(b n) \exp(\mathrm{inn}(b^{n-1})\mathbb{R}) \to \exp(b^{n}\mathbb{R})

    of the canonical dg-algebra morphism W(b n1)CE(b n)\mathrm{W}(b^{n-1}\mathbb{R}) \leftarrow \mathrm{CE}(b^n \mathbb{R}) is a fibration in [CartSp smooth op,sSet] proj[\mathrm{CartSp}_{\mathrm{smooth}}^{\mathrm{op}}, \mathrm{sSet}]_{\mathrm{proj}} that presents the point inclusion *B n+1{*} \to \mathbf{B}^{n+1}\mathbb{R} in SmoothGrpd\mathrm{Smooth}\infty \mathrm{Grpd}.

Definition

Let B n diff,smp[CartSp smooth op,sSet]\mathbf{B}^n \mathbb{R}_{diff,smp} \in [CartSp_{smooth}^{op}, sSet] be the simplicial presheaf defined by

B n diff,smp:(U,[k]){Ω si,vert (U×Δ k) A vert CE(b n1) Ω si (U×Δ k) A W(b n1)}, \mathbf{B}^n \mathbb{R}_{diff,smp} : (U,[k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega_{si}^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(b^{n-1}\mathbb{R}) } \right\} \,,

where on the right we have the set of commuting diagrams in dgAlg as indicated.

Remark

This means that an element of B n diff,smp(U)[k]\mathbf{B}^n \mathbb{R}_{diff,smp}(U)[k] is a smooth nn-form AA (with sitting instants) on U×Δ kU \times \Delta^k such that its curvature (n+1)(n+1)-form dAd A vanishes when restricted in all arguments to vector fields tangent to Δ k\Delta^k. We may write this condition as dAΩ si 1,(U×Δ k)d A \in \Omega^{\bullet \geq 1, \bullet}_{si}(U \times \Delta^k).

Observation

There are canonical morphisms

B n diff,smp curv smp dRB n smp B n smp \array{ \mathbf{B}^n \mathbb{R}_{diff,smp} &\stackrel{curv_{smp}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{smp} \\ \downarrow \\ \mathbf{B}^n \mathbb{R}_{smp} }

in [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet], where the vertical map is given by remembering only the top horizontal morphism in the above square diagram, and the horizontal morphism is given by forming the pasting composite

curv smp:{Ω si,vert (U×Δ k) A vert CE(b n1) Ω si (U×Δ k) A W(b n1)}{Ω si,vert (U×Δ k) A vert CE(b n1) 0 Ω si (U×Δ k) A W(b n1) CE(b n)}. curv_{smp} : \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega_{si}^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(b^{n-1}\mathbb{R}) } \right\} \;\; \mapsto \;\; \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(b^{n-1}\mathbb{R}) &\leftarrow& 0 \\ \uparrow && \uparrow && \uparrow \\ \Omega_{si}^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(b^{n-1}\mathbb{R}) &\stackrel{}{\leftarrow}& CE(b^{n}\mathbb{R}) } \right\} \,.
Proposition

This span is a presentation in [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet] of the universal curvature characteristics curv:B n dRB n+1curv : \mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} in SmoothGrpdSmooth \infty Grpd.

Proof

We need to produce a fibration resolution of the point inclusion *B n+1 smp* \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}_{smp} in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj} and then show that the above is the ordinary pullback of this along dRB n+1 smpB n+1 smp\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{smp} \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}_{smp} .

We claim that this is achieved by the morphism

(U,[k]):{Ω si (U×Δ k)W(b n1)}{Ω si (U×Δ k)W(b n1)CE(b n)}. (U,[k]) : \{ \Omega^\bullet_{si}(U \times \Delta^k) \leftarrow W(b^{n-1} \mathbb{R}) \} \mapsto \{ \Omega^\bullet_{si}(U \times \Delta^k) \leftarrow W(b^{n-1} \mathbb{R}) \leftarrow CE(b^{n} \mathbb{R}) \} \,.

Here the simplicial presheaf on the left is that which assigns the set of arbitrary nn-forms (with sitting instants but not necessarily closed) on U×Δ kU \times \Delta^k and the map is simply given by sending such an nn-form AA to the (n+1)(n+1)-form d dRAd_{dR} A.

It is evident that the simplicial presheaf on the left resolves the point: since there is no condition on the forms every form on U×Δ kU \times \Delta^k is in the image of the map of the normalized chain complex of a form on U×Δ k+1U \times \Delta^{k+1}: such is given by any form that is, up to a sign, equal to the given form on one nn-face and 0 on all the other faces. Clearly such forms exist.

Moreover, this morphism is a fibration in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}, for instanxce because its image under the normalized chains complex functor is a degreewise surjection, by the Poincare lemma.

Now we observe that we have over each (U,[k])(U,[k]) a double pullback diagram in Set

{Ω si,vert (U×Δ k) A vert CE(b n1) Ω si (U×Δ k) A W(b n1)} {Ω si,vert (U×Δ k) W(b n1) id Ω si (U×Δ k) W(b n1)} {Ω si,vert (U×Δ k) 0 Ω si (U×Δ k) CE(b n)} {Ω si,vert (U×Δ k) CE(b n) id Ω si (U×Δ k) CE(b n)} {Ω si,vert (U×Δ k) 0 Ω si (U×Δ k) 0} {Ω si,vert (U×Δ k) CE(b n) Ω si (U×Δ k) 0}, \array{ \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(b^{n-1}\mathbb{R}) } \right\} &\to& \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{}{\leftarrow}& W(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow^{\mathrlap{id}} \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& W(b^{n-1} \mathbb{R}) } \right\} \\ \downarrow && \downarrow \\ \left\{ \array{ \Omega^\bullet_{si, vert}(U \times \Delta^k) &\stackrel{}{\leftarrow}& 0 \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& CE(b^{n} \mathbb{R}) } \right\} &\to& \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n} \mathbb{R}) \\ \uparrow && \uparrow^{\mathrlap{id}} \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& CE(b^{n} \mathbb{R}) } \right\} \\ \downarrow && \downarrow \\ \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& 0 \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& 0 } \right\} &\to& \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(b^{n} \mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{}{\leftarrow}& 0 } \right\} } \,,

hence a coresponding pullback diagram of simplicial presheaves, that we claim is a presentation for the defining double (∞,1)-pullback

B n * curv dRB n+1 B n+1 * B n+1 \array{ \mathbf{B}^n \mathbb{R} &\to& * \\ {}^{\mathllap{curv}}\downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} &\to& \mathbf{\flat} \mathbf{B}^{n+1} \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}^{n+1} \mathbb{R} }

for curvcurv.

The bottom square is the one we already discussed for the de Rham coefficients. Since the top right vertical morphism is a fibration, also the top square is a homotopy pullback and hence exhibits the defining (,1)(\infty,1)-pullback for curv.

Corollary

The degreewise map

(1) +1 Δ :B n diff,smpB n diff,chn (-1)^{\bullet+1} \int_{\Delta^\bullet} : \mathbf{B}^n \mathbb{R}_{diff,smp} \to \mathbf{B}^n \mathbb{R}_{diff,chn}

that sends an nn-form AΩ n(U×Δ k)A \in \Omega^n(U \times \Delta^k) and its curvature dAd A to (1) k+1(-1)^{k+1} times its fiber integration ( Δ kA, Δ kdA)(\int_{\Delta^k} A, \int_{\Delta^k} d A) is a weak equivalence in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}.

Proof

Since under homotopy pullbacks a weak equivalence of diagrams is sent to a weak equivalence. See the analagous argument above.

The canonical form on a simplicial Lie group

Above we discussed the canonical differential form on smooth \infty-groups GG for the special cases where a) GG is a Lie group and b) where GG is a circle Lie n-group. These are both in turn special cases of the situation where GG is a Lie simplicial group. This we discuss now.

Proposition

For GG a Lie simplicial group, the flat de Rham coefficient object dRBG\mathbf{\flat}_{dR}\mathbf{B}G is presented by the simplicial presheaf which in degree kk is given by Ω flat 1(,𝔤 k)\Omega^1_{flat}(-, \mathfrak{g}_k), where 𝔤 k=Lie(G k)\mathfrak{g}_k = Lie(G_k) is the Lie algebra of G kG_k.

Proof

Let

Ω flat 1(,𝔤 )//G =(Ω flat 1(,𝔤 )×C (,G )Ω flat 1(,𝔤 )) \Omega^1_{flat}(-,\mathfrak{g}_\bullet) //G_\bullet = \left( \Omega^1_{flat}(-,\mathfrak{g}_\bullet) \times C^\infty(-,G_\bullet) \stackrel{\to}{\to} \Omega^1_{flat}(-,\mathfrak{g}_\bullet) \right)

be the presheaf of simplicial groupoids which in degree kk is the groupoid of Lie-algebra valued forms with values in G kG_k from above. As in the above discussion there we have that under the degreewise nerve this is a degreewise fibrant resolution of presheaves of bisimplicial sets

N(Ω flat 1(,𝔤 )//G )N*//G =NB(G disc) N \left( \Omega^1_{flat}(-,\mathfrak{g}_\bullet) // G_\bullet \right) \to N *//G_\bullet = N B (G_{disc})_\bullet

of the standard presentation of the delooping of the discrete group underlying GG. By the discussion at bisimplicial set we know that under taking the diagonal

diag:sSet ΔsSet diag : sSet^\Delta \to sSet

the object on the right is a presentation for dRBG\mathbf{\flat}_{dR} \mathbf{B}G, because

diagNB(G disc) W¯(G disc)BG. diag N B (G_{disc})_\bullet \stackrel{\simeq}{\to} \bar W (G_{disc}) \simeq \mathbf{\flat}\mathbf{B}G \,.

Now observe that the morphism

diag(NΩ flat 1(,𝔤 )//G )diagN*//G disc diag (N \Omega^1_{flat}(-,\mathfrak{g}_\bullet) // G_\bullet ) \to diag N *// G_{disc}

is a global fibration. This is in fact true for every morphism of the form

diagN(S //G )diag*//G diag N (S_\bullet//G_\bullet) \to diag *//G_\bullet

for S //G *//G S_\bullet//G_\bullet \to *//G_\bullet a simlicial action groupoid projection with GG a simplicial group acting on a Kan complex SS: we have that

(diagN(S//G)) k=S k×(G k) × k. (diag N (S//G))_k = S_k \times (G_k)^{\times_k} \,.

On the second factor the horn filling condition is simply that of the identity map diagNBGdiagNBGdiag N B G \to diag N B G which is evidently solvable, whereas on the first factor it amounts to S*S \to * being a Kan fibration, hence to SS being Kan fibrant.

But the simplicial presheaf Ω flat 1(,𝔤 )\Omega^1_{flat}(-,\mathfrak{g}_\bullet) is indeed Kan fibrant: for a given UCartSpU \in CartSp we may use parallel transport to (non-canonically) identify

Ω flat 1(U,𝔤 k)SmoothMfd *(U,G k), \Omega^1_{flat}(U, \mathfrak{g}_k) \simeq SmoothMfd_*(U, G_k) \,,

where on the right we have smooth functions that send the origin of UU to the neutral element. But since G G_\bullet is Kan fibrant and has smooth global fillers (by the discussion at simplicial group one can give algebraic formulas for the fillers, which translate into smooth manps) als SmoothMfd *(U,G )SmoothMfd_*(U,G_\bullet) is Kan fibrant.

In summary this means that the defining homotopy pullback

dRBG:=BG× BG*\mathbf{\flat}_{dR} \mathbf{B}G := \mathbf{\flat} \mathbf{B}G \times_{\mathbf{B}G} *

is presented by the ordinary pullback of simplicial presheaves

diagNΩ flat 1(,𝔤 )×diagNBG *=Ω 1(,𝔤 ). diag N \Omega^1_{flat}(-,\mathfrak{g}_\bullet) \times diag N B G_\bullet * = \Omega^1(-, \mathfrak{g}_\bullet) \,.
Proposition

For GG a simplicial Lie group the canonical differential form

θ:G dRBG \theta : G \to \mathbf{\flat}_{dR} \mathbf{B}G

is presented in terms of the above presentation for dRBG\mathbf{\flat}_{dR} \mathbf{B}G by the morphisms of simplicial presheaves

θ k:G kΩ flat 1(,𝔤 k) \theta_k : G_k \to \Omega^1_{flat}(-, \mathfrak{g}_k)

which is the presheaf-incarnation of the Maurer-Cartan form of the ordinary Lie group G kG_k.

Proof

Continuing with the strategy of the previous proof we find a resolution of *BG* \to \mathbf{\flat} \mathbf{B}G by applying the construction of The canonical form on a Lie group degreewise and then applying diagNdiag N.

The defining homotopy pullback

G * dR BG \array{ G &\stackrel{}{\to}& * \\ \downarrow && \downarrow \\ \mathbf{\flat}_{dR} &\to& \mathbf{\flat} \mathbf{B}G }

for θ\theta is this way presented by the ordinary pullback

G diagN(Ω flat 1(,𝔤 )) triv//G ) Ω flat 1(,𝔤 ) diagN(Ω flat 1(,𝔤 )//G ) \array{ G_\bullet &\stackrel{}{\to}& diag N \left( \Omega^1_{flat}(-, \mathfrak{g}_\bullet))_{triv} // G_\bullet \right) \\ \downarrow && \downarrow \\ \Omega^1_{flat}(-, \mathfrak{g}_\bullet) &\to & diag N (\Omega^1_{flat}(-,\mathfrak{g}_\bullet)//G_\bullet) }

of simplicial presheaves, where Ω flat 1(,𝔤 k)\Omega^1_{flat}(-,\mathfrak{g}_\k) is the set of flat 𝔤\mathfrak{g}-valued forms AA equipped with a gauge transformation 0gA0 \stackrel{g}{\to} A. As in the above proof one finds that the right vertical morphism is a fibration, hence indeed a resolution of the point inclusion. The pullback is degreewise that from the case of ordinary Lie groups and thus the result follows.

We can now give a simplicial description of the canonical curvature form θ:B nU(1) dRB n+1U(1)\theta : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1) that above we obtained by a chain complex model:

Example

The canonical form on the circle Lie n-group

θ:B n1U(1) dRB nU(1) \theta : \mathbf{B}^{n-1}U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)

is presented by the simplicial map

Ξ(U(1)[n1])Ξ(Ω cl 1()[n1]) \Xi( U(1)[n-1] ) \to \Xi( \Omega^1_{cl}(-)[n-1] )

which is simply the Maurer-Cartan form on U(1)U(1) in degree nn.

The equivalence to the model we obtained before is given by noticing the equivalence in hypercohomology of chain complexes of abelian sheaves

Ω cl 1()[n](Ω 1()d dRd dRΩ cl n()) \Omega^1_{cl}(-)[n] \simeq (\Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-) )

on CartSp.

Flat Ehresmann connections

We discuss the realization of the general abstract concept of flat Ehresmann infinity-connections realized in SmoothGrpdSmooth\infty Grpd. We show that when applied to an ordinary Lie group this reproduces the traditional notion of Ehresmann connection.

(…)

Differential cohomology

We discuss the intrinsic differential cohomology in SmoothGrpdSmooth \infty Grpd

We first expose the simple special case of ordinary U(1)U(1)-principal bundles with connection in more detail. Then we turn to the general case.

Circle 0- and circle 1-bundle with connection

Circle bundles with connection

Before discussing the full theorem, it is instructive to start by looking at the special case n=1n=1 in some detail, which is about ordinary U(1)U(1)-principal bundles with connection.

This contains in it already all the relevant structure of the general case, but the low categorical degree is more transparently written out and will allow us to pause to highlight some maybe noteworthy aspects of the situation, such as the phenomenon of pseudo-connections below.

In terms of the Dold-Kan correspondence the object BU(1)H\mathbf{B}U(1) \in \mathbf{H} is modeled in [CartSp op,sSet][CartSp^{op}, sSet] by

BU(1)=Ξ(C (,U(1))0). \mathbf{B}U(1) = \Xi(\; C^\infty(-,U(1)) \to 0 \;) \,.

Accordingly we have for the double delooping the model

B 2U(1)=Ξ(C (,U(1))00) \mathbf{B}^2 U(1) = \Xi( \; C^\infty(-,U(1)) \to 0 \to 0 \;)

and for the universal principal 2-bundle

EBU(1)=Ξ(C (,U(1))IdC (,U(1))0). \mathbf{E}\mathbf{B}U(1) = \Xi( \; C^\infty(-,U(1)) \stackrel{Id}{\to} C^\infty(-, U(1)) \to 0 \; ) \,.

In this notation we have also the constant presheaf

B 2U(1)=Ξ(constU(1)00). \mathbf{\flat} \mathbf{B}^2 U(1) = \Xi( \; const U(1) \to 0 \to 0 \; ) \,.

Above we already found the model

dRB 2U(1)=Ξ(0Ω 1()d dRΩ cl 2()). \mathbf{\flat}_{dR} \mathbf{B}^2 U(1) = \Xi(0 \to \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-)) \,.

In order to compute the differential cohomology H diff(,BU(1))\mathbf{H}_{diff}(-,\mathbf{B}U(1)) by an ordinary pullback in sSet we also want to resolve the curvature characteristic morphism BU(1) dRB 2U(1)\mathbf{B}U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1) by a fibration. We claim that this may be obtained by choosing the resolution BU(1)BU(1) diff,chn\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B} U(1)_{diff,chn} given by

BU(1) diff:=Ξ(C (,U(1))Ω 1()d dRIdΩ 1()) \mathbf{B}U(1)_{diff} := \Xi( \; C^\infty(-,U(1)) \oplus \Omega^1(-) \stackrel{d_{dR} \oplus Id}{\to} \Omega^1(-) \; )

with the morphism curv:B diffU(1) dRB 2U(1)curv : \mathbf{B}_{diff}U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^2 U(1) given by

C (,U(1))Ω 1() d dR+Id Ω 1() p 2 d dR Ω 1() d dR Ω cl 2(). \array{ C^\infty(-,U(1)) \oplus \Omega^1(-) &\stackrel{d_{dR} + Id}{\to}& \Omega^1(-) \\ \downarrow^{\mathrlap{p_2}} && \downarrow^{\mathrlap{d_{dR}}} \\ \Omega^1(-) &\stackrel{d_{dR}}{\to}& \Omega^2_{cl}(-) } \,.

By the Poincare lemma applied to each Cartesian space, this is indeed a fibration.

In the next section we give the proof of this (simple) claim. Here in the warmup phase we instead want to discuss the geometric interpretation of this resolution, along the lines of the section curvature characteristics of 1-bundles in the survey-part.

Proposition

We have the following geometric interpretation of the above models:

dRB 2U(1):U{U * Π 2(U) B 2U(1)}={Π 2(U)B 2U(1)} \mathbf{\flat}_{dR} \mathbf{B}^2 U(1) : U \mapsto \left\{ \array{ U &\to& * \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}^2 U(1) } \right\} = \left\{ \mathbf{\Pi}_2(U) \to \mathbf{B}^2 U(1) \right\}

and

BU(1) diff:U{U BU(1) Π 2(U) BINN(U(1))}. \mathbf{B}U(1)_{diff} : U \mapsto \left\{ \array{ U &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}INN(U(1)) } \right\} \,.

And in this presentation the morphism curv:B diffU(1)B 2U(1)curv : \mathbf{B}_{diff}U(1) \to \mathbf{B}^2 U(1) is given over UCartSpU \in CartSp by forming the pasting composite

U BU(1) underlyingcocycle Π 2(U) BINN(U(1)) connection Π 2(U) B 2U(1) curvature \array{ U &\to& \mathbf{B}U(1) &&& underlying\;cocycle \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}INN(U(1)) &&& connection \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}^2 U(1) &&& curvature }

and picking the lowest horizontal morphism.

Here the terms mean the following:

  • INN(U(1))INN(U(1)) is the 2-group Ξ(U(1)U(1))\Xi(U(1) \to U(1)), which is a groupal model for the universal U(1)-principal bundle EU(1)\mathbf{E}U(1);

  • Π 2(U)\mathbf{\Pi}_2(U) is the path 2-groupoid with homotopy class of 2-dimensional paths as 2-morphisms

  • the groupoids of diagrams in braces have as objects commuting diagrams in [CartSp op,sSet][CartSp^{op}, sSet] as indicated, and horizontal 2-morphisms fitting into such diagrams as morphisms.

Using the discussion at 2-groupoid of Lie 2-algebra valued forms (SchrWalII) we have the following:

  1. For XX a smooth manifold, morphisms in [CartSp op,2Grpd][CartSp^{op}, 2Grpd] of the form tra A:Π 2(X)EBU(1)tra_A : \Pi_2(X) \to \mathbf{E}\mathbf{B}U(1) are in bijection with smooth 1-forms AΩ 1(X)A \in \Omega^1(X): the 2-functor sends a path in XX to the the parallel transport of AA along that path, and sends a surface in XX to the exponentiated integral of the curvature 2-form F A=dAF_A = d A over that surface. The Bianchi identity dF A=0d F_A = 0 says precisely that this assignment indeed descends to homotopy classes of surfaces, which are the 2-morphisms in Π 2(X)\Pi_2(X).

  2. Moreover 2-morphisms of the form (λ,α):tra Atra A(\lambda,\alpha) : tra_A \to \tra_{A'} in [CartSp op,2Grpd][CartSp^{op}, 2Grpd] are in bijection with pairs consisting of a λC (X,U(1))\lambda \in C^\infty(X,U(1)) and a 1-form αΩ 1(X)\alpha \in \Omega^1(X) such that A=A+d dRλαA' = A + d_{dR} \lambda - \alpha.

  3. And finally 3-morphisms h:(λ,α)(λ,α)h : (\lambda, \alpha) \to (\lambda', \alpha') are in bijection with hC (X,U(1))h \in C^\infty(X,U(1)) such that λ=λh\lambda' = \lambda \cdot h and α=α+d dRh\alpha' = \alpha + d_{dR} h.

By the same reasoning we find that the coefficient object for flat B 2U(1)\mathbf{B}^2 U(1)-valued differential cohomology is

B 2U(1)=[Π 2(),B 2U(1)]=Ξ(C (,U(1))d dRΩ 1()d dRΩ cl 2()). \mathbf{\flat}\mathbf{B}^2 U(1) = [\Pi_2(-), \mathbf{B}^2U(1)] = \Xi( C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-) ) \,.

So by the above definition of differential cohomology in H\mathbf{H} we find that BU(1)\mathbf{B}U(1)-differential cohomology of a paracompact smooth manifold XX is given by choosing any good open cover {U iX}\{U_i \to X\}, taking C({U i})C(\{U_i\}) to be the Cech nerve, which is then a cofibrant replacement of XX in [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov} and forming the ordinary pullback

H diff(X,BU(1)) H dR 2(X) [CartSp op,sSet](C({U i}),B diffU(1)) curv [CartSp op,sSet](C({U i}), dRB 2U(1)) \array{ \mathbf{H}_{diff}(X,\mathbf{B}U(1)) &\to& H^2_{dR}(X) \\ \downarrow && \downarrow \\ [CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1)) &\stackrel{curv}{\to}& [CartSp^{op},sSet](C(\{U_i\}), \flat_{dR}\mathbf{B}^2 U(1)) }

(because the bottom vertical morphism is a fibration, by the fact that our model for B diffU(1) dRB 2U(1)\mathbf{B}_{diff} U(1) \to \flat_{dR}\mathbf{B}^2 U(1) is a fibration, that C({U i})C(\{U_i\}) is cofibrant and using the axioms of the sSet-enriched model category [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj}).

Observations

A cocycle in [CartSp op,sSet](C({U i}),B diffU(1))[CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1)) is

  1. a collection of functions

    (g ijC (U iU j,U(1))) (g_{i j } \in C^\infty(U_i \cap U_j, U(1)))

    satsifying g ijg jk=g ikg_{i j} g_{j k} = g_{i k} on U iU jU kU_i \cap U_j \cap U_k;

  2. a collection of 1-forms

    (A iΩ 1(U i)) (A_i \in \Omega^1(U_i))
  3. a collection of 1-forms

    (a ijΩ (U iU j)) (a_{i j} \in \Omega^(U_i \cap U_j))

    such that

    A j=A i+d dRg ij+a ij A_j = A_i + d_{dR} g_{i j} + a_{i j}

    on U iU jU_i \cap U_j and

    a ij+a jk=a ik a_{i j} + a_{j k} = a_{i k}

    on U iU jU kU_i \cap U_j \cap U_k.

The curvature-morphism takes such a cocycle to the cocycle

(dA i,a ij,) (d A_i, a_{i j}, )

in the above model [CartSp op,sSet](C({U i}), dRB 2U(1))[CartSp^{op},sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)) for intrinsic de Rham cohomology.

Every cocycle with nonvanishing (a ij)(a_{i j}) is in [C({U i}),B diffU(1)][C(\{U_i\}), \mathbf{B}_{diff}U(1)] coboundant to one with vanishing (a ij)(a_{i j})

Proof

The first statements are effectively the definition and the construction of the above models. The last statement is as in the above discussion of our model for ordinary de Rham cohomology: given a cocycle with non-vanishing closed a ija_{i j}, pick a partition of unity (ρ iC (X))(\rho_i \in C^\infty(X)) subordinate to the chosen cover and the coboundary given by ( i 0ρ i 0a i 0i)(\sum_{i_0} \rho_{i_0} a_{i_0 i}). This connects (A i,a ij,g ij)(A_i,a_{i j}, g_{i j}) with the cocycle (A i,a ij,g ij)(A'_i, a'_{i j}, g_{i j}) where

A i=A i+ i 0ρ i 0a i 0i A'_i = A_i + \sum_{i_0} \rho_{i_0} a_{i_0 i}

and

a ij =A jA idg ij =a ij+ i 0(a i 0ia i 0j) =0. \begin{aligned} a'_{i j} & = A'_j - A'_i - d g_{i j} \\ & = a_{i j} + - \sum_{i_0}( a_{i_0 i} - a_{i_0 j} ) \\ & = 0 \end{aligned} \,.

So in total we have found the following story:

  1. In order to compute the curvature characteristic form of a Cech cohomology cocycle g:C({U i})BU(1)g : C(\{U_i\}) \to \mathbf{B}U(1) of a U(1)U(1)-principal bundle, we first lift it

    B diffU(1) (g,) C({U i}) g BU(1) \array{ && \mathbf{B}_{diff}U(1) \\ & {}^{\mathllap{(g,\nabla)}}\nearrow & \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1) }

    to an equivalent B diffU(1)\mathbf{B}_{diff}U(1)-cocycle, and this amounts to putting (the Cech-representatitve of) a pseudo-connection on the U(1)U(1)-principal bundle.

  2. From that lift the desired curvature characteristic is simply projected out

    B diffU(1) curv dRB 2U(1) (g,) C({U i}) g BU(1), \array{ && \mathbf{B}_{diff}U(1) &\stackrel{curv}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^2 U(1) \\ & {}^{\mathllap{(g,\nabla)}}\nearrow & \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1) } \,,

    and we find that it lives in the sheaf hypercohomology that models ordinary de Rham cohomology.

  3. Therefore we find that in each cohomology class of curvatures, there is at least one representative which is an ordinary globally defined 2-form. Moreover, the pseudo-connections that map to such a representative are precisely the genuine connections, those for which the (a ij)(a_{i j})-part of the cocycle vaishes.

So we see that ordinary connections on ordinary circle bundles are a means to model the homotopy pullback

H diff(X,BU(1)) H dR 2(X) H(X,BU(1)) H dR(X,BU(1)) \array{ \mathbf{H}_{diff}(X,\mathbf{B}U(1)) &\to& H_{dR}^2(X) \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}U(1)) &\to& \mathbf{H}_{dR}(X,\mathbf{B}U(1)) }

in a 2-step process: first the choice of a pseudo-connection realizes the bottom horizontal morphism as an anafunctor, and then second the restriction imposed by forming the ordinary pullback chooses from all pseudo-connections precisely the genuine connections.

The general version of this story is discussed in detail at differential cohomology in an (∞,1)-topos – Local (pseudo-)connections.

Circle bundles with pseudo-connection

In the above discussion of extracting ordinary connections on ordinary U(1)U(1)-principal bundles from the abstract topos-theoretic definition of differential cohomology, we argued that a certain homotopy pullback may be computed by choosing in the Cech-hypercohomology of the complex of sheaves (Ω 1()d dRΩ cl 2())(\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-)) over a manifold XX those cohomology representatives that happen to be represented by globally defined 2-forms on XX. We saw that the homotopy fiber of pseudo-connections over these 2-forms happened to have connected components indexed by genuine connections.

But by the general abstract theory, up to isomorphism the differential cohomology computed this way is guaranteed to be independent of all such choices, which only help us to compute things.

To get a feeling for what is going on, it may therefore be useful to re-tell the analgous story with pseudo-connections that are not genuine connections.

By the very fact that BU(1)B diffU(1)\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B}_{diff}U(1) is a weak equivalence, it follows that every pseudo-connection is equivalent to an ordinary connection as cocoycles in [CartSp op,sSet](C({U i}),B diff(G))[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}_{diff}(G)).

If we choose a partition of unity (ρ iC (X,))(\rho_i \in C^\infty(X,\mathbb{R})) subordinate to the cover {U iX}\{U_i \to X\}, then we can construct the corresponding coboundary explicitly:

let (A ig ij,a ij)(A_i g_{ij}, a_{i j}) be an arbitrary pseudo-connection cocycle. Consider the Cech-hypercohomology coboundary given by ( i 0ρ i 0a i 0i,0)(\sum_{i_0} \rho_{i_0} a_{i_0 i}, 0). This lands in the shifted cocycle

(A i:=A i+ i 0ρ i 0a i 0i,g ij,a ij), (A'_i := A_i + \sum_{i_0} \rho_{i_0} a_{i_0 i}, g_{i j}, a'_{i j}) \,,

and we can find the new pseudo-components a ija'_{i j} by

a ij=A jA id dRg ij. a'_{i j} = A'_j - A'_i - d_{dR} g_{i j} \,.

Using the computation

i 0ρ i 0(a i 0ia i 0j = i 0ρ i 0(a ii 0+a i 0j = i 0ρ i 0a ij =a ij \begin{aligned} \sum_{i_0} \rho_{i_0} (a_{i_0 i} - a_{i_0 j} &= - \sum_{i_0} \rho_{i_0} (a_{i i_0} + a_{i_0 j} \\ & = \sum_{i_0} \rho_{i_0} a_{i j} \\ & = a_{i j} \end{aligned}

we find that these indeed vanish.

The most drastic example for this is a lift \nabla of a cocycle g=(g ij)g = (g_{i j}) in

B diffU(1) C({U i}) g BU(1) \array{ && \mathbf{B}_{diff} U(1) \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1) }

is one which takes all the ordinary curvature forms to vanish identically

=(A i:=0,g ij,a ij). \nabla = (A_i := 0, g_{i j}, a_{i j}) \,.

This fixes the pseudo-components to be a ij=dg ija_{i j} = - d g_{i j}. By the above discussion, this pseudo-connection with vanishing connection 1-forms is equivalent, as a pseudo-connection, to the ordinary connection cocycle with connection forms (A i:= i 0ρ i 0dg i 0i)(A_i := \sum_{i_0} \rho_{i_0} d g_{i_0 i}). This is a standard formula for equipping U(1)U(1)-principal bundles with Cech cocycle (g ij)(g_{i j}) with a connection.

Circle 0-bundle

We saw above that the intrinsic coefficient object dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) yields ordinary de Rham cohomology in degree n>1n \gt 1. For n=1n = 1 we have that dRBU(1)\mathbf{\flat}_{dR} \mathbf{B}U(1) is given simply by the 0-truncated sheaf of 1-forms, Ω 1():CartSp opSetsSet\Omega^1(-) : CartSp^{op} \to Set \hookrightarrow sSet. Accordingly we have for XX a paracompact smooth manifold

H(X, dRBU(1))=Ω cl 1(X) \mathbf{H}(X, \mathbf{\flat}_{dR}\mathbf{B}U(1)) = \Omega^1_{cl}(X)

instead of H dR 1(X)H^1_{dR}(X).

There is a good reason for this discrepancy: for n1n \geq 1 the object dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) is the recipient of the intrinsic curvature characteristic morphism

curv B n1U(1):B n1U(1) dRB nU(1). curv_{\mathbf{B}^{n-1} U(1)} : \mathbf{B}^{n-1} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1) \,.

For XB n1U(1)X \to \mathbf{B}^{n-1} U(1) a cocycle (an (n2)(n-2)-gerbe without connection), the cohomology class of the composite XB n1U(1) dRB nU(1)X \to \mathbf{B}^{n-1} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1) is precisely the obstruction to the existence of a flat extension XB n1U(1)B n1U(1)X \to \mathbf{\flat} \mathbf{B}^{n-1} U(1) \to \mathbf{B}^{n-1} U(1) for the original cocycle.

For n=2n = 2 this is the usual curvature 2-form of a line bundle, for n=3n = 3 it is curvature 3-form of a bundle gerbe, etc. But for n=1n = 1 we have that the original cocycle is just a map of spaces

f:XU(1). f : X \to U(1) \,.

This can be understoody as a cocycle for a groupoid principal bundle, for the 0-truncated groupoid with U(1)U(1) as its space of objects. Such a cocycle extends to a flat cocycle precisely if ff is constant as a function. The corresponding curvature 1-form is d dRfd_{dR} f and this is precisely the obstruction to constancy of ff already, in that ff is constant if and only if d dRfd_{dR} f vanishes. Not (necessarily) if it vanishes in de Rham cohomology .

This is the simplest example of a general statement about curvatures of higher bundles: the curvature 1-form is not subject to gauge transformations.

Circle nn-bundles with connection

Recall the definition of the intrinsic differential cohomology on XSmoothGrpdX \in Smooth \infty Grpd with coefficients in U(1)U(1) as the (∞,1)-pullback

H diff(X,B nU(1)) H dR(X,B n+1U(1)) H(X,B nU(1)) curv H dR(X,B n+1U(1)) \array{ \mathbf{H}_{diff}(X,\mathbf{B}^n U(1)) &\to & H_{dR}(X,\mathbf{B}^{n+1} U(1)) \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}^n U(1)) &\stackrel{curv}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1)) }

in ∞Grpd, where the morphism on the right picks one base point in each connected component.

Theorem

For XSmoothMfdSmoothGrpdX \in SmoothMfd \hookrightarrow Smooth \infty Grpd a paracompact smooth manifold we have

H diff(X,B nU(1))(H(X,(n+1) D ))× Ω cl n+1(X)H dR n+1int(X). H_{diff}(X,\mathbf{B}^n U(1)) \simeq \left( \;\; H(X,\mathbb{Z}(n+1)_D^\infty) \;\; \right) \times_{\Omega_{cl}^{n+1}(X)} H_{dR}^{n+1}_{int}(X) \,.

Here on the right we have the subset of Deligne cocycles that picks for each integral de Rham cohomology class of XX only one curvature form representative.

If we make use of the explicit presentation of SmoothGrpdSmooth \infty Grpd by the model structure on simplicial presheaves [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc} and the explicit presentation dRB n+1U(1) chn\mathbf{\flat}_{\mathrm{dR}} \mathbf{B}^{n+1}U(1)_{chn} for dRB n+1U(1)\mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) by ordinary differential forms, as above we may replace the right morphism in this pullback by Ω cl n+1(X)H dR(X,B n+1U(1))\Omega^{n+1}_{cl}(X) \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}U(1)) and consider the (∞,1)-pullback

H diff(X,B nU(1)) Ω cl n+1(X) H(X,B nU(1)) curv H dR(X,B n+1U(1)) \array{ \mathbf{H}'_{diff}(X,\mathbf{B}^n U(1)) &\to & \Omega^{n+1}_{cl}(X) \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}^n U(1)) &\stackrel{curv}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1)) }
Theorem

For XSmoothMfdSmoothGrpdX \in SmoothMfd \hookrightarrow Smooth \infty Grpd a paracompact smooth manifold we have that

H diff(X,B nU(1))H(X,(n+1) D ) H'_{diff}(X,\mathbf{B}^n U(1)) \simeq \;\; H(X,\mathbb{Z}(n+1)_D^\infty) \;\;

is the ordinary Deligne cohomology of XX in degree n+1n+1.

Proof

Choose a differentiably good open cover {U iX}\{U_i \to X\} and let C({U i})XC(\{U_i\}) \to X in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} be the corresponding Cech nerve projection, a cofibrant resolution of XX.

Since the above model curv chn:B diff nU(1) chn dRB n+1U(1) chncurv_{chn} : \mathbf{B}_{diff}^n U(1)_{chn} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{chn} for the intrinsic curv:B diff nU(1) dRB n+1U(1)curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) is a fibration and C({U i})C(\{U_i\}) is cofibrant, also

[Cartp op,sSet](C({U i}),B diff nU(1) chn)[Cartp op,sSet](C({U i}), dRB nU(1) chn) [Cartp^{op}, sSet](C(\{U_i\}), \mathbf{B}^n_{diff}U(1)_{chn}) \to [Cartp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^n U(1)_{chn})

is a Kan fibration by the fact that [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} is an simplicial model category. Therefore the homotopy pullback is computed as an ordinary pullback.

By the above discussion of de Rham cohomology we have that we can assume the morphism H dR n+1(X)[CartSp op,sSet](C({U i}), dRB chn n+1)H_{dR}^{n+1}(X) \to [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^{n+1}_{chn}) picks only cocycles represented by globally defined closed differential forms FΩ cl n+1(X)F \in \Omega^{n+1}_{cl}(X).

By the nature of the chain complexes curv chn:B diff nU(1) chn dRB n+1U(1) chncurv_{chn} : \mathbf{B}_{diff}^n U(1)_{chn} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{chn}, we see that the elements in the fiber over such a globally defined (n+1)(n+1)-form FF are precisely the cocycles with values only in the “upper row complex” of B diff nU(1) chn\mathbf{B}_{diff}^{n}U(1)_{chn}

C (,U(1))d dRΩ 1()d dRd dRΩ n() C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-)

such that FF is the differential of the last term.

This is the complex of sheaves that defines Deligne cohomology in degree (n+1)(n+1).

Abstract properties

We discuss how the general abstract definition of differential cohomology in SmoothGrpdSmooth \infty Grpd reproduces on general grounds the abstract properties of the traditional definition of ordinary differential cohomology.

Proposition

For XX a smooth manifold, the cohomology classes H diff(X,B nU(1))H'_{diff}(X, \mathbf{B}^n U(1)) of the cocycle \infty-groupoid H diff(X,B nU(1))\mathbf{H}'_{diff}(X, \mathbf{B}^n U(1)) defined by the homotopy pullback

H diff(X,B nU(1) ch) Ω cl n+1(X) H(X,B nU(1)) curv H(X, dRB n+1U(1)) \array{ \mathbf{H}'_{diff}(X, \mathbf{B}^n U(1)_{ch}) &\to & \Omega^{n+1}_{cl}(X) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}^n U(1)) &\stackrel{curv}{\to}& \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)) }

fit, with their canonical abelian group structure, into the two characteristic short exact sequences of ordinary differential cohomology:

  • 0Omega n(X)/Ω cl n(X)H diff(X,B nU(1))H n(X,)00 \to Omega^{n}(X)/\Omega^{n}_{cl}(X) \to H'_{diff}(X, \mathbf{B}^n U(1)) \to H^n(X, \mathbb{Z}) \to 0.

  • 0H n1(X,U(1))H diff(X,B nU(1))Ω cl,int n+1(X)00 \to H^{n-1}(X, U(1)) \to H'_{diff}(X, \mathbf{B}^n U(1)) \to \Omega^{n+1}_{cl, int}(X) \to 0

Proof

For the first sequence, the characteristic class exact sequence, use this general proposition from the discussion at cohesive (∞,1)-topos . This says that over the vanishing curvature form 0Ω cl n+1(X)0 \in \Omega_{cl}^{n+1}(X) we have the short exact sequence

0Ω cl n(X)/Ω cl,int n(X)H diff n(X,U(1))H n+1(X,)0, 0 \to \Omega_{cl}^n(X)/\Omega_{cl,int}^{n}(X) \to H_{diff}^n(X, U(1)) \to H^{n+1}(X, \mathbb{Z}) \to 0 \,,

where we used that H smooth(X,B nU(1))H n+1(X,)H_{smooth}(X, \mathbf{B}^n U(1)) \simeq H^{n+1}(X, \mathbb{Z}) on the paracompact space XX and that H dR n(X)/Ω cl,int n(X)=Ω cl n(X)/Ω cl,int n(X)H_{dR}^n(X)/\Omega^n_{cl,int}(X) = \Omega^n_cl(X)/\Omega^n_{cl, int}(X) because exact forms are in particular closed integral forms (with periods 00 \in \mathbb{N}). Looking at H diffH'_{diff} instead of H diffH_{diff}, we get in the fibers one such contribution per closed (n+1)(n+1)-form Λ\Lambda trivial in de Rham cohomology, hence per arbitrary nn-form ω\omega with dω=Λd \omega = \Lambda

0Prod Λ{ωdω=Λ}/Ω cl,int n(X)H diff n(X,U(1))H n+1(X,)0. 0 \to \Prod_{\Lambda}\{\omega | d \omega = \Lambda \}/\Omega_{cl,int}^{n}(X) \to H'_{diff}^n(X, U(1)) \to H^{n+1}(X, \mathbb{Z}) \to 0 \,.

But this is the fiber sequence in question.

For the second sequence, the curvature exact sequence, this general proposition from the discussion at cohesive (∞,1)-topos , which implies that we have a short exact sequence

0H flat n(X,B nU(1))H diff n(X,U(1))Ω cl,int n+1(X)0. 0 \to H_{flat}^n(X, \mathbf{B}^n U(1)) \to H_{diff}^n(X, U(1)) \to \Omega_{cl,int}^{n+1}(X) \to 0 \,.

Then use prop. 1, which says that H flat n(X,B nU(1))H n(X,U(1) disc)H_{flat}^n(X, \mathbf{B}^n U(1)) \simeq H^{n}(X, U(1)_{disc}).

Presentation by exponentiated \infty-Lie algebras

(…)

Proposition

The morphism given by fiber integration of differential forms over the simplex factor fits into a diagram

B nU(1) diff,simp curv smp dRB n+1U(1) smp Δ Δ B nU(1) diff,chn curv chn dRB n+1U(1) chn, \array{ \mathbf{B}^n U(1)_{diff,simp} &\stackrel{curv_{smp}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{smp} \\ \downarrow^{\mathrlap{\int}_{\Delta^\bullet}} && \downarrow^{\mathrlap{\int}_{\Delta^\bullet}} \\ \mathbf{B}^n U(1)_{diff,chn} &\stackrel{curv_{chn}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{chn} } \,,

where the vertical morphisms are weak equivalences.

Proposition

Fiber integration induces a weak equivalence

Δ :B n diff,simpB n diff,chn \int_{\Delta^\bullet} : \mathbf{B}^n \mathbb{R}_{diff,simp} \stackrel{\simeq}{\to} \mathbf{B}^n \mathbb{R}_{diff, chn}
Proof

Observe that B n diff,simp\mathbf{B}^n \mathbb{R}_{diff,simp} is the pullback of dRB n+1 smpB n+1 smp\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{smp} \to \mathbf{\flat}\mathbf{B}^{n+1} \mathbb{R}_{smp} along the evident forgetful morphism from

(U,[k]){Ω (U×Δ k)W(b n1)}. (U,[k]) \mapsto \{\Omega^\bullet(U \times \Delta^k) \leftarrow W(b^{n-1} \mathbb{R})\} \,.

This forgetful morphism is evidently a fibration (because it is a degreewise surjection under Dold-Kan), hence this pullback models the homotopy fiber of dRB n+1B n+1\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}. Since by the above fiber integration gives a weak equivalence of pulback diagrams the claim follows.

Definition

Write B nU(1) conn,simpB nU(1) diff,simp\mathbf{B}^n U(1)_{conn,simp} \hookrightarrow \mathbf{B}^n U(1)_{diff,simp} for the sub-presheaf which over (U,[k])(U,[k]) is the set of those forms ω\omega on U×Δ kU \times \Delta^k such that the curvature dωd \omega has no leg along Δ k\Delta^k.

Corollary

Under fiber integration over simplices, B nU(1) conn,simp\mathbf{B}^n U(1)_{conn,simp} is quasi-isomorphic to the Deligne cohomology-complex.

B nU(1) conn,simp Δ U(1)(n) D connection X^ B nU(1) diff,simp Δ B nU(1) diff,chn pseudoconnection dRB n+1U(1) smp Δ dRB n+1U(1) chn curvature. \array{ && \mathbf{B}^n U(1)_{conn,simp} &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}& U(1)(n)_D^\infty &&& connection \\ & \nearrow & \downarrow && \downarrow \\ \hat X &\to& \mathbf{B}^n U(1)_{diff,simp} &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}& \mathbf{B}^n U(1)_{diff,chn} &&& pseudo-connection \\ & \searrow & \downarrow && \downarrow \\ && \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{smp} &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{chn} &&& curvature } \,.

In summary this gives us the following alternative perspective on connections on B n1U(1)\mathbf{B}^{n-1}U(1)-principal ∞-bundles: such a connection is a cocycle with values in the B n\mathbf{B}^n \mathbb{Z}-quotient of the (n+1)(n+1)-coskeleton of the simplicial presheaf which over (U,[k])(U,[k]) is the set of diagrams of dg-algebras

C (U)Ω (Δ k) CE(b n1) underlyingcocycle Ω (U)Ω (Δ k) W(b n1) connection Ω (U)C (Δ k) CE(b n) curvature \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^k) &\leftarrow& CE(b^{n-1}\mathbb{R}) &&& underlying\;cocycle \\ \uparrow && \uparrow \\ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) &\leftarrow& W(b^{n-1}\mathbb{R}) &&& connection \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes C^\infty(\Delta^k) &\leftarrow& CE(b^n \mathbb{R}) &&& curvature }

where the restriction to the top morphism is the underlying cocycle and the restriction to the bottom morphism the curvature form.

The generalization to such diagram cocycles from b n1b^{n-1}\mathbb{R} to general ∞-Lie algebras 𝔤\mathfrak{g} we discuss below in ∞-Lie algebra valued connections.

Over unoriented base objects

The higher holonomy of a circle nn-bundle with connection is well defined only over oriented smooth manifolds. In the unorientable or even unoriented case, extra structure is needed to define it.

See orientifold for more.

Chern-Weil homomorphism and \infty-connections

We discuss the general abstract notion of Chern-Weil homomorphism and ∞-connections realized in SmoothGrpdSmooth \infty Grpd.

Recall that for ASmoothGrpdA \in Smooth \infty Grpd a smooth \infty-groupoid regarded as a coefficient object for cohomology, for instance the delooping A=BGA = \mathbf{B}G of an ∞-group GG we have abstractly that

  • a characteristic class on AA with coefficients in the circle Lie n-group is represented by a morphism

    c:AB nU(1); \mathbf{c} : A \to \mathbf{B}^n U(1) \,;
  • the (unrefined) Chern-Weil homomorphism induced from this is the differential characteristic class given by the composite

    c dR:AcB nU(1)curv dRB n+1 {\mathbf{c}_{dR}} : A \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}

    with the universal curvature characteristic on B nU(1)\mathbf{B}^n U(1), or rather: is the morphism on cohomology

    H Smooth 1(X,G):=π 0SmoothGrpd(X,BG)π 0((c dR) *)π 0SmoothGrpd(X, dRB n+1)H dR n+1(X) H^1_{Smooth}(X,G) := \pi_0 Smooth\infty Grpd(X, \mathbf{B}G) \stackrel{\pi_0({(\mathbf{c}_{dR})_*})}{\to} \pi_0 Smooth\infty Grpd(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}) \simeq H_{dR}^{n+1}(X)

    induced by this.

  • the ∞-connections with coefficients in AA are the cocycles in the ∞-groupoid SmoothGrpd(X,A) connSmooth \infty Grpd(X,A)_{conn} that universally lifts these differential classs in de Rham cohomology to full differential cohomology, in that it universally fills the diagrams

    SmoothGrpd(X,A) conn c^ SmoothGrpd(X,B nU(1)) diff SmoothGrpd(X,A) c dR SmoothGrpd(X,B n+1U(1)) dR. \array{ Smooth \infty Grpd(X,A)_{conn} &\stackrel{\hat {\mathbf{c}}}{\to}& Smooth\infty Grpd(X, \mathbf{B}^n U(1))_{diff} \\ \downarrow && \downarrow \\ Smooth \infty Grpd(X,A) &\stackrel{{\mathbf{c}_{dR}}}{\to}& Smooth \infty Grpd(X,\mathbf{B}^{n+1}U(1))_{dR} } \,.

Above we have discussed a presentation of the universal curvature class B n dRB n+1\mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R} by a span

B n diff,smp curv smp dRB n+1 smp B n smp \array{ \mathbf{B}^n \mathbb{R}_{diff,smp} &\stackrel{curv_{smp}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}_{smp} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n \mathbb{R}_{smp} }

in the model structure on simplicial presheaves [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}, given by maps of smooth families of differential forms.

We now insert this in the above general abstract definition of the \infty-Chern-Weil homomorphism to deduce a presentation of that in terms of smooth families of ∞-Lie algebroid valued differential forms.

The main step is the construction of a well-suited composite of two spans of morphisms of simplicial presheaves (of two ∞-anafunctors): we consider presentations of characteristic classes c:BGB nU(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1) in the image of the ∞-Lie integration map and presented by trunactions and quotients of morphisms of simplicial presheaves of the form

exp(𝔤)exp(μ)exp(b n1). \array{ \exp(\mathfrak{g}) \stackrel{\exp(\mu)}{\to} \exp(b^{n-1}\mathbb{R}) } \,.

Then, using the above, the composite differential characteristic class c dR\mathbf{c}_{dR} is presented by the zig-zag

B n diff,smp curv smp dRB n+1 smp exp(𝔤) exp(μ) B n smp \array{ && \mathbf{B}^n \mathbb{R}_{diff,smp} &\stackrel{curv_{smp}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}_{smp} \\ && \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \mathbf{B}^n \mathbb{R}_{smp} }

of simplicial presheaves. In order to efficiently compute which morphism in SmoothGrpdSmooth \infty Grpd this presents we need to construct – preferably naturally in the L-∞ algebra 𝔤\mathfrak{g} – a simplicial presheaf exp(𝔤) diff\exp(\mathfrak{g})_{diff} that fills this diagram as follows:

exp(𝔤) diff exp(μ,cs) B n diff,smp curv smp dRB n+1 smp exp(𝔤) exp(μ) B n smp. \array{ \exp(\mathfrak{g})_{diff} &\stackrel{\exp(\mu,cs)}{\to}& \mathbf{B}^n \mathbb{R}_{diff,smp} &\stackrel{curv_{smp}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}_{smp} \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \mathbf{B}^n \mathbb{R}_{smp} } \,.

Given this, exp(𝔤) diff,smp\exp(\mathfrak{g})_{diff,smp} serves as a new resolution of exp(𝔤)\exp(\mathfrak{g}) for which the composite differential characteristic class is presented by the ordinary composite of morphisms of simplicial presheaves curv smpexp(μ,cs)curv_{smp}\circ \exp(\mu, cs).

This object exp(𝔤) diff\exp(\mathfrak{g})_{diff} we shall see may be interpreted as the coefficient for pseudo ∞-connections with values in 𝔤\mathfrak{g}.

There is however still room to adjust this presentation such as to yield in each cohomology class special nice cocycle representatives. This we will achieve by finding naturally a subobject exp(𝔤) connexp(𝔤) diff\exp(\mathfrak{g})_{conn} \hookrightarrow \exp(\mathfrak{g})_{diff} whose inclusion is an isomorphism on connected components and restricted to which the morphism curv smpexp(μ,cs)curv_{smp} \circ \exp(\mu,cs) yields nice representatives in the de Rham hypercohomology encoded by dRB n+1 smp\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{smp}, namely globally defined differential forms. On this object the differential characteristic classes we will show factors naturally through the refinements to differential cohomology, and hence exp(𝔤) conn\exp(\mathfrak{g})_{conn} is finally identified as a presentation for the the coefficient object for ∞-connections with values in 𝔤\mathfrak{g}.

\infty-Pseudo-connections

We discuss presentations for the differential characteristic classes in the image of ∞-Lie integration.

Let 𝔤L CEdgAlg op\mathfrak{g} \in L_\infty \stackrel{CE}{\hookrightarrow} dgAlg^{op} be an L-∞ algebra.

Definition

A L-∞ algebra cocycle on 𝔤\mathfrak{g} in degree nn is a morphism

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

to the line Lie n-algebra.

Observation

Every L L_\infty-algebra cocycle induces canonically a morphism of simplicial presheaves of exponentiated L-∞-algebras

exp(μ):exp(𝔤)exp(b n1) \exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{n-1}\mathbb{R})

given componentwise by postcomposition with the image of μ\mu under CE()CE(-)

exp(μ)(U,[k]) :(Ω si,vert (U×Δ k)ACE(𝔤)) (Ω si,vert (U×Δ k)ACE(𝔤)CE(μ)CE(b n1)). \begin{aligned} \exp(\mu)(U, [k]) & : (\Omega^\bullet_{si,vert}(U \times \Delta^k) \stackrel{A}{\leftarrow} CE(\mathfrak{g})) \\ & \mapsto (\Omega^\bullet_{si,vert}(U \times \Delta^k) \stackrel{A}{\leftarrow} CE(\mathfrak{g}) \stackrel{CE(\mu)}{\leftarrow} CE(b^{n-1}\mathbb{R})) \end{aligned} \,.
Definition

The Weil algebra W(𝔤)dgAlgW(\mathfrak{g}) \in dgAlg is the unique representative of the free dg-algebra on the chain complex 𝔤 *[1]\mathfrak{g}_\bullet^*[1] underlying 𝔤\mathfrak{g} such that the canonical projection 𝔤 *[1]𝔤 *[2]𝔤 *[1]\mathfrak{g}_\bullet^*[1] \oplus \mathfrak{g}_\bullet^*[2] \to \mathfrak{g}_\bullet^*[1] extends to a dg-algebra homomorphism

CE(𝔤)W(𝔤). CE(\mathfrak{g}) \leftarrow W(\mathfrak{g}) \,.
Definition

For 𝔤L \mathfrak{g} \in L_\infty define the simplicial presheaf exp(𝔤) diff[CartSp smooth op,sSet]\exp(\mathfrak{g})_{diff} \in [CartSp_{smooth}^{op}, sSet] by

exp(𝔤) diff:(U,[k]){Ω si,vert (U×Δ k) CE(𝔤) Ω (U×Δ k) W(𝔤)}, \exp(\mathfrak{g})_{diff} : (U, [k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\leftarrow& W(\mathfrak{g}) } \right\} \,,

where on the left we have the set of commuting diagrams in dgAlg as indicated, with the vertical morphisms being the canonical projections.

Observation

For 𝔤=b n1\mathfrak{g} = b^{n-1}\mathbb{R} the line Lie n-algebra, this subsumes the previous definition.

Proposition

The canonical projection

exp(𝔤) diffexp(𝔤) \exp(\mathfrak{g})_{diff} \to \exp(\mathfrak{g})

is an acyclic fibration in [CartSp smooth op,sSet] proj[CartSp_{smooth}^{op}, sSet]_{proj}.

Moreover, for every L L_\infty-algebra cocycle it fits into a commuting diagram

exp(𝔤) diff exp(μ) diff exp(b n1) diff = B n diff,smp exp(𝔤) exp(μ) exp(b n1) = B n smp \array{ \exp(\mathfrak{g})_{diff} &\stackrel{\exp(\mu)_{diff}}{\to}& \exp(b^{n-1}\mathbb{R})_{diff} &=& \mathbf{B}^n \mathbb{R}_{diff,smp} \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \exp(b^{n-1}\mathbb{R}) &=& \mathbf{B}^n \mathbb{R}_{smp} }

for some morphism exp(μ) diffexp(\mu)_{diff}.

Proof

Both claims follow from the free property of the Weil algebra.

For the first, we need to show that for all UU \in CartSp we have lifts in all diagrams of the form

Δ[k] exp(𝔤) diff,smp(U) Δ[k] exp(𝔤)(U). \array{ \partial \Delta[k] &\to& \exp(\mathfrak{g})_{diff,smp}(U) \\ \downarrow && \downarrow \\ \Delta[k] &\to& \exp(\mathfrak{g})(U) } \,.

The bottom morphism is a collection of differential forms on U×Δ kU \times \Delta^k (with sitting instants), satisfying a flatness condition for their d Δ kd_{\Delta^k}-differentials. The top morphism is a collection of forms on U×Δ kU \times \partial \Delta^k with no flatness constraint except that those with no component along UU coincide with the restriction of those of the bottom morphism to the boundary Δ k\partial \Delta^k. These latter extend to a unique lift to the interior of the simplex. The remaining forms may be smoothly interpolated for instance to 0 in the interior of the simplex, while keeping at least one leg along UU. Since for the top morphism there is no condition on the differentials, any choice will do.

For the second claim, let U(CE(μ))U(CE(\mu)) be the underlying morphism on chain complexes of μ\mu. Then we have the free dg-algebra homomorphism FU(CE(μ)):W(b n1)W(𝔤)F U (CE(\mu)) : W(b^{n-1}) \to W(\mathfrak{g}) fitting into the commutative diagram

CE(𝔤) CE(μ) CE(b n1) W(𝔤) W(b n1). \array{ CE(\mathfrak{g}) &\stackrel{CE(\mu)}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{}{\leftarrow}& W(b^{n-1}\mathbb{R}) } \,.

Pasting-precomposition with this diagram yields a morhism exp(μ) diff\exp(\mu)_{diff} as desired.

Definition

Let GSmoothGrpdG \in Smooth \infty Grpd be an n-group given by Lie integration of an L-∞ algebra 𝔤\mathfrak{g}, in that the delooping object BG\mathbf{B}G is presented by the (n+1)(n+1)-coskeleton simplicial presheaf cosk n+1exp(𝔤)\mathbf{cosk}_{n+1}\exp(\mathfrak{g}).

Then for X[CartSp smooth,sSet] projX \in [CartSp_{smooth}, sSet]_{proj} any object and X^\hat X a cofibrant resolution, we say that

[CartSp smooth op,sSet](X^,cosk n+1exp(𝔤) diff) [CartSp_{smooth}^{op},sSet](\hat X, \mathbf{cosk}_{n+1}\exp(\mathfrak{g})_{diff})

is the Kan complex of pseudo n-connections on GG-principal ∞-bundles.

\infty-Connections

We discuss presentations in [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet] of the the intrinsic notion of ∞-connections in SmoothGrpdSmooth \infty Grpd.

Let still 𝔤 CE()dgAlg op\mathfrak{g} \in _\infty \stackrel{CE(-)}{\hookrightarrow} dgAlg^{op} be an L-∞ algebra.

Definition

An invariant polynomial on 𝔤\mathfrak{g} is an element W(𝔤)\langle - \rangle \in W(\mathfrak{g}) such that both \langle - \rangle \in \wedge^\bullet as well as d W(𝔤)d_{W(\mathfrak{g})}\; \langle - \rangle are elements in the graded subalgebra generated by the shifted generators 𝔤 *[1]W(𝔤)\mathfrak{g}^*[1] \hookrightarrow W(\mathfrak{g});

Write inv(𝔤)W(𝔤)inv(\mathfrak{g}) \hookrightarrow W(\mathfrak{g}) for the sub-dg-algebra of invariant polynomials.

Observation

For the line Lie n-algebra we have

inv(b n1)CE(b n). \mathrm{inv}(b^{n-1}\mathbb{R}) \simeq CE(b^n \mathbb{R}) \,.

This allows us to identify an invariant polynomial \langle - \rangle of degree n+1n+1 with a morphism

inv(𝔤)inv(b n1) inv(\mathfrak{g}) \stackrel{\langle - \rangle}{\leftarrow} inv(b^{n-1}\mathbb{R})

in dgAlg.

Definition

We say an invariant polynomial \langle - \rangle on 𝔤\mathfrak{g} is in transgression with an L-∞ algebra cocycle μ:𝔤b n1\mu : \mathfrak{g} \to b^{n-1} \mathbb{R} if there is a morphism cs:W(b n1)W(𝔤)cs : W(b^{n-1}\mathbb{R}) \to W(\mathfrak{g}) such that we have a commuting diagram

CE(𝔤) μ CE(b n1) W(𝔤) cs W(b n1) inv(𝔤) inv(b n1) =CE(b n). \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle- \rangle}{\leftarrow}& inv(b^{n-1}\mathbb{R}) & = CE(b^n \mathbb{R}) } \,.

We say that cscs is a Chern-Simons element exhibiting the transgression between μ\mu and \langle - \rangle.

We say that an L L_\infty-algebra cocycle is transgressive if it is in transgression with some invariant polynomial.

Observation

We have

  1. There is a transgressive cocycle for every invariant polynomial.

  2. Any two L L_\infty-algebra cocycles in transgression with the same invariant polynomial are cohomologous.

  3. Every decomposable invariant polynomial (the wedge product of two non-vanishing invariant polynomials) transgresses to a cocycle cohomologous to 0.

Proof
  1. By the fact that the Weil algebra is free, its cochain cohomology vanishes and hence the definition property d W(𝔤)=0d_{W(\mathfrak{g})} \langle -\rangle = 0 implies that there is some element csW(𝔤)cs \in W(\mathfrak{g}) such that d W(𝔤)cs=d_{W(\mathfrak{g})} cs = \langle - \rangle. Then the image of cscs along the canonical dg-algebra homomorphism W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}) is d CE(𝔤)d_{CE(\mathfrak{g})}-closed hence is a cocycle on 𝔤\mathfrak{g}. This is by construction in transgression with \langle - \rangle.

  2. Let cs 1cs_1 and cs 2cs_2 be Chern-Simons elements for the to given L L_\infty-algebra cocycles. Then by assumption d (𝔤)(cs 1cs 2)=0d_{(\mathfrak{g})} (cs_1 - cs_2) = 0 . By the acyclicity of W(𝔤)W(\mathfrak{g}) there is then λW(𝔤)\lambda \in W(\mathfrak{g}) such that cs 1=cs 2+d W(𝔤)λcs_1 = cs_2 + d_{W(\mathfrak{g})} \lambda.
    Since W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}) is a dg-algebra homomorphism this implies that also μ 1=μ 2+d CE(𝔤)λ CE(𝔤)\mu_1 = \mu_2 + d_{CE(\mathfrak{g})} \lambda|_{CE(\mathfrak{g})}.

  3. Given two nontrivial invariant polynomials 1\langle - \rangle_1 and 2\langle - \rangle_2 let cs 1W(𝔤)cs_1 \in W(\mathfrak{g}) be any element such that d W(𝔤)cs 1= 1d_{W(\mathfrak{g})}cs_1 = \langle - \rangle_1. Then cs 1,2:=cs 1 2cs_{1,2} := cs_1 \wedge \langle -\rangle_2 satisfies d W(𝔤)cs 1,2= 1 2d_{W(\mathfrak{g})} cs_{1,2} = \langle - \rangle_1 \wedge \langle -\rangle_2. By the first observation the restriction of cs 1,2cs_{1,2} to CE(𝔤)CE(\mathfrak{g}) is therefore a cocycle in transgression with 1 2\langle - \rangle_1 \wedge \langle -\rangle_2. But by the definition of invariant polynomials the restriction of 2\langle - \rangle_2 vanishes, and hence so does that of cs 1,2cs_{1,2}. The claim the follows with the second point above.

Definition

Define the simplicial presheaf exp(𝔤) ChW[CartSp smooth op,sSet]\exp(\mathfrak{g})_{ChW} \in [CartSp_{smooth}^{op}, sSet] by the assignment

exp(𝔤) ChW:(U,[k]){Ω si,vert (U×Δ k) A vert CE(𝔤) Ω si (U×Δ k) A W(𝔤) Ω (U) F A inv(𝔤)}, \exp(\mathfrak{g})_{ChW} : (U , [k]) \mapsto \left\{ \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k ) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k ) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A\rangle}{\leftarrow}& inv(\mathfrak{g}) } \right\} \,,

where on the right we have the set of horizontal morphisms in dgAlg making commuting diagrams with the canonical vertical morphisms as indicated.

We call F A\langle F_A \rangle the curvature characteristic forms of AA.

Let

exp(𝔤) diff (exp(μ i,cs i)) i iexp(b n i1) diff ((curv i) smp) i dRB smp n i exp(𝔤) \array{ \exp(\mathfrak{g})_{diff} & \stackrel{(\exp(\mu_i,cs_i))_i}{\to} & \prod_{i} \exp(b^{n_i-1}\mathbb{R})_{diff} &\stackrel{((curv_i)_{smp})}{\to}& \prod_i \mathbf{\flat}_{dR}\mathbf{B}^{n_i}_{smp} \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) }

be the presentation, as above, of the product of all differential refinements of characteristic classes on exp(𝔤)\exp(\mathfrak{g}) induced from Lie integration of transgressive L-∞ algebra cocycles.

Observation

We have that exp(𝔤) ChW\exp(\mathfrak{g})_{ChW} is the pullback in [CartSp smooth op,sSet][CartSp_{smooth}^{op}, sSet] of the globally defined closed forms along the curvature characteristics induced by all transgressive L L_\infty-algebra cocycles:

exp(𝔤) ChW exp(μ,cs) n iΩ cl n i+1() exp(𝔤) diff,smp (curv i) i i dRB n i+1 smp exp(𝔤). \array{ \exp(\mathfrak{g})_{ChW} &\stackrel{\exp(\mu,cs)}{\to}& \prod_{n_i} \Omega^{n_i + 1}_{cl}(-) \\ \downarrow && \downarrow \\ \exp(\mathfrak{g})_{diff,smp} &\stackrel{({curv}_i)_i}{\to}& \prod_i \mathbf{\flat}_{dR} \mathbf{B}^{n_i + 1} \mathbb{R}_{smp} \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) } \,.
Proof

By the above proposition we have that the bottom horizontal morphims sends over each (U,[k])(U,[k]) and for each ii an element

Ω si,vert (U×Δ k) A vert CE(𝔤) Ω si (U×Δ k) A W(𝔤) \array{ \Omega^\bullet_{si,vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet_{si}(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) }

of exp(𝔤)(U) k\exp(\mathfrak{g})(U)_k to the composite

(Ω si (U×Δ k)AW(𝔤)cs iW(b n i1)inv(b n i)=CE(b n i))) \left( \; \Omega^\bullet_{si}(U \times \Delta^k) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{cs_i}{\leftarrow} W(b^{n_i-1} \mathbb{R}) \stackrel{}{\leftarrow} inv(b^{n_i} \mathbb{R}) = CE(b^{n_i}\mathbb{R})) \; \right)
=(Ω si (U×Δ k)F A iCE(b n i)) = \left( \; \Omega^\bullet_{si}(U \times \Delta^k) \stackrel{\langle F_A\rangle_i}{\leftarrow} CE(b^{n_i}\mathbb{R}) \; \right)

regarded as an element in dRB smp n i+1(U) k\mathbf{\flat}_{dR} \mathbf{B}^{n_i+1}_{smp}(U)_k. The right vertical morphism Ω n i+1(U) dRB n i+1 smp(U)\Omega^{n_i + 1}(U) \to \mathbf{\flat}_{dR}\mathbf{B}^{n_i+1}\mathbb{R}_{smp}(U) from the constant simplicial set of closed (n i+1)(n_i+1)-forms on UU picks precisely those of these elements for which F A\langle F_A\rangle is a basic form on the U×Δ kU \times \Delta^k-bundle in that it is in the image of the pullback Ω (U)Ω si (U×Δ k)\Omega^\bullet(U) \to \Omega^\bullet_{si}(U \times \Delta^k).

Remark

This shows that exp(𝔤) ChW\exp(\mathfrak{g})_{ChW} serves as a convenient object on which the differential characteristic classes of exp(𝔤)\exp(\mathfrak{g}) are supported.

For more see connection on a smooth principal ∞-bundle.

Higher holonomy and Chern-Simons functional

References

For standard references on differential geometry and Lie groupoids see there.

The (,1)(\infty,1)-topos SmoothGrpdSmooth \infty Grpd is discussed in section 3.3 of

A discussion of smooth \infty-groupoids as (,1)(\infty,1)-sheaves on CartSpCartSp and the presentaton of the \infty-Chern-Weil homomorphism on these is in

For references on Chern-Weil theory in Smooth∞Grpd and connection on a smooth principal ∞-bundle, see there.

The results on differentiable Lie group cohomology used above are in

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

and

which parallels

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

A review is in section 4 of

Classification of topological principal 2-bundles is discussed in

and the generalization to classification of smooth principal 2-bundles is in

Revised on October 10, 2012 13:51:48 by Urs Schreiber (82.169.65.155)