nLab
Lie infinity-groupoid

Idea

The notion of $\mathrm{\infty }$-Lie groupoid is the generalization of the notion of Lie group and Lie groupoid from category theory to higher category theory and homotopy theory:

an $\mathrm{\infty }$-Lie groupoid is an ∞-groupoid that is smooth in some sense.

Reminder: Lie groupoids and differentiable stacks

Before indicating the idea of $\mathrm{\infty }$-Lie groupoids in more detail, recall some aspects of the theory of ordinary Lie groupoids.

An ordinary Lie groupoid is usually understood to be a groupoid internal to Diff, possibly with further extra conditions on its structure morphisms. The literature on Lie groupoids is familiar with the fact that it is often useful to regard these internal groupoids after embedding them into the more general context of stacks on the site Diff of all smooth manifolds: there they are called differentiable stacks.

Regarding a groupoid internal to manifolds $𝒢$ as a stack $𝒢:{\mathrm{Diff}}^{\mathrm{op}}\to$ Grpd means encoding it in terms of the groupoids of smooth families of objects and morphisms inside it:

• to the point $*\in \mathrm{Diff}$ it assigns the underlying bare groupoid $𝒢(*)$, the groupoid $𝒢$ with its smooth structure forgotten;

• to a manifold $U\in \mathrm{Diff}$ it assigns the groupoid $𝒢(U)$ whose set of objects is the set $𝒢(U{)}_0:={\mathrm{Hom}}_{\mathrm{Diff}}(U,{𝒢}_0)$ of smooth $U$-parameterized objects in $U$, and whose set of morphisms $𝒢(U{)}_1:={\mathrm{Hom}}_{\mathrm{Diff}}(U,{𝒢}_1)$ is the set of smooth $U$-parameterized morphisms in $𝒢$.

Not every stack on Diff comes from a groupoid $𝒢$ internal to Diff this way. For instance the (stackification of the) groupoid of Lie-algebra valued forms for some Lie group $G$ is a non-concrete stack, which can never be represented by an internal groupoid. Still, most operations that one may want to apply to internal groupoids also make sense for general stacks on Diff. Indeed, some operations that one may want to apply to internal groupoids take values not in internal groupoids, but in more general stacks: the 2-category ${\mathrm{Sh}}_{(\mathrm{2,1})}(\mathrm{Diff})$ of all stacks on $\mathrm{Diff}$ is formally more well behaved than the sub-category of differentiable stacks inside it. One useful way to formalize all the nice structure that ${\mathrm{Sh}}_{(\mathrm{2,1})}(\mathrm{Diff})$ has is to see that this is a 2-topos.

Therefore it is quite useful to think of every stack on $\mathrm{Diff}$ as encoding a smooth groupoid and to think of the study of Lie groupoids as being the theory of the 2-topos ${\mathrm{Sh}}_{(\mathrm{2,1})}(\mathrm{Diff})$. The groupoids internal to $\mathrm{Diff}$ are special nice objects in this 2-topos: the geometric stacks.

For this reason, here we shall find it useful to adopt the term Lie groupoid for a general objects in ${\mathrm{Sh}}_{(\mathrm{2,1})}(\mathrm{Diff})$ and to speak of Lie groupoids represented in smooth manifolds or of geometric stacks if we mean groupoids internal to manifolds, under the embedding indicated above.

$\mathrm{\infty }$-Lie groupoids as $(\mathrm{\infty }\mathrm{,1})$-sheaves / $\mathrm{\infty }$-stacks

As we generalize from groupoids to ∞-groupoids, the notion of stack/(2,1)-sheaf generalizes to that of ∞-stack/(∞,1)-sheaf. Therefore we shall define an $\mathrm{\infty }$-Lie groupoid to be an (∞,1)-sheaf on a site of smooth test spaces.

Notice that for the definition of the smooth structure on a diffeological space or on an ordinary Lie groupoid it is not in fact necessary to regard these as objects tested by objects in all of Diff: since smoothness is a local property, it is entirely sufficient to know around every point in the set of k-morphisms of the $\mathrm{\infty }$-groupoid what all the extensions of this point to a ball -shaped smooth family of points around that point are. This is precisely what an ∞-stack on CartSp encodes, the category of just Cartesian spaces and smooth functions between them. A discussion of the difference or not between $\mathrm{\infty }$-stacks on Diff and on CartSp see the section below

We shall think of the (∞,1)-topos ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ as being the context in wich ∞-Lie theory takes place. As before, inside this large context only some nice objects correspond to internal ∞-groupoids in Diff or in Diffeol. These geometric ∞-stacks are the concrete or geometric $\mathrm{\infty }$-Lie groupoids inside more general objects. One way to make precise the notion of geometric ∞-stack with respect to a chosen notion of geometric is to adopt the concept of geometry (for structured (∞,1)-toposes).

There are other sites on wich one may want a smooth $\mathrm{\infty }$-groupoid to be modeled on. For instance instead of testing only with smooth manifolds, one may want to test with smooth loci. An ordinary sheaf on the category $𝕃$ of smooth loci is a generalized smooth space as considered in synthetic differential geometry. Accordingly, an $\mathrm{\infty }$-stack on $𝕃$ may be thought of as a smooth $\mathrm{\infty }$-Lie groupoid to which synthetic differential geometry applies.

The key difference of $𝕃$ to Diff is that the former contains smooth infinitesimal spaces. Therefore an $\mathrm{\infty }$-Lie groupoid modeled on $𝕃$ may have spaces of k-morphisms that have infinitesimal extension in some direction. Notably one obtains a notion of $\mathrm{\infty }$-Lie groupoids for which all $k$-morphisms are infinitesimal in a precise sense. It sturns out that such infinitesimal $\mathrm{\infty }$-Lie groupoids may be identified with ∞-Lie algebroids: generalizations to higher category theory of Lie algebras and Lie algebroids.

The $(\mathrm{\infty }\mathrm{,1})$-topos of $\mathrm{\infty }$-Lie groupoids

We discuss in more detail some properties of the (∞,1)-topos ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ of (∞,1)-sheaves on CartSp.

Definition

Let $C=$ CartSp equipped with the structure of a site by the coverage of good open covers.

Write

$$\mathrm{\infty }\mathrm{LieGrpd}:={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})\stackrel{\stackrel{L}{\leftarrow }}{\hookrightarrow }{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$$\infty LieGrpd := Sh_{(\infty,1)}(CartSp) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(CartSp)

for the corresponding (∞,1)-category of (∞,1)-sheaves.

Remark

By the discussion at Cech localizaton at a coverage this is modeled by the left Bousfield localization of $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ at Cech nerves of covering families.

$$\begin{array}{ccc}{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})& \stackrel{\stackrel{L}{\leftarrow }}{\hookrightarrow }& {\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})\\ {\uparrow }^{\simeq }& & {\uparrow }^{\simeq }\\ ([{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}{)}^{\circ }& \stackrel{\stackrel{𝕃\mathrm{Id}}{\leftarrow }}{\underset{ℝ\mathrm{Id}}{\to }}& ([{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}{)}^{\circ }\end{array}.$$\array{ Sh_{(\infty,1)}(CartSp) &\stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}& PSh_{(\infty,1)}(CartSp) \\ \uparrow^{\simeq} && \uparrow^{\simeq} \\ ([CartSp^{op}, sSet]_{proj,cov})^\circ &\stackrel{\overset{\mathbb{L} Id}{\leftarrow}}{\underset{\mathbb{R} Id}{\to}}& ([CartSp^{op}, sSet]_{proj})^{\circ} } \,.

Lemma

Let $X$ be a smooth manifold, regarded as an object in $\mathrm{sPSh}(C)$. Let $\{U_i\to X\}$ be a good open cover of $X$ and $C(\{U_i\})$ the corresponding Cech nerve. Then

$$C(\{U_i\})\stackrel{\simeq }{\to }X$$C(\{U_i\}) \stackrel{\simeq}{\to} X

is a weak equivalence in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$ and in fact a cofibrant replacement for $X$.

Proof

The morphism ${\coprod }_{i}j(U_i)\to X$ of which $C(\{U_i\})$ is the Cech nerve of a local epimorphism in that for every $j(V)\to X$ there exists a covering family $\{V_i\to V\}$ and lifts ${\sigma }_i$

$$\begin{array}{ccc}j(V_i)& \stackrel{{\sigma }_i}{\to }& \coprod _{i}j(U_i)\\ \downarrow & & \downarrow \\ j(V)& \to & X\end{array}.$$\array{ j(V_i) &\stackrel{\sigma_i}{\to}& \coprod_i j(U_i) \\ \downarrow && \downarrow \\ j(V) &\to& X } \,.

Namely take $\{V_i\to V\}$ to be simply (any open refinement of) the open cover of $X$ pulled back to $V$.

By the discussion at Cech localization at Grothendieck (pre)topologies this implies that $C(\{U_i\})\to X$ is a weak equivalence in $\mathrm{sPSh}(C{)}_{\mathrm{proj},\mathrm{cov}}$.

Moreover, since the cover is good the Cech nerve $C(\{U_i\})$ is degreewise a coproduct of representables. By the discussion at cofibrant objects this implies that it is cofibrant.

Remark

This fact related to the classical nerve theorem which asserts that the simplicial set obtained by contracting in $C(\{U_i\})$ all copies of Cartesian spaces to the point is a model for the homotopy type of $X$.

More on that below in the discussion of ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ as a locally ∞-connected (∞,1)-topos…

$\mathrm{\infty }$-Connectedness

We discuss that $\mathrm{\infty }\mathrm{LieGrpd}$ is an ∞-connected (∞,1)-topos and recall the notions of intrinsic de Rham objects induced from that.

Lemma

The $(\mathrm{\infty }\mathrm{,1})$-topos $\mathrm{\infty }\mathrm{LieGrpd}$ is an ∞-connected (∞,1)-topos.

This means that the global section geometric morphism is essential in that we have a triple of adjoint (∞,1)-functors

$$(\Pi ⊣\mathrm{LConst}⊣\Gamma ):\mathrm{\infty }\mathrm{LieGrpd}\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd}$$(\Pi \dashv LConst \dashv \Gamma) : \infty LieGrpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd

and that $\mathrm{LConst}$ is a full and faithful (∞,1)-functor.

Notice that this is the $\mathrm{\infty }$-analog of the statement that $\mathrm{Sh}(\mathrm{CartSp})$ is a connected topos, as discussed in detail at diffeological space.

Proof

The proof can be found at path ∞-groupoid -- Unstructured homotopy ∞-groupoid.

Lemma

In addition the (∞,1)-functor $\Pi$ preserves finite products.

Proof

The proof can be found at path ∞-groupoid -- Unstructured homotopy ∞-groupoid.

Path $\mathrm{\infty }$-groupoids and flat objects

We make the usual definitions in an ∞-connected (∞,1)-topos as described in more detail at path ∞-groupoid and at differential cohomology in an (∞,1)-topos:

Definition

Write

$$(\Pi ⊣♭):=(\mathrm{LConst}\circ \Pi ⊣\mathrm{LConst}\circ \Gamma ):\mathrm{\infty }\mathrm{LieGrpd}\overleftarrow{\to }\mathrm{\infty }\mathrm{LieGrpd}$$(\mathbf{\Pi} \dashv \mathbf{\flat}) := (LConst \circ \Pi \dashv LConst \circ \Gamma) : \infty LieGrpd \stackrel{\leftarrow}{\to} \infty LieGrpd

for the composite adjunction. For $X,A\in \mathrm{\infty }\mathrm{LieGrpd}$ we call ${\Pi }_{\inf }$ the Lie path ∞-groupoid of $X$ and we call ${♭}_{\inf }A$ the flat $\mathrm{\infty }$-Lie groupoid of $A$.

Coefficients for $\mathrm{\infty }$-Lie algebra valued differential forms

For $X\in \mathrm{\infty }\mathrm{LieGrpd}$ we write ${\Pi }_{\mathrm{dR}}(X)$ for the homotopy cofiber of the unit $X\to \Pi (X)$, i.e. for the pushout

$$\begin{array}{ccc}X& \to & *\\ \downarrow & ⇙& \downarrow \\ \Pi (X)& \to & {\Pi }_{\mathrm{dR}}(X)\end{array}$$\array{ X &\to& * \\ \downarrow &\swArrow& \downarrow \\ \mathbf{\Pi}(X) &\to& \mathbf{\Pi}_{dR}(X) }

in $\mathrm{\infty }\mathrm{LieGrpd}$.

For $*\to A\in \mathrm{\infty }\mathrm{LieGrpd}$ a pointed object, we write ${♭}_{\mathrm{dR}}A$ for the homotopy fiber of the counit $♭A\to A$, i.e. for the pullback

$$\begin{array}{ccc}{♭}_{\mathrm{dR}}A& \to & ♭A\\ \downarrow & ⇙& \downarrow \\ *& \to & A\end{array}$$\array{ \mathbf{\flat}_{dR}A &\to& \mathbf{\flat}A \\ \downarrow &\swArrow& \downarrow \\ * &\to& A }

in $\mathrm{\infty }\mathrm{LieGrpd}$\,.

The canonical form on an $\mathrm{\infty }$-Lie group $G$

For $G\in \mathrm{\infty }\mathrm{LieGrpd}$ an ∞-group with delooping $BG$ consider the double (∞,1)-pullback diagram

$$\begin{array}{ccc}G& \to & *\\ {}^{\theta }\downarrow & & \downarrow \\ {♭}_{\mathrm{dR}}BG& \to & ♭BG\\ \downarrow & & \downarrow \\ *& \to & BG\end{array}.$$\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 } \,.

The bottom square is an (∞,1)-pullback by definition. By the pasting law for $(\mathrm{\infty }\mathrm{,1})$-pullbacks, the top square being an $(\mathrm{\infty }\mathrm{,1})$-pullback implies that the outer rectangle is, too, which identifies $G$ as the top pullback.

This induces a canonical element in the $G$-valued intrinsic de Rham cohomology of $G$:

$$(\theta :G\to {♭}_{\mathrm{dR}}BG)\in H_{\mathrm{dR}}(G,BG).$$(\theta : G \to \mathbf{\flat}_{dR} \mathbf{B}G) \in \mathbf{H}_{dR}(G, \mathbf{B}G) \,.

This we may identify with the $\mathrm{\infty }$-groupoid analog of the Maurer-Cartan form on a Lie group $G$.

The vertical form on a $G$-principal $\mathrm{\infty }$-bundle

For $P\to X$ the $G$-principal ∞-bundle classified by a morphism $X\to BG$ in $\mathrm{\infty }\mathrm{LieGrpd}$, for each point $x:*\to X$ the pasting diagram of (∞,1)-pullback squares

$$\begin{array}{ccccc}G\simeq P_x& \to & P& \to & *\\ {}^{{\theta }_x}\downarrow & & \downarrow & & \downarrow \\ {♭}_{\mathrm{dR}}BG& \to & \mathrm{At}(P)& \to & ♭BG\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & X& \to & BG\end{array}$$\array{ G \simeq P_x &\to& P &\to& * \\ {}^{\mathllap{\theta}_x}\downarrow && \downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}G &\to& At(P) &\to& \mathbf{\flat} \mathbf{B}G \\ \downarrow && \downarrow && \downarrow \\ * &\to& X &\to& \mathbf{B}G }

exhibits the canonical $𝔤$-valued vertical intrinsic form

$$({\theta }_x:P_x\to {♭}_{\mathrm{dR}}BG)\in H_{\mathrm{dR}}(P_x,BG)$$(\theta_x : P_x \to \mathbf{\flat}_{dR}\mathbf{B}G) \in \mathbf{H}_{dR}(P_x, \mathbf{B}G)

on the fiber $P_x$ of $P$ over $x$.

Geometric realization

Proposition

Let $X$ be a simplicial manifold that is degreewise paracompact, regarded as a simplicial diffeological space, hence as an object in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$, hence as an object in $\mathrm{\infty }\mathrm{LieGrpd}$.

Write $\mid X\mid \in$ Top for the geometric realization of simplicial topological spaces. Then

$$\Pi (X)\simeq \mid X\mid \in \mathrm{Top}\simeq \mathrm{\infty }\mathrm{Grpd}.$$\Pi(X) \simeq |X| \in Top \simeq \infty Grpd \,.

Proof

The (∞,1)-functor $\Pi :\mathrm{\infty }\mathrm{LieGrpd}\to \mathrm{\infty }\mathrm{Grpd}$ is the left derived functor of ${\lim }_{\to }:[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}\to {\mathrm{sSet}}_{\mathrm{Quillen}}$. Use the above cofibrant replacement for $X$ degreewise with Dugger’s general description of projective cofibrant objects in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ to compute the cofibrant replacement, then apply ${\lim }_{\to }$ and use that the colimit of a representable is the point. The statement then is degreewise the classical nerve theorem.

A detailed proof can be found at path ∞-groupoid -- Unstructured homotopy ∞-groupoid.

Proposition

Let $G$ be a well sectioned simplicial topological group?. Write, as usual for simplicial groups, $BG:=\overline{W}G$ for its delooping.

Regard $BG$ in the canonical way as an object of $[\mathrm{CartSp},\mathrm{sSet}]$. Let $X_{•}\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ be any other simplicial topological space and let $X\to BG$ be a morphism. Then:

on such morphisms geometric realization of simplicial spaces $X_{•}\mapsto \mid X_{•}\mid :={\int }^{[n]}{\Delta }_{\mathrm{Top}}^n\times X_n\in \mathrm{Top}$ preserves homotopy fibers (up to weak equivalence).

Proof

In unpublished notes, Danny Stevenson and David Roberts show that under geometric realization of simplicial topological spaces the universal simplicial principal bundle (see there) $EG:=WG\to \overline{W}G$ maps to the universal $\mid G\mid$-principal bundle $ℰ\mid G\mid \to ℬ\mid G\mid$ in Top.

But (as described at homotopy fiber and generalized universal bundle) the universal bundle is a means to compute homotopy fibers: the ordinary pullback

$$\begin{array}{ccc}{P}_{•}& \to & WG\\ \downarrow & & \downarrow \\ X_{•}& \to & \overline{W}G\end{array}$$\array{ P_\bullet &\to& W G \\ \downarrow && \downarrow \\ X_\bullet &\to& \bar W G }

computes the homotopy fiber of $X_{•}\to \overline{W}G$. Since geometric realization of simplicial spaces preserves pullbacks (see there), this is sent to the pullback square

$$\begin{array}{ccc}\mid P_{•}\mid & \to & ℰG\\ \downarrow & & \downarrow \\ \mid X_{•}\mid & \to & ℬG\end{array}$$\array{ |P_\bullet| &\to& \mathcal{E}G \\ \downarrow && \downarrow \\ |X_\bullet| &\to& \mathcal{B}G }

in Top. Again, this computes the homotopy fiber of the bottom morphism, up to equivalence.

Corollary

The functor $\Pi :{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})\to \mathrm{\infty }\mathrm{Grpd}$ preserves homotopy fibers of morphisms represented by degreewise paracompact simplicial topological spaces $X_{•}\to BG$ as above.

may need to polish the technical assumptions…

One application of this result is in the construction of lifts of Whitehead towers in ∞Grpd $\simeq$ Top to $\mathrm{\infty }\mathrm{LieGrpd}$. This we discuss in the section Smooth Whitehead towers.

Relation to $(\mathrm{\infty }\mathrm{,1})$-sheaves on all manifolds

In the literature on Lie groupoids and differentiable stacks, these are traditionally conceived as stacks on the site Diff of all smooth manifolds. As mentioned above, for the purpose of encoding a smooth structure on a groupoid the category Diff regarded as a category of test objects is larger than necessary. After all, every manifold is, by definition, itself patched together from Cartesian spaces, and passing to sheaves or stacks on a site really just means that one allows objects patched together from the objects in the site, so that one could just as well take the site to be just that of Cartesian spaces in the first place.

More precisely, we have an equivalence of categories between the categories of sheaves on CartSp and on Diff

$$\mathrm{Sh}(\mathrm{Diff})\stackrel{\simeq }{\to }\mathrm{Sh}(\mathrm{CartSp})$$Sh(Diff) \stackrel{\simeq}{\to} Sh(CartSp)

induced from the full and faithful functor $\mathrm{CartSp}\hookrightarrow \mathrm{Diff}$. Under this equivalence a sheaf on all of $\mathrm{Diff}$ is simply restricted to just the subcategory CartSp.

To see this, notice that every smooth manifold $X$ admits a good cover $\{U_i\to X\}$, where each $U_i$ is diffeomorphic to a Cartesian space (essentially by definition of manifold). By the sheaf condition, the value $A(X)$ of a sheaf on $X$ is determined by its value on these $U_i$. Hence the sheaf on Diff is already entirely determined by its restriction to CartSp.

An analogous discussion holds for (∞,1)-sheaves on these sites, which we illustrate by the following standard example.

Let $G$ be a Lie group. We shall write

• $BG:\mapsto G\mathrm{TrivBund}(U):=({\mathrm{Hom}}_{\mathrm{Diff}}(U,G)\overrightarrow{\to }*)$ ;

• $G\mathrm{Bund}:X\mapsto G\mathrm{Bund}(X)$;

for the functorial assignments of groupoids to smooth manifolds, where in the last case we assign the groupoid of $G$-principal bundles and in the first case the groupoid of trivial $G$-principal bundles.

Now let $X$ be any smooth manifold. We want to compute the groupoid of smooth $G$-principal bundles as the hom-object $X\to BG$ in the (∞,1)-category of (∞,1)-sheaves on Diff or CartSp. In order to present that (∞,1)-category, we shall make use of its model category-theoretic presentation in terms of the model structure on simplicial presheaves $\mathrm{sPSh}(C{)}_{\mathrm{proj},\mathrm{loc}}$. Then in order to compute the derived hom-space in question, we need to

1. find a cofibrant replacement $Y\stackrel{\simeq }{\to }X$ of $X$;

2. find a fibrant replacement $A\stackrel{\simeq }{\to }B$ of $A$.

3. compute the ordinary enriched hom-object

   $H(X,BG)=\mathrm{sPSh}(Y,B)$

The point now is that the kind of work one has to do to achieve this differs from $\mathrm{sPSh}(\mathrm{CartSp}{)}_{\mathrm{proj},\mathrm{loc}}$ and $\mathrm{sPSh}(\mathrm{Diff}{)}_{\mathrm{proj},\mathrm{loc}}$. But the outcome is the same:

1. The approach traditionally used in the literature is, essentially, this: in $\mathrm{sPSh}(\mathrm{Diff}{)}_{\mathrm{proj},\mathrm{loc}}$ the manifold $X$ is a representable object, of course. This means it is already cofibrant and we can simply take $Y=X$.

   On the other hand, in this model structure the presheaf $BG$ is far from being fibrant. But $G\mathrm{Bund}$, the closure of $BG$ under descent, is of course its fibrant replacement, and the canonical inclusin morpism $BG\to G\mathrm{Bund}$ is a weak equivalence.

   So finally we can compute

   $$H(X,BG)={\mathrm{sPSh}}_{\mathrm{Diff}}(X,G\mathrm{Bund})$$\mathbf{H}(X,\mathbf{B}G) = sPSh_{Diff}(X,G Bund)

   simply by the Yoneda lemma as

   $$\cdots \simeq G\mathrm{Bund}(X).$$\cdots \simeq G Bund(X) \,.

   In conclusion here no work had to be done on the cofibrant replacement, while lots of work has to be done on the fibrant replacement. Notice that in order to compute the groupoid of $G$-principal bundles on just $X$, we here first computed the corresponding groupoid for each and every manifold, in that we first computed the full stack $G\mathrm{Bund}$. (Of course in this simple example this is not really a big deal, but it should be clear that for $G$ generalized to any smooth ∞-group and hence $G\mathrm{Bund}$ to the $\mathrm{\infty }$-stack of $G$-principal ∞-bundles, it does become quite a big deal).

2. Now consider the same situation, but in $\mathrm{sPSh}(\mathrm{CartSp}{)}_{\mathrm{proj},\mathrm{loc}}$. Here the technicalities reverse:

   Now $X$ is (in general) no longer representable, hence it is in general no longer cofibrant. We need to pass to a cofibrant replacement $Y\to X$ instead. Such can be obtained for instance by taking $Y$ to be the Cech nerve of a good cover of $X$.

   On the other hand, now $BG$ is already fibrant! Because the fibrancy condition is that it satisfies descent along Cech nerves $C(\{U_i\})$ of covers $\{U_i\to U\}$ of objects $U$ in CartSp. But since every $G$-principal bundle on a Cartesian space is necessarily equivalent to the trivial one, we have that

   $$BG(U)=G\mathrm{TrivBund}(U)\simeq G\mathrm{Bund}(U)\simeq {\mathrm{sPSh}}_{\mathrm{CartSp}}(C(\{U_i\}),BG)$$\mathbf{B}G(U) = G TrivBund(U) \simeq G Bund(U) \simeq sPSh_{CartSp}(C(\{U_i\}), \mathbf{B}G)

   because $U$ is topologically contractible. So $BG$ does satisfy descent – not on Diff, but on CartSp.

   In conclusion, here we may compute the hom-object as

   $$H(X,BG)\simeq {\mathrm{sPSh}}_{\mathrm{CartSp}}(C(\{U_i\}),BG).$$\mathbf{H}(X,\mathbf{B}G) \simeq sPSh_{CartSp}(C(\{U_i\}), \mathbf{B}G) \,.

   On the right this is just the Cech cohomology of $X$ with values in $BG$ and hence indeed

   $$\simeq G\mathrm{Bund}(X).$$\simeq G Bund(X) \,.

Strict $\mathrm{\infty }$-Lie groupoids

Many $\mathrm{\infty }$-Lie groupoids appearing in practice are (equivalent) to objects in sub-(∞,1)-categories of ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ of much stricter $\mathrm{\infty }$-Lie groupoids. These subcategories typically offer convenient and desireable contexts for formulating and proving statements about special cases of general $\mathrm{\infty }$-Lie groupoids. Therefore it is of interest to have various notions of strict $\mathrm{\infty }$-Lie groupoids inside all of them.

One well-known such notion is given by the Dold-Kan correspondence. This identifies chain complexes of abelian groups with strict and strictly symmetric monoidal $\mathrm{\infty }$-groupoids.

Dropping the condition on symmetric monoidalness we obtain a more general such inclusion, a kind of non-abelian Dold-Kan correspondence:

the identification of crossed complexes of groupoids as precisely the strict ∞-groupoids. This has been studied in particular in nonabelian algebraic topology.

So we have a sequence of inclusions

$$\begin{array}{ccccc}\mathrm{ChainCplx}& \hookrightarrow & \mathrm{CrsCpl}& \hookrightarrow & \mathrm{KanCplx}\\ \downarrow & & \downarrow & & \downarrow \\ \mathrm{StrAb}\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}& \hookrightarrow & \mathrm{Str}\mathrm{\infty }\mathrm{Grpd}& \hookrightarrow & \mathrm{\infty }\mathrm{Grpd}\end{array}$$\array{ ChainCplx &\hookrightarrow& CrsCpl &\hookrightarrow& KanCplx \\ \downarrow && \downarrow && \downarrow \\ StrAb Str\infty Grpd &\hookrightarrow& Str \infty Grpd &\hookrightarrow& \infty Grpd }

of strict $\mathrm{\infty }$-groupoids into all $\mathrm{\infty }$-groupoids. See also the cosmic cube of higher category theory.

Among the special tools for handling $\mathrm{\infty }$-stacks on $\mathrm{CartSp}$ that factor at some point through the above inclusion are the following:

• restriction to abelian sheaf cohomology – Using the fact that the objects of ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ are modeled by simplicial presheaves symmetric monoidal $\mathrm{\infty }$-Lie groupoids are identified under the Dold-Kan correspondence with $\mathbb{N}$-graded chain complexes of sheaves. To these the rich set of tools for abelian sheaf cohomology apply.

• descent for strict $\mathrm{\infty }$-groupoid valued sheaves – There is a good theory pf descent for (presheaves) with values in strict $\mathrm{\infty }$-groupoids (more restrictive than the fully general theory but more general than abelian sheaf cohomology). This goes back to Ross Street and its relation to the full theory has been clarified by Dominic Verity in Verity09.

It should be noticed that for $\mathrm{\infty }$-stacks of $\mathrm{\infty }$-groupoids the intuition from the homotopy hypothesis no longer quite applies, necessarily. For instance under geometric realization $\Pi :\mathrm{\infty }\mathrm{LieGrpd}\to \mathrm{\infty }\mathrm{Grpd}$ already strict $\mathrm{\infty }$-groupoid-valued presheaves exhaust all homotopy types in ∞Grpd $\simeq$ Top: because already all 0-truncated objects (set-values sheaves) exhaust all homotopy types, as the geometric geometric realization does not produces the categorical homotopy groups in an (∞,1)-topos, but the geometric homotopy groups in an (∞,1)-topos.

Descent for strict $\mathrm{\infty }$-Lie groupoids

We state a useful theorem for the computation of descent for presheaves with values in strict ∞-groupoids. Recall the standard terminology for descent, i.e. for the $(\mathrm{\infty }\mathrm{,1})$-categorical sheaf-condition:

For $U\in C$ a representable (here $C=$ CartSp for our purposes), $Y,A\in [C^{\mathrm{op}},\mathrm{sSet}]$ simplicial presheaves and $p:Y\to U$ a morphism, we say that $A$ satisfies descent along $p$ or equivalently that $A$ is a $p$-local object if the canonical morphism

$$A(U)\stackrel{=}{\to }[C^{\mathrm{op}},\mathrm{sSet}](U,A)\to [C^{\mathrm{op}},\mathrm{sSet}](Y,A)$$A(U) \stackrel{=}{\to} [C^{op}, sSet](U,A) \to [C^{op}, sSet](Y,A)

is a weak equivalence. Here the first equality is the enriched Yoneda lemma. By the co-Yoneda lemma we may decompose $Y$ into itsw cells as

$$Y={\int }^{[n]\in \Delta }\Delta [n]\cdot Y_n,$$Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n \,,

where in the integrand we have the tensoring of $[C^{\mathrm{op}},\mathrm{sSet}]$ over sSet. Using that the enriched hom-functor sends coends to ends, the enriched hom-functor on the right we may equivalently write out as an end

$$\begin{array}{rl}[C^{\mathrm{op}},\mathrm{sSet}](Y,A)& =[C^{\mathrm{op}},\mathrm{sSet}]({\int }^{[n]\in \Delta }\Delta [n]\cdot Y_n,A)\\ & ={\int }_{[n]\in \Delta }[C^{\mathrm{op}},\mathrm{sSet}](\Delta [n]\cdot Y_n,A)\\ & ={\int }_{[n]\in \Delta }\mathrm{sSet}(\Delta [n],[C^{\mathrm{op}},\mathrm{sSet}](Y_n,A))\\ & ={\int }_{[n]\in \Delta }\mathrm{sSet}(\Delta [n],A(Y_{•}))\\ & =:\mathrm{Desc}(Y,A)\end{array}$$\begin{aligned} [C^{op}, sSet](Y,A) & = [C^{op}, sSet](\int^{[n] \in \Delta} \Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta}[C^{op}, sSet](\Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], [C^{op}, sSet](Y_n, A)) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], A(Y_\bullet)) \\ & =:Desc(Y,A) \end{aligned}

(equality signs denote isomorphisms), where in the second but last line we again used the tensoring of simplicial presheaves $[C^{\mathrm{op}},\mathrm{sSet}]$ over sSet.

In the last line we have the totalization of the cosimplicial simplicial object

$$A(Y_{•}):\Delta \to \mathrm{sSet},$$A(Y_\bullet) : \Delta \to sSet \,,

sometimes called the descent object of $A$ relative to $Y$, even though in this case it is really nothing but the hom-object of $Y$ into $A$. If $A$ is fibrant and $Y$ cofibrant, then $\mathrm{Desc}(Y,A)$ is a Kan complex: the descent $\mathrm{\infty }$-groupoid .

Now suppose that $𝒜:C^{\mathrm{op}}\to \mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$ is a presheaf with values in strict ∞-groupoids. In the context of strict $\mathrm{\infty }$-groupoids the standard $n$-simplex is given by the $n$th oriental $O(n)$. This allows to perform a construction that looks like a descent object in $\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$:

Definition

(Ross Strees)

The descent object for $𝒜\in [C^{\mathrm{op}},\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}]$ relative to $Y\in [C^{\mathrm{op}},\mathrm{sSet}]$ is

$$\mathrm{Desc}(Y,𝒜):={\int }_{[n]\in \Delta }\mathrm{Str}\mathrm{\infty }\mathrm{Cat}(O(n),𝒜(Y_n))\in \mathrm{Str}\mathrm{\infty }\mathrm{Grpd},$$Desc(Y,\mathcal{A}) := \int_{[n] \in \Delta} Str\infty Cat(O(n), \mathcal{A}(Y_n)) \;\in Str \infty Grpd \,,

where the end is taken in $\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$.

This objects had been suggested by Ross Street to be the right descent object for strict $\mathrm{\infty }$-category-valued presheaves in Street03

Under the ω-nerve functor $N_O:\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}\to \mathrm{sSet}$ this yields a Kan complex $N_0\mathrm{Desc}(Y,𝒜)$. On the other hand, applying the $\omega$-nerve directly to $𝒜$ yields a simplicial presheaf $N_O𝒜$ to which the above simplicial descent applies.

The following theorem asserts that under certain conditions both notions coincide.

Theorem

(Dominic Verity)

If $𝒜:C^{\mathrm{op}},\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$ and $Y:C^{\mathrm{op}}\to \mathrm{sSet}$ are such that $N_O𝒜(Y_{•}):\Delta \to \mathrm{sSet}$ is fibrant in the Reedy model structure $[\Delta ,{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{Reedy}}$, then

$$N_O\mathrm{Desc}(Y,𝒜)\stackrel{\simeq }{\to }\mathrm{Desc}(Y,N_O𝒜)$$N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A})

is a weak homotopy equivalence of Kan complexes.

This is proven in Verity09.

Corollary

If $Y\in [C^{\mathrm{op}},\mathrm{sSet}]$ is such that $Y_{•}:\Delta \to [C^{\mathrm{op}},\mathrm{Set}]\hookrightarrow [C^{\mathrm{op}},\mathrm{sSet}]$ is cofibrant in $[\Delta ,[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}{]}_{\mathrm{Reedy}}$ then for $𝒜:C^{\mathrm{op}}\to \mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$ we have

$$N_O\mathrm{Desc}(Y,𝒜)\stackrel{\simeq }{\to }\mathrm{Desc}(Y,N_O𝒜).$$N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A}) \,.

Proof

If $Y_{•}$ is Reedy cofibrant, then by definition the canonical morphisms

$$\underrightarrow{\lim }(([n]\stackrel{+}{\to }[k])\mapsto Y_k)\to Y_n$$\lim_{\to}( ([n] \stackrel{+}{\to} [k]) \mapsto Y_k ) \to Y_n

are cofibrations in $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$. Since the latter is an ${\mathrm{sSet}}_{\mathrm{Quillen}}$ enriched model category and $N_O𝒜$ is fibrant, it follows that the hom-functor $[C^{\mathrm{op}},\mathrm{sSet}](-,N_O𝒜)$ sends cofibrations to fibrations, so that

$$N_O𝒜(Y_n)\to \underleftarrow{\lim }([n]\stackrel{+}{\to }[k]\mapsto N_O𝒜(Y_k))$$N_O\mathcal{A}(Y_n) \to \lim_{\leftarrow}( [n]\stackrel{+}{\to} [k] \mapsto N_O\mathcal{A}(Y_k))

is a Kan fibration. But this says that $N_O𝒜(Y_{•})$ is Reedy fibrant, so that the assumption of Verity’s theorem is met.

Corollary

For $Y$ the Cech nerve of a good open cover $\{U_i\to X\}$ of a manifold $X$ and any $𝒜:{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$ we have that

$$[C^{\mathrm{op}},\mathrm{sSet}](Y,N_O𝒜)\simeq N_O\mathrm{Desc}(Y_{•},𝒜).$$[C^{op}, sSet](Y,N_O \mathcal{A}) \simeq N_O Desc(Y_\bullet, \mathcal{A}) \,.

Proof

By the above is sufices to note that $Y_{•}$ is cofibrant in $[{\Delta }^{\mathrm{op}},[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}{]}_{\mathrm{Reedy}}$ if $Y$ is the Cech nerve of a good open cover. By the assumption of good open cover we have that $Y$ is degreewise a coproduct of representables and that the inclusion of all degenerate $n$-cells into all $n$-cells is a full inclusion into a coproduct, i.e. an inlusion of the form

$$\coprod _{i\in I}{U}_i\to \coprod _j{U}_{j\in J}$$\coprod_{i \in I} U_i \to \coprod_j U_{j \in J}

induced from an inclusion of subsets $I\hookrightarrow J$. Since all representables are cofibrant in $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ such an inclusion is a cofibration.

In conclusion we find that for determining the $\mathrm{\infty }$-stack condition for strict $\mathrm{\infty }$-Lie groupoids we may equivalently use Street’s formula for strict $\mathrm{\infty }$-groupid valued presheaves. This is sometimes useful for computations in low categorical degree.

Circle $n$-Lie groupoids

Write $U(1)=S^1=ℝ/\mathbb{Z}$ for the abelian Lie group called the circle group or 1-dimensional unitary group.

Delooping

Write $\Xi :{\mathrm{Ch}}_{•}\to \mathrm{sAb}\to \mathrm{sSet}$ for the Dold-Kan correspondence functor and with convenient abuse of notation use the same symbols for its extension $\Xi :[{\mathrm{CartSp}}^{\mathrm{op}},{\mathrm{Ch}}_{•}]\to [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ to presheaves.

Write

$$U(1)[n]=[\cdots \to 0\to C^{\mathrm{\infty }}(-,U(1))\to 0\to \cdots \to 0]$$U(1)[n] = [\cdots \to 0 \to C^\infty(-,U(1)) \to 0 \to \cdots \to 0]

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

Theorem

The presheaf $\Xi (U(1)[n])\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$ is a fibrant model of the $n$-fold delooping of the group object $U(1)$ in $\mathrm{\infty }\mathrm{LieGrpd}$.

Proof

The fibrant objects in question are those presheaves that are degreewise Kan complexes and that satisfy descent along good open covers of Cartesian spaces.

For the first, notice that all objects in the image of the Dold-Kan map are Kan complexes.

For $n=0$ the second condition says that the ordinary presheaf ${\mathrm{Hom}}_{\mathrm{Diff}}(-,U(1)):{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Set}\to \mathrm{sSet}$ is in fact a sheaf, which clearly it is.

For $n\ge 1$ the integral cohomology of a Cartesian space ${ℝ}^k$ vanishes and therefore every $U(1)$-cocycle in degree $n$ is trivializable. The automorphism of the trivial $n$-cocycle are precisely the $(n-1)$-cocycles. Continuing this way, one finds that the $n$-groupoid of $n$-cocycle is equivalent to the $n$-fold delooping of the group of 0-cocycle, which is $C^{\mathrm{\infty }}(ℝ,U(1))$. This is the value of $\Xi (U(1)[n])$ on ${ℝ}^k$. Hence the descent condition is satisfied.

(Notice as before that here it is crucial that the site we use is CartSp and not all of Diff.)

Now consider the delooping statement by induction. We need to show that for $n\ge 1$ the loop space object $\Omega \Xi (U(1)[n])\simeq \Xi (U(1)[n-1])$. Since ∞-stackification preserves fniite limits, it is sufficient to compute the homotopy pullback of $*\to \Xi (U(1)[n])\leftarrow *$ in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$.

Therefor we take a fibrant replacement of the morphism $*\to \Xi (U(1)[n])$ to be

$$\begin{array}{c}\Xi [C^{\mathrm{\infty }}(-,U(1))\stackrel{\mathrm{Id}}{\to }{C}^{\mathrm{\infty }}(-,U(1))\to 0\to \cdots \to 0]\\ \downarrow \\ \Xi [C^{\mathrm{\infty }}(-,U(1))\to 0\to 0\to \cdots \to 0]\end{array}.$$\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 surjective, hence a projective fibration, hence its image under $\Xi$ is a projective fibration. So the homotopy pullback in question is the ordinary pullback

$$\begin{array}{ccc}\mathrm{Xi}[0\to C^{\mathrm{\infty }}(-,U(1))\to 0\to \cdots \to 0]& \to & \Xi [C^{\mathrm{\infty }}(-,U(1))\stackrel{\mathrm{Id}}{\to }{C}^{\mathrm{\infty }}(-,U(1))\to 0\to \cdots \to 0]\\ \downarrow & & \downarrow \\ \Xi [0\to 0\to 0\to \cdots \to 0]\to \Xi [C^{\mathrm{\infty }}(-,U(1))\to 0\to 0\to \cdots \to 0]\end{array},$$\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 $[{\mathrm{CartSp}}^{\mathrm{op}},{\mathrm{Ch}}_{•}]$ and then using that $\Xi$ is a right part of a Quillen adjunction, hence right adjoint and hence preserves products.

Definition

We therefore write $B^{n}U(1)\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ for $\Xi (U(1)[n])$. This may be called the circle Lie $(n+1)$-group.

Differential coefficients

We now describe the Lie $n$-groupoids ${♭}_{\mathrm{dR}}{B}^{n}U(1)$ and ${\Pi }_{\mathrm{dR}}{♭}_{\mathrm{dR}}{B}^{n}U(1)$ induced from $B^{n}U(1)$ as discussed at ∞-connectedness.

A fibrant representative in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$ of $♭B^{n}U(1)$ is

$$\Xi [C^{\mathrm{\infty }}(-,U(1))\stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^1(-)\stackrel{d_{\mathrm{dR}}}{\to }\cdots \to {\Omega }_{\mathrm{cl}}^n(-)].$$\Xi[C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)] \,.

and of ${♭}_{\mathrm{dR}}{B}^{n}U(1)$ is

$$\Xi [0\to {\Omega }^1(-)\stackrel{d_{\mathrm{dR}}}{\to }\cdots \to {\Omega }_{\mathrm{cl}}^n(-)].$$\Xi[0 \to \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)] \,.

Proof

Since the global section $\Gamma$ amounts to evaluation on the point ${ℝ}^0$ and since conmstant simplicial presheaves on CartSp satisfy descent, we have that $♭B^{n}U(1)$ is represented by $\Xi [\mathrm{const}U(1)\to 0\to \cdots \to 0]$. This is weakly equivalent to $\Xi [C^{\mathrm{\infty }}(-,U(1))\stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^1(-)\stackrel{d_{\mathrm{dR}}}{\to }\cdots \stackrel{d_{\mathrm{dR}}}{\to }{\Omega }_{\mathrm{cl}}^n(-)]$ by the Poincare lemma applied to each Cartesian space (using the same standard logic that proves the de Rham theorem).

And so a fibration representing the counit $♭B^{n}U(1)\to B^{n}U(1)$ is the image under $\Xi$ of

$$\begin{array}{ccccccc}{C}^{\mathrm{\infty }}(-,U(1))& \to & {\Omega }^1(-)& \to & \cdots & \to & {\Omega }_{\mathrm{cl}}^n(-)\\ \downarrow & & \downarrow & & & & \downarrow \\ C^{\mathrm{\infty }}(-,U(1))& \to & 0& \to & \cdots & \to & 0\end{array}.$$\array{ C^\infty(-,U(1)) &\to& \Omega^1(-) &\to & \cdots &\to& \Omega^n_{cl}(-) \\ \downarrow && \downarrow && && \downarrow \\ C^\infty(-, U(1)) &\to& 0 &\to& \cdots &\to& 0 } \,.

The pullback

$$\begin{array}{ccc}\Xi [0\to {\Omega }^1(-)\to \cdots \to {\Omega }_{\mathrm{cl}}^n(-)]& \to & \Xi [C^{\mathrm{\infty }}(-,U(1))\to {\Omega }^1(-)\to \cdots \to {\Omega }_{\mathrm{cl}}^n(-)]\\ \downarrow & & \downarrow \\ \Xi [0\to 0\to \cdots \to 0]& \to & \Xi [C^{\mathrm{\infty }}(-,U(1))\to 0\to \cdots \to 0]\end{array}$$\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] }

of this is a homotopy pullback and since fibrantions are stable under pullback, we find that the top left is indeed fibrant.

Remark

We observe that the complex of sheaves $♭B^{n}U(1)$ is that which defines flat Deligne cohomology, while that of ${♭}_{\mathrm{dR}}{B}^{n}U(1)$ is that which computes de Rham cohomology in degree $n$.

Also notice that the complex that defines ${♭}_{\mathrm{dR}}{B}^{n}U(1)$ has the following description in terms of hom-spaces of ∞-Lie algebroids.

Lie groups

Let $G$ be a Lie group, regarded as an object of $H:=\mathrm{\infty }\mathrm{LieGrpd}$.

Delooping

Proposition

A fibrant representative of the delooping object $BG$ in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ is given by the nerve of the one-objec Lie groupoid

$$N(G\overrightarrow{\to }*)$$N(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 $\{U_i\to {ℝ}^n\}$ of Cartesian spaces, because the descent $\mathrm{\infty }$-groupoid $\mathrm{sPSh}(C(\{U_i\}),BG)$ is $G\mathrm{Bund}({ℝ}^n)\simeq G\mathrm{TrivBund}({ℝ}^n)$.

To show that $BG$ is indeed the delooping object of $G$ it is sufficient, due to the fact that fact that ∞-stackification preserves finite limits to exhibit a homotopy pullback $G\simeq *{\times }_{BG}*$ in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$.

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

$$\begin{array}{ccc}G& \to & N(G\times G\stackrel{\stackrel{p_1\cdot p_2}{\to }}{\underset{p_1}{\to }}G)\\ \downarrow & & \downarrow \\ *& \to & N(G\overrightarrow{\to }*)\end{array}$$\array{ G &\to& N(G\times G \stackrel{\overset{p_1 \cdot p_2}{\to}}{\underset{p_1}{\to}} G) \\ \downarrow && \downarrow \\ * &\to& N(G \stackrel{\to}{\to} *) }

as discussed at universal principal ∞-bundle.

The universal $G$-principal bundle

The universal G-principal bundle is a replacement of the point inclusion $*\to BG$ by a fibration $EG\to BG$.

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

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

which is the action groupoid $G//G$ of $G$ 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 $G\to EG$: it is an example of a groupal model for universal principal ∞-bundles.

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

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

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

$$(G\stackrel{\mathrm{Id}}{\to }G\overrightarrow{\to }*).$$(G \stackrel{Id}{\to}G \stackrel{\to}{\to} * ) \,.

$G$-principal bundles

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

Proposition

Let $X$ be a paracompact smooth manifold. The ordinary first nonabelian cohomology of $X$ with coefficients in $G$ coincided with the intrinsic cohomology of $\mathrm{\infty }\mathrm{Lie}\mathrm{Grpd}$

$$H^1(X,G)\simeq {\pi }_0\mathrm{\infty }\mathrm{LieGrpd}(X,BG)$$H^1(X,G) \simeq \pi_0 \infty LieGrpd(X, \mathbf{B}G)

and the $G$-principal bundle $P\to X$ corresponding to a cocycle $X\to BG$ in $\mathrm{\infty }\mathrm{LieGrpd}$ 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 $X$ in the model $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$ for $\mathrm{\infty }\mathrm{LieGrpd}$ is civen by the Cech nerve $C\{U_i\}$ of a good open cover $\{U_i\to X\}$. It follows that ${\pi }_0\mathrm{\infty }\mathrm{LieGrpd}(X,BG)$ is the Cech cohomology of $X$ with coefficients in $G$ (see there for details).

Concretely, a cocycle

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

is canonically identified with a transition function

$$g:\coprod _{i,j}{U}_i\cap U_j\to G$$g : \coprod_{i,j} U_i \cap U_j \to G

satisfying on $U_i\cap U_j\cap U_k$ the cocycle condition $g_{ij}{g}_{jk}=g_{ik}$.

From this we can compute the homotopy fiber of $g:C(\{U_i\})\to BG$ by forming the ordinaty pullback of the fibrant replacement $EG\to BG$ of the point inclusion $*\to BG$, where $mathbEG=B{G}^I{\times }_{BG}*$ is the smooth groupoid

$$EG=\{\begin{array}{ccc}& & •\\ & {}^{g_1}\swarrow & & {\searrow }^{g_2}\\ *& & \stackrel{h}{\to }& & *\end{array}\mid g_1,g_2,h\in G,g_2=h{g}_1\}.$$\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 $\hat{P}$ in

$$\begin{array}{ccc}\hat{P}& \to & EG\\ \downarrow & & \downarrow \\ C(\{U_i\})& \stackrel{g}{\to }& BG\end{array}$$\array{ \hat P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}G }

to be the smooth Lie groupoid

$$\hat{P}=(\coprod _{i,j}{U}_i\cap U_j\times G\stackrel{\stackrel{p_2\times g_{i,j}\cdot p_3}{\to }}{\underset{p_1\times P_3}{\to }}\coprod _i{U}_i\times G)$$\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.

$$\hat{P}=\left\{\begin{array}{ccc}& & •\\ & {}^{g_1}\swarrow & & {\searrow }^{g_2}\\ *& & \stackrel{g_{ij}(x)}{\to }& & *\\ \\ (x,i)& & \to & & (x,j)\end{array}\right\}.$$\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 $\hat{P}\to P$ is objectwise a surjective and full and faithful functor.

Differential coefficients

For $G$ an ordinary Lie group, we give a concrete representative for the $\mathrm{\infty }$-Lie groupoid $♭BG=\mathrm{LConst}\Gamma BG$ in terms of Lie algebra-valued differential forms.

Let $\Xi :\mathrm{CrsdCplx}\to \mathrm{KanCplx}$ now denote the inclusion of crossed complexes into all Kan complexes/∞-groupoids.

Proposition

The $\mathrm{\infty }$-Lie groupoid $♭BG\in \mathrm{\infty }\mathrm{LieGrpd}$ has a fibrant representative in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$ given by

$$♭BG=\Xi [C^{\mathrm{\infty }}(-,G)\times {\Omega }_{\mathrm{flat}}^1(-,𝔤)\stackrel{\stackrel{{\mathrm{Ad}}_{p_1}(p_2)+p_1^{-1}d{p}_1}{\to }}{\underset{p_2}{\to }}{\Omega }_{\mathrm{flat}}^1(-,𝔤)],$$\mathbf{\flat}\mathbf{B}G = \Xi[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})] \,,

where $𝔤$ is the Lie algebra of $G$.

This is the groupoid of Lie-algebra valued forms restricted to flat forms.

In other words, a $U={ℝ}^n$-parameterized family of objects of $♭BG$ is given by a Lie-algebra valued 1-form $A\in {\Omega }^1(U)\otimes 𝔤$ whose curvature 2orm $F_A=d_{\mathrm{dR}}A+[A,\wedge A]=0$ vanishes, and a $U$-parameterized family of morphisms $g:A\to A\prime$ is given by a smooth function $g\in C^{\mathrm{\infty }}(U,G)$ such that $A\prime ={\mathrm{Ad}}_{g}A+g^{-1}dg$, where ${\mathrm{Ad}}_{g}A=g^{-1}Ag$ is the adjoint action of $G$ on its Lie algebra, and where $g^{-1}dg:=g^{*}\theta$ is the pullback of the Maurer-Cartan form on $G$ along $g$.

Proof

By the above discussion we have that the object in question is

$$𝕃\mathrm{LConst}\circ ℝ\Gamma BG,$$\mathbb{L}LConst \circ \mathbb{R}\Gamma \mathbf{B}G \,,

the image of $BG$ under the right derived functor of global sections and the left derived functor of constant ∞-stacks. But since $BG$ is fibrant in $[{\mathrm{CartSp}}^{\mathrm{op}},{\mathrm{sSet}}_{\mathrm{Quillen}}{]}_{\mathrm{proj},\mathrm{cov}}$ and every object in ${\mathrm{sSet}}_{\mathrm{Quillen}}$ is cofibrant, this is simply

$$\cdots =\mathrm{Lconst}\circ \Gamma BG=N(\mathrm{const}G\overrightarrow{\to }*).$$\cdots = Lconst \circ \Gamma \mathbf{B}G = N( const G \stackrel{\to}{\to} *) \,.

So first we have to show that this is equivalent to the Lie groupoid of flat Lie-algebra valued 1-forms. There is an evident morphism

$$N(\mathrm{const}G\overrightarrow{\to }*)\to N(G\times {\Omega }_{\mathrm{flat}}^1(-,𝔤)\overrightarrow{\to }{\Omega }_{\mathrm{flat}}^1(-,𝔤))$$N( const G \stackrel{\to}{\to} *) \to N( G \times \Omega^1_{flat}(-,\mathfrak{g}) \stackrel{\to}{\to} \Omega^1_{flat}(-,\mathfrak{g}))

that sends the single object to the trivial 1-form. We claim that this is objectwise an equivalence of groupoids: it is essentially surjective since every flat $𝔤$-valued 1-form on the contractible ${ℝ}^n$ is of the form $gd{g}^{-1}$ for some function $g:{ℝ}^n\to G$ (let $g(x)=P\exp ({\int }_0^x)A$ be the parallel transport of $A$ along any path from the origin to $x$). Since the gauge automorphism of the trivial $𝔤$-valued 1-form are precisely given by the constant $G$-valued functions, this is also objectwise a full and faithful functor.

Finally we need to show that $N(G\times {\Omega }_{\mathrm{flat}}^1(-,𝔤)\overrightarrow{\to }{\Omega }_{\mathrm{flat}}^1(-,𝔤))$ is fibrant in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$. This can be seen by observing that this sheaf is the coefficient object that in Cech cohomology computes $G$-principal bundles with flat connection and then reasoning as above: every $G$-principal bundle with flat connection is equivalent to a trivial $G$-principal bundle whose connection is given by a globally defined $𝔤$-valued 1-form. Morphisms between these are precisely $G$-valued functions that act on the 1-forms by gauge transformations as in the groupoid of Lie-algebra valued forms.

A detailed discussion of how this arises concretely from the formula $[{\Pi }_1(-),BG]$ for the right adjoint of $\Pi =\mathrm{LConst}\circ \Pi$ and how it is the coefficient object for smooth flat $G$-principal bundles is in SchrWalI.

Corollary

The object ${♭}_{\mathrm{dR}}BG$ of flat intrinsic de Rham coefficients of $BG$ is represented in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ by the 0-truncated sheaf of flat $𝔤$-valued forms

$$U\mapsto {\Omega }_{\mathrm{flat}}^1(U,𝔤).$$U \mapsto \Omega^1_{flat}(U,\mathfrak{g}) \,.

Proof

The diagram

$$\begin{array}{ccc}{\Omega }_{\mathrm{flat}}^1(-,𝔤)& \to & \Xi [C^{\mathrm{\infty }}(-,G)\times {\Omega }_{\mathrm{flat}}^1(-,G)\overrightarrow{\to }{\Omega }_{\mathrm{flat}}^1(-,𝔤)]\\ \downarrow & & \downarrow \\ *& \to & \Xi [C^{\mathrm{\infty }}(-,G)\overrightarrow{\to }*]\end{array}$$\array{ \Omega^1_{flat}(-,\mathfrak{g}) &\to& \Xi[C^\infty(-,G)\times \Omega^1_{flat}(-,G) \stackrel{\to}{\to} \Omega^1_{flat}(-,\mathfrak{g})] \\ \downarrow && \downarrow \\ {*} &\to& \Xi[C^\infty(-,G) \stackrel{\to}{\to} *] }

is a pullback diagram in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ with all objects fibrant and the right vertical morphism being a fibration. Therefore this is a homotopy pullback. By the above statements and since ∞-stackification preserves finite limits, this also models the (∞,1)-pullback

$$\begin{array}{ccc}{♭}_{\mathrm{dR}}BG& \to & ♭BG\\ \downarrow & & \downarrow \\ *& \to & BG\end{array}.$$\array{ \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat} \mathbf{B}G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G } \,.

Remark

Writing $TU$ for the tangent Lie algebroid of $U$ the flat de Rham object of $BG$ may be also be written as

$${♭}_{\mathrm{dR}}BG:U\mapsto \mathrm{Hom}(TU,𝔤),$$\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

$${♭}_{\mathrm{dR}}BG:U\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),({\Omega }^{•}(U),d_{\mathrm{dR}})),$$\mathbf{\flat}_{dR} \mathbf{B}G : U \mapsto Hom_{dgAlg}(CE(\mathfrak{g}),(\Omega^\bullet(U), d_{dR})) \,,

So far we have discussed the object ${♭}_{\mathrm{dR}}BG$ for $G$ a Lie group. By the general logic of intrinsic ∞-Lie algebroids the object ${\Pi }_{\mathrm{dR}}{♭}_{\mathrm{dR}}BG$ is the intrinsic incarnation in $\mathrm{\infty }\mathrm{LieGrpd}$ of the Lie algebra $𝔤$, defined to be the $(\mathrm{\infty }\mathrm{,1})$-pushout

$$\begin{array}{ccc}{♭}_{\mathrm{dR}}BG& \to & *\\ \downarrow & & \downarrow \\ \Pi {♭}_{\mathrm{dR}}BG& \to & {\Pi }_{\mathrm{dR}}BG\end{array}.$$\array{ \mathbf{\flat}_{dR}\mathbf{B}G &\to& * \\ \downarrow && \downarrow \\ \mathbf{\Pi} \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\Pi}_{dR} \mathbf{B}G } \,.

There are some issues here with finding the right cofibrant replacement of this diagram that computes the correct $(\mathrm{\infty }\mathrm{,1})$-pushout by an ordinary pushout in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$. We describe now some ordinary such pushout, and discuss its relation to the proper $(\mathrm{\infty }\mathrm{,1})$-pushout later.

Proposition

For $X:{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Set}$ a sheaf, write

$$\tilde{\Pi }(X):U\mapsto \mathrm{Hom}(U\times {\Delta }_{\mathrm{Diff}}^{•},X)$${\tilde \mathbf{\Pi}}(X) : U \mapsto Hom(U \times \Delta^\bullet_{Diff}, X)

for the simplicial presheaf of paths in $X$. (By the discussion at path ∞-groupoid this is constructed similar to the path model for $\Pi (X)$, but without any cofibrant replacement thrown in.)

Then the pushout

$$\begin{array}{ccc}{♭}_{\mathrm{dR}}BG& \to & *\\ \downarrow & & \downarrow \\ \Pi {♭}_{\mathrm{dR}}BG& \to & \exp (𝔤)\end{array}$$\array{ \mathbf{\flat}_{dR} \mathbf{B}G &\to& * \\ \downarrow && \downarrow \\ \mathbf{\Pi}\mathbf{\flat}_{dR} \mathbf{B}G &\to& \exp(\mathfrak{g}) }

is the presheaf

$$\exp (𝔞):U\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),C^{\mathrm{\infty }}(U)\otimes {\Omega }^1({\Delta }_{\mathrm{diff}}^{•})).$$\exp(\mathfrak{a}) : U \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), C^\infty(U)\otimes \Omega^1(\Delta^\bullet_{diff})) \,.

For more discussion of this and its relevance, see the section Lie integrated ∞-Lie groupoids below.

The canonical form on $G$

The following proposition asserts that the abstract $(\mathrm{\infty }\mathrm{,1})$-topos-theoretic definition of the canonical $𝔤$-valued form on an $\mathrm{\infty }$-Lie group $G$ given above reduces indeed to the ordinary notion of Maurer-Cartan form when $G$ is an ordinary Lie group.

Recall from the discussion of differential coefficients above that the $\mathrm{\infty }$-Lie groupoid ${♭}_{\mathrm{dR}}BG$ is modeled by the 0-truncated simplicial sheaf of flat $𝔤$-valued forms.

Proposition

For $G$ a Lie group, the canonical morphism $G\to {♭}_{\mathrm{dR}}BG$ is modeled in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ by the morphism of presheaves

$${\mathrm{Hom}}_{\mathrm{Diff}}(-,G)\to {\Omega }_{\mathrm{flat}}^1(-,𝔤)$$Hom_{Diff}(-,G) \to \Omega^1_{flat}(-,\mathfrak{g})

given by

$$(g:U\to G)\mapsto (g^{*}\theta =:g^{-1}dg),$$(g : U \to G) \mapsto (g^* \theta =: g^{-1} d g) \,,

where $\theta$ is the Maurer-Cartan form on $G$.

Remark. By the general identification of differential forms on presheaves/diffeological spaces, this morphism is indeed the Maurer-Cartan form $\theta$ on $G$.

Proof

We need to compute the double (∞,1)-pullback diagram

$$\begin{array}{ccc}G& \to & *\\ \downarrow & & \downarrow \\ {♭}_{\mathrm{dR}}BG& \to & ♭BG\\ \downarrow & & \downarrow \\ *& \to & BG\end{array}.$$\array{ G &\to& * \\ \downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat} \mathbf{B}G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G } \,.

In the above discussion of differential coefficients we already modeled the lower $(\mathrm{\infty }\mathrm{,1})$-pullback square by the ordinary pullback in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ of the presheaf that assigns to $U$ the groupoid of flat Lie-algebra valued forms on $U$:

$$♭BG:U\mapsto \{A\stackrel{g}{\to }{\mathrm{Ad}}_{g}A+g^{*}\theta \mid A\in {\Omega }_{\mathrm{flat}}^1(U,𝔤),g\in C^{\mathrm{\infty }}(U,G)\}.$$\mathbf{\flat} \mathbf{B}G : U \mapsto \{ A \stackrel{g}{\to} Ad_g A + g^* \theta | A \in \Omega^1_{flat}(U,\mathfrak{g}), g \in C^\infty(U,G) \} \,.

We need to form the homotopy pullback of the point in this – which is the vanishing form $A=0$. A standard fibrant replacement of $*\to ♭BG$ (as discussed at generalized universal bundle) for this is given by the presheaf

$$U\mapsto \left\{\begin{array}{ccc}& & A_0=0\\ & {}^{g_1}\swarrow & & {\searrow }^{g_2}\\ A_1& & \stackrel{h}{\to }& & A_2\end{array}\right\},$$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 in ${♭}_{\mathrm{dR}}BG(U)$ is a morphism from $(g_1,A_1)$ to $(g_2,A_2)$.

The pullback of this along the above model for ${♭}_{\mathrm{dR}}BG\to ♭BG$ is the 0-truncated sheaf

$$U\mapsto \left\{\begin{array}{c}{A}_0=0\\ {\downarrow }^g\\ g^{*}\theta \end{array}\right\}.$$U \mapsto \left\{ \array{ A_0 = 0 \\ \downarrow^{\mathrlap{g}} \\ g^* \theta } \right\} \,.

First of all we see that this is indeed weakly equivalent (indeed isomorphic) to $G$, as it sould be. But the point is that we see from the above pullback that the projection $G\to {♭}_{\mathrm{dR}}BG$ is modeled by the morphisms of presheaves

$${\mathrm{Hom}}_{\mathrm{Diff}}(-,G)\to {\Omega }_{\mathrm{flat}}^1(-,𝔤)$$Hom_{Diff}(-,G) \to \Omega^1_{flat}(-,\mathfrak{g})

which is the codomain evaluation of the above cone morphisms:

$$(0\stackrel{g}{\to }{g}^{*}\theta )\mapsto (g^{*}\theta =g^{-1}dg).$$(0 \stackrel{g}{\to} g^* \theta) \mapsto (g^* \theta = g^{-1} d g) \,.

The universal $G$-connection on the universal $G$-principal bundle

under construction

We have seen above that the universal $G$-principal bundle $EG$ is itself naturally modeled as a Lie 2-group. In the next section Differential coefficients for Lie 2-groups we discuss Lie 2-groups and the canonical differential forms with values in a Lie 2-algebra on these. We shall now discuss how, in a sense, for the Lie 2-group $EG$ this universal form is the universal Ehresmann connection on the universal $G$-principal bundle. The reader not familiar with the section Differential coefficients for Lie 2-groups should skip this section here to come back later. This section here is a corollary or special case or example application of that section.

The universal pseudo-connection

The Lie 2-group $EG$ is the one coming from the crossed module $(G\stackrel{\mathrm{Id}}{\to }G)$. Its Lie 2-algebra is accordingly that given by the differential crossed module $(𝔤\stackrel{\mathrm{Id}}{\to }𝔤)$. The Chevalley-Eilenberg algebra of this Lie 2-algebra is the Weil algebra of the Lie algebra $𝔤$.

In section Differential coefficients for Lie 2-groups we find a replacement for $♭BEG$ and ${♭}_{\mathrm{dR}}BEG$ that induces a realization of the Maurer-Cartan form on the Lie 2-group $EG$ in terms of a span

$$\begin{array}{ccc}{E}_{\mathrm{diff}}G& \stackrel{{\theta }_{EG}}{\to }& {♭}_{\mathrm{dR}}BEG\\ {\downarrow }^{\simeq }\\ EG\end{array}$$\array{ \mathbf{E}_{diff} G &\stackrel{\theta_{\mathbf{E}G}}{\to}& \mathbf{\flat}_{dR}\mathbf{B E}G \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{E}G }

realized as follows. The Lie groupoid $E_{\mathrm{diff}}(G)$ is given by

$$E_{\mathrm{diff}}G:U\mapsto \{\begin{array}{ccc}& & (\mathrm{0,0})\\ & {}^{(g_1,a_1)}\swarrow & {}^{f^{-1}}⇙& {\searrow }^{(g_2,a_2)}\\ (A_1=g_1^{-1}d{g}_1+g_1^{-1}{a}_1{g}_1,F_{A_1})& & \stackrel{(e,\lambda )}{\to }& & (A_2=g_2^{-1}d{g}_2+g_2^{-1}{a}_2{g}_2,F_{A_2})\end{array}\mid \begin{array}{c}{A}_i\in {\Omega }^1(U,𝔤)\\ g_i\in C^{\mathrm{\infty }}(U,G)\\ f\in C^{\mathrm{\infty }}(U,G)\end{array}\},$$\mathbf{E}_{diff}G : U \mapsto \left\{ \array{ && (0,0) \\ & {}^{\mathllap{(g_1,a_1)}}\swarrow & {}^{\mathllap{f^{-1}}}\swArrow & \searrow^{\mathrlap{(g_2,a_2)}} \\ (A_1 = g_1^{-1} d g_1 + g_1^{-1}a_1 g_1, F_{A_1}) &&\stackrel{(e,\lambda)}{\to}&& (A_2 = g_2^{-1} d g_2 + g_2^{-1}a_2 g_2, F_{A_2}) } \;\;\; | \;\;\; \array{ A_i \in \Omega^1(U,\mathfrak{g}) \\ g_i \in C^\infty(U,G) \\ f \in C^\infty(U,G) } \right\} \,,

where the cone on the right is a 2-morphism in the model for $♭BEG$ for the 2-group $EG$ as described here and constitutes a morphism from the object

$$\begin{array}{c}0\\ {\downarrow }^{(g_1,a_1)}\\ (A_1=g_1^{-1}d{g}_1+a_1,F_{A_1})\end{array}$$\array{ 0 \\ \downarrow^{\mathrlap{(g_1,a_1)}} \\ (A_1 = g_1^{-1} d g_1 + a_1, F_{A_1}) }

to the object

$$\begin{array}{c}0\\ {\downarrow }^{(g_2,a_2)}\\ (A_2=g_2^{-1}d{g}_2+a_2,F_{A_2})\end{array}.$$\array{ 0 \\ \downarrow^{\mathrlap{(g_2,a_2)}} \\ (A_2 = g_2^{-1} d g_2 + a_2, F_{A_2}) } \,.

From the description of the resolution for $♭BEG$ in Differential coefficients for Lie 2-groups we have that in such a morphism the label $g_1$, $g_2$ and $f$ are related by

$$g_2=f{g}_1.$$g_2 = f g_1 \,.

A little inspection shows that all the rest of the data is already fixed by this. Therefore the evident forgetful furnctor $E_{\mathrm{diff}}G\to EG$ is clearly over each $U$ an essentially surjective and full and faithful functor, hence indeed a weak equivalence.

The canonical form ${\theta }_{EG}$ itself is given by projection onto the codomain of these cones, as for the canonical form on $G$ discussed above. This way we find that the canonical forms on $G$ and on $E(G)$ fit into a diagram

$$\begin{array}{ccc}G& \stackrel{{\theta }_G}{\to }& {♭}_{\mathrm{dR}}BG\\ \downarrow & & \downarrow \\ E_{\mathrm{diff}}G& \stackrel{{\theta }_{EG}}{\to }& {♭}_{\mathrm{dR}}BEG\end{array}.$$\array{ G &\stackrel{\theta_G}{\to}& \mathbf{\flat}_{dR} \mathbf{B}G \\ \downarrow && \downarrow \\ \mathbf{E}_{diff}G &\stackrel{\theta_{\mathbf{E}G}}{\to}& \mathbf{\flat}_{dR} \mathbf{B E}G } \,.

Observe that the $G$-action on $EG$ lifts immediately to an action on the slightly bigger model $E_{\mathrm{diff}}G$, where it is still principal: the only element that leaves any objects or morphisms in $E_{\mathrm{diff}}G$ fixed is the neutral element.

Explicitly, we have that over $U\in \mathrm{CartSp}$ an element $h\in G(U)=C^{\mathrm{\infty }}(U,G)$ acts on a morphism given by a cone as above by

$$\left(\begin{array}{ccc}& & (\mathrm{0,0})\\ & {}^{(g_1,a_1)}\swarrow & {}^{f^{-1}}⇙& {\searrow }^{(g_2,a_2)}\\ (A_1=g_1^{-1}d{g}_1+g_1^{-1}{a}_1{g}_1,F_{A_1})& & \stackrel{(e,\lambda )}{\to }& & (A_2=g_2^{-1}d{g}_2+g_2^{-1}{a}_2{g}_2,F_{A_2})\end{array}\right)\mapsto \left(\begin{array}{ccc}& & (\mathrm{0,0})\\ & {}^{(g_{1}h,a_1)}\swarrow & {}^{(fh{)}^{-1}}⇙& {\searrow }^{(g_{2}h,a_2)}\\ ((g_{1}h{)}^{-1}d(g_{1}h)+(g_{1}h{)}^{-1}{a}_1{g}_{1}h,h^{-1}{F}_{A_1}h)& & \stackrel{(e,\lambda )}{\to }& & ((g_{2}h{)}^{-1}d(g_{2}h)+(g_{2}h{)}^{-1}{a}_2{g}_{2}h,h^{-1}{F}_{A_1}h)\end{array}\right),$$\left( \array{ && (0,0) \\ & {}^{\mathllap{(g_1,a_1)}}\swarrow & {}^{\mathllap{f^{-1}}}\swArrow & \searrow^{\mathrlap{(g_2,a_2)}} \\ (A_1 = g_1^{-1} d g_1 + g_1^{-1}a_1 g_1, F_{A_1}) &&\stackrel{(e,\lambda)}{\to}&& (A_2 = g_2^{-1} d g_2 + g_2^{-1}a_2 g_2, F_{A_2}) } \right) \;\; \mapsto \;\; \left( \array{ && (0,0) \\ & {}^{\mathllap{(g_1 h,a_1)}}\swarrow & {}^{\mathllap{(f h)^{-1}}}\swArrow & \searrow^{\mathrlap{(g_2 h,a_2)}} \\ ((g_1 h)^{-1} d (g_1 h) + (g_1 h)^{-1}a_1 g_1 h, h^{-1}F_{A_1} h) &&\stackrel{(e,\lambda)}{\to}&& ((g_2 h)^{-1} d (g_2 h) + (g_2 h)^{-1}a_2 g_2 h, h^{-1}F_{A_1} h) } \right) \,,

hence in particular by gauge transformations of the connection forms

$$A_i\mapsto h^{-1}{A}_{i}h+h^{-1}{A}_{i}h$$A_i \mapsto h^{-1} A_i h + h^{-1} A_i h

and of the curvatures

$$F_{A_i}\mapsto h^{-1}{F}_{A_i}h.$$F_{A_i} \mapsto h^{-1} F_{A_i} h \,.

It follows that the coequalizer $B_{\mathrm{diff}}G$ of

$$G\times EG\overrightarrow{\to }EG$$G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G

is the presheaf which over $U\in \mathrm{CartSp}$ coequalizes the maps given by

$$(h,(g,a))\mapsto (g,a)$$(h,(g,a)) \mapsto (g,a)

and

$$(h,(g,a))\mapsto (g\cdot h,a),$$(h,(g,a)) \mapsto (g\cdot h,a) \,,

respectively. This is the presheaf

$$B_{\mathrm{diff}}G:U\mapsto \{a_1\stackrel{f,\lambda }{\to }(a_2=f^{-1}(a_1+d)f-\lambda )\mid \begin{array}{c}{a}_i,\lambda \in {\Omega }^1(U,𝔤)\\ f\in C^{\mathrm{\infty }}(U,G)\end{array}\}.$$\mathbf{B}_{diff}G : U \mapsto \left\{ a_1 \stackrel{f,\lambda}{\to} ( a_2 = f^{-1}(a_1 + d)f - \lambda ) \;\;\; | \;\;\; \array{ a_i, \lambda \in \Omega^1(U,\mathfrak{g}) \\ f \in C^\infty(U,G) } \right\} \,.

The projection map $E_{\mathrm{diff}}G\to B_{\mathrm{diff}}G$ is on objects given by

$$(g,a)\mapsto a.$$(g,a) \mapsto a \,.

The evident forgetful functor $BG\stackrel{\simeq }{\leftarrow }{B}_{\mathrm{diff}}G$ is, similarly to the previous argument, a weak equivalence.

Therefore the morphism of Lie groupoids

$$E_{\mathrm{diff}}G\to B_{\mathrm{diff}}G$$\mathbf{E}_{diff}G \to \mathbf{B}_{diff}G

that we have obtained is indeed another model for the universal $G$-principal bundle, somewhat bigger than the canonical one.

Cocycles with values in $B_{\mathrm{diff}}G$.

For $\{U_i\to X\}$ a good open cover of a paracompact smooth manifold $X$ and $C(\{U_i\})$ the Cech groupoid we have that a morphism

$$X\stackrel{\simeq }{\leftarrow }C(\{U_i\})\stackrel{(A_i{g}_{ij},{\lambda }_{ij})}{\to }{B}_{\mathrm{diff}}G$$X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \stackrel{(A_i g_{i j}, \lambda_{i j})}{\to} \mathbf{B}_{diff}G

of groupoid-valued presheaves is a $G$-Cech cocycle on $X$ together with on each patch $U_i$ a choice of a Lie-algebra valued 1-form $A_i\in {\Omega }^1(U_i,𝔤)$. These 1-forms do not need to satisfy any condition on double overlaps (yet): by the above characterization the failure of the $(A_i)$ to satisfy the usual Cech-de Rham cocycle condition for a connection on a bundle is measure on double overlaps by the forms

$${\lambda }_{ij}=A_j-g_{\mathrm{ij}}^{-1}(A_i+d)g_{\mathrm{ij}}.$$\lambda_{i j} = A_j - g_{ij}^{-1}(A_i + d)g_{ij} \,.

Such data is sometimes called a pseudo-connection . By itself it contains no information (and otherwise $B_{\mathrm{diff}}G\to BG$ would not be a weak equivalence) but it does serve to model the universal curvature characteristic form as a 2-anafunctor out of $BG$. We will see below that in the homotopy fibers of the morphism that this induces on cocycles, we do find the genuine (non-pseudo) connection forms.

The universal curvature characteristic forms

In order to obtain the universal curvature characteristic forms of the universal pseudo-connection we need to find the universal coefficient object in the bottom right of

$$\begin{array}{ccc}G& \stackrel{{\theta }_G}{\to }& {♭}_{\mathrm{dR}}BG\\ \downarrow & & \downarrow \\ E_{\mathrm{diff}}G& \stackrel{{\theta }_{EG}}{\to }& {♭}_{\mathrm{dR}}BEG\\ \downarrow & & \downarrow \\ B_{\mathrm{diff}}G& \to & {♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG\end{array}.$$\array{ G &\stackrel{\theta_G}{\to}& \mathbf{\flat}_{dR} \mathbf{B}G \\ \downarrow && \downarrow \\ \mathbf{E}_{diff}G &\stackrel{\theta_{\mathbf{E}G}}{\to}& \mathbf{\flat}_{dR}\mathbf{B E}G \\ \downarrow && \downarrow \\ \mathbf{B}_{diff} G &\to& \mathbf{\flat}_{dR}^{inv} \mathbf{B E}G } \,.

This is the coequalizer of the two composite maps

$$G\times EG\stackrel{\stackrel{{\theta }_{EG}\circ \rho }{\to }}{\underset{{\theta }_{EG}\circ p_1}{\to }}{♭}_{\mathrm{dR}}EG.$$G \times \mathbf{E}G \stackrel{\overset{\theta_{\mathbf{E}G}\circ \rho}{\to}}{\underset{\theta_{\mathbf{E}G}\circ p_1}{\to}} \mathbf{\flat}_{dR} \mathbf{E}G \,.

The bottom one picks the codomain of the cones depicted above, and the top one first acts with $G$ as described above, and then picks the codomain. On objects the bottom is given by

$$(h,(g,a,F_A))\mapsto (A:=g^{-1}dg+g^{-1}ag,F_A)$$(h, (g,a, F_A)) \mapsto (A:= g^{-1} d g + g^{-1} a g, F_A)

whereas the top map is

$$(h,(g,a))\mapsto (h^{-1}Ah+h^{-1}dh,h^{-1}{F}_{A}h).$$(h, (g,a)) \mapsto (h^{-1} A h + h^{-1} d h, h^{-1} F_A h) \,.

So the action divides out gauge transformations.

We therefore obtain elements in the coequalizer from the curvature characteristic forms: if $A$ and $A\prime$ are related by a gauge transformation, then for each invariant polynomial $P_i$ we have $P(F_A)=P_{F_{A\prime }}$.

So define ${♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG$ to be the Lie groupoid whose

• $U$-parameterized families of objects are collections $(P_i(F_A))$ of curvature characteristic forms (for some connection form $A\in {\Omega }^1(U,𝔤)$ which is not part of the data)

• $U$-parameterized families of morphisms are collections of Chern-Simons forms $({\mathrm{CS}}_i(A,A\prime ))$ modulo exact forms interpolating between these.

Then evaluation of curvature in invariant polynomials yields yields a morphism

$${♭}_{\mathrm{dR}}B\mathrm{INN}(G)\to {♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG$$\mathbf{\flat}_{dR} \mathbf{B} INN(G) \to \mathbf{\flat}^{inv}_{dR} \mathbf{B E}G

where $i$ ranges over the generators of invariant polynomials and $n_i$ is the degree of the $i$th generator, that fits into the above diagram for $Q$.

This morphism sends over $U\in$ CartSp the form $A\in {\Omega }^1(U,𝔤)$ to a collection ${\sum }_i{P}_i(F_A)$ of curvature characteristic forms and sends any morphism between two such forms (which, recall, need not be a gauge transformation of forms but may involve a shift of conneciton forms) to the Chern-Simons form interpolating between these, which is indeed well defined modulo exact forms

$${\mathrm{curv}}_G:((A,F_A)\stackrel{\lambda }{\to }(A\prime ,F_{A\prime }))\mapsto \sum _i{P}_i(F_A)\stackrel{\sum _i\mathrm{CS}(A,A\prime )}{\to }\sum _i{P}_i(F_{A\prime }).$$curv_G : \left( (A, F_A) \stackrel{\lambda}{\to} (A', F_{A'}) \right) \;\;\; \mapsto \;\;\; \sum_i P_i(F_A) \stackrel{\sum_i CS(A,A')}{\to} \sum_i P_i(F_{A'}) \,.

The universal connection

In total we find a model for the universal (pseudo-)Ehresmann connection on the universal $G$-principal bundle in terms of a diagram

$$\begin{array}{cccccc}G& \stackrel{{\theta }_G}{\to }& {♭}_{\mathrm{dR}}BG& & & \mathrm{vertical}\mathrm{form}\\ \downarrow & & \downarrow \\ E_{\mathrm{diff}}G& \stackrel{{\theta }_{EG}}{\to }& {♭}_{\mathrm{dR}}BEG& & & \mathrm{connection}\mathrm{and}\mathrm{curvature}\\ \downarrow & & \downarrow \\ B_{\mathrm{diff}}G& \stackrel{{\mathrm{char}}_G}{\to }& {♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG& & & \mathrm{curvature}\mathrm{characteristic}\mathrm{class}\end{array}.$$\array{ G &\stackrel{\theta_G}{\to}& \mathbf{\flat}_{dR} \mathbf{B}G &&& vertical\; form \\ \downarrow && \downarrow \\ \mathbf{E}_{diff}G &\stackrel{\theta_{\mathbf{E}G}}{\to}& \mathbf{\flat}_{dR} \mathbf{B E}G &&& connection \; and \; curvature \\ \downarrow && \downarrow \\ \mathbf{B}_{diff} G &\stackrel{char_G}{\to}& \mathbf{\flat}^{inv}_{dR} \mathbf{B E}G &&& curvature\; characteristic\; class } \,.

The morphism

$${\mathrm{char}}_G:B{G}_{\mathrm{diff}}\to {♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG$$char_G : \mathbf{B}G_{diff} \to \mathbf{\flat}^{inv}_{dR} \mathbf{B E}G

that arises from pushing down the canonical form/universal pseudo-connection ${\theta }_{EG}$ on $EG$ is the universal curvature characteristic form . We notice that cohomology with coefficients in ${♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG$ sits in the de Rham cohomology ${\prod }_i{H}_{\mathrm{dR}}^{n_i}(X)$ and that every cohomology class in there has a representative

$$\begin{array}{ccc}C(\{U_i\})& \to & {♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG\\ {}^{\simeq }\downarrow \\ X\end{array}$$\array{ C(\{U_i\}) &\to& \mathbf{\flat}_{dR}^{inv} \mathbf{B E}G \\ {}^{\mathllap{\simeq}}\downarrow \\ X }

given by a collection of globally defined forms, in other words we have representatives for each cohomology class that extend as

$$\begin{array}{ccc}C(\{U_i\})& \to & {♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG\\ {}^{\simeq }\downarrow & \nearrow \\ X\end{array}.$$\array{ C(\{U_i\}) &\to& \mathbf{\flat}_{dR}^{inv} \mathbf{B E}G \\ {}^{\mathllap{\simeq}}\downarrow & \nearrow \\ X } \,.

Picking one such representative for each class yields gives a morphism

$$\prod _i{H}_{\mathrm{dR}}^{n_i}(X)\to H(X,{♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG).$$\prod_i H^{n_i}_{dR}(X) \to \mathbf{H}(X,\mathbf{\flat}_{dR}^{inv} \mathbf{B E}G) \,.

By the general reasoning of differential cohomology in an (∞,1)-topos we have that the differential cocycles proper are those in the homotopy pullback $H_{\mathrm{diff}}(X,BG)$ of this morphism along the curvature class morphism

$$\begin{array}{ccc}{H}_{\mathrm{diff}}(X,BG)& \to & \prod _i{H}_{\mathrm{dR}}^{n_i}(X)\\ \downarrow & & \downarrow \\ H(X,BG)& \stackrel{{\mathrm{char}}_G}{\to }& H(X,{♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG)\end{array}.$$\array{ \mathbf{H}_{diff}(X,\mathbf{B}G) &\to& \prod_i H^{n_i}_{dR}(X) \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}G) &\stackrel{char_G}{\to}& \mathbf{H}(X,\mathbf{\flat}_{dR}^{inv} \mathbf{B E}G) } \,.

Using that the quotient map is a Kan fibration, we find hat the bottom morphism evaluated on a Cech nerve is a Kan fibration, so that this homotopy pullback is computed as the ordinary pullback of cocycles.

It follows that a differential cocycle is a pseudo-connection on a bundle, that does satisfy the condition that the connection forms $(A_i)$ induce on double overlaps exact Chern-Simons forms interpolating between their curvature characteristic forms. This is solved in particular by proper connections.

This way the connection cocycle condition is imposed after all on the differential cocycles, even though the universal conneciton is a pseudo-connection.

Differential K-theory classes

But notice that in the above differential cohomology cocycle groupoid $H_{\mathrm{diff}}(X,BG)$ we have coboundaries that are more general than usual gauge transformations of connections:

Again by the nature of ${♭}_{\mathrm{dR}}^{\mathrm{inv}}BEG$ we have that a coboundary between $(A_i,g_{ij})$ and $(A{\prime }_i,g{\prime }_{ij})$ is a transformation such that the interpolating Chern-Simons forms of the curvature characteristic forms are exact.

This equivalence relation is that defining Simons-Sullivan structured bundles. For $G=U$ the unitary group these represent classes in the differential K-theory of $X$.

… (under construction)…

Strict Lie 2-groups

Let now $G=\Xi [G_2\to G_1]$ be a strict Lie 2-group coming from a smooth crossed module $G_2\stackrel{\delta }{\to }{G}_1$ with action $\alpha :G_1\to \mathrm{Aut}(G_2)$.

Delooping

Proposition

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

$$\Xi [G_2\to G_1\overrightarrow{\to }*].$$\Xi[G_2 \to G_1 \stackrel{\to}{\to} *] \,.

Proof

As above for Lie groups.

Differential coefficients

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

Write $[{𝔤}_2\stackrel{{\delta }_{*}}{\to }{𝔤}_1]$ for the corresponding differential crossed module with action ${\alpha }_{*}:{𝔤}_1\to \mathrm{der}({𝔤}_2)$ corresponding to the Lie strict 2-group crossed module $(G_2\stackrel{\delta }{\to }{G}_1)$ with action $\alpha :G_1\to \mathrm{Aut}(G_2)$.

Proposition

The Lie 2-groupoid $♭B[G_2\stackrel{\delta }{\to }{G}_1]$ is represented in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ by the Lie 2-groupoid which on $U\in \mathrm{CartSp}$ s the following 2-groupoid:

• An object is a pair

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

  such that

  $${\delta }_{*}B-dA+[A\wedge A]=0$$\delta_* B - d A + [A \wedge A] = 0

  and

  $$dB+[A\wedge B]=0.$$d B + [A \wedge B] = 0 \,.

• A 1-morphism $(g,a):(A,B)\to (A\prime ,B\prime )$ is a pair

  $$g\in C^{\mathrm{\infty }}(U,G_1),a\in {\Omega }^1(U,{𝔤}_2)$$g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)

  such that

  $$A\prime =g^{-1}Ag+g^{-1}dg+g^{-1}{\delta }_{*}ag$$A' = g^{-1} A g + g^{-1} d g + g^{-1} \delta_* a g

  and

  $$B\prime ={\alpha }_{g^{-1}}(B+da+[a\wedge a]+{\alpha }_{*}(A\wedge a)).$$B' = \alpha_{g^{-1}}( B + d a + [a \wedge a] + \alpha_*(A \wedge a) ) \,.

  The composite of two 1-morphisms

  $$(A,B)\stackrel{(g_1,a_1)}{\to }(A\prime ,B\prime )\stackrel{(g_2,a_2)}{\to }(A″,B″)$$(A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to} (A'', B'')

  is given by the pair

  $$(g_1{g}_2,a_1+({\alpha }_{g_2}{)}_{*}{a}_2).$$(g_1 g_2, a_1 + (\alpha_{g_2})_* a_2) \,.

• a 2-morphism $f:(\lambda ,a)\to (\lambda \prime ,a\prime )$ is a function

  $$f\in C^{\mathrm{\infty }}(U,G_2)$$f \in C^\infty(U,G_2)

  such that

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

  and

  $$a\prime =f^{-1}df+f^{-1}af+f^{-1}(r_f^{-1}\circ {\alpha }_f{)}_{*}(a)f$$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 ${♭}_{\mathrm{dR}}B[G_2\to G_1]$ is as the one above, discarding the piece $C^{\mathrm{\infty }}(-,G_1)$ in the 1-morphisms and the piece f$C^{\mathrm{\infty }}(-,G_2)$ in the 2-morphismms.

Proof

Form the defining pullback as before. (…)

Examples

The String Lie 2-group

There is a strict Lie 2-group model for the string Lie 2-group.

Proposition

The string Lie 2-algebra in the form ${𝔤}_{\mu }$ is equivalent to the strict Lie 2-algebra given by the differential crossed module $\hat{\Omega }𝔤\to P𝔤$, where…

(…)

We now describe three different but equivalent strict Lie 2-group models for the String 2-group.

(…)

Connections

(…)

The Fivebrane Lie 6-group

There is a strict Lie 6-group model of the fivebrane Lie 6-group?.

There is a strict Lie 2-group model for the string Lie 2-group

Connections

(…)

General $\mathrm{\infty }$-Lie groupoids

We now discuss general, not-necessarily strict $\mathrm{\infty }$-Lie groupoids.

Lie-integrated $\mathrm{\infty }$-Lie groupoids

We discuss here $\mathrm{\infty }$-Lie groupoids that arise as the Lie integration of an ∞-Lie algebroid.

Lie integration

Definition

(forms on the simplex)

Write ${\Delta }_{\mathrm{Diff}}:\Delta \to \mathrm{CartSp}\hookrightarrow \mathrm{sPSh}(\mathrm{CartSp})$ for the cosimplicial object of standard smooth simplices.

Write ${\Omega }_{\mathrm{si}}^{•}({\Delta }_{\mathrm{Diff}}^n)$ for the dg-algebra of those differential forms on ${\Delta }_{\mathrm{Diff}}^n$ that have the property that for every $k\in \mathbb{N}$ every $k$-face of ${\Delta }_R^n$ has an open neighbourhood of its boundary such that the form restricted to that face is constant in the direction perpendicular to that boundary.

Definition

Let $𝔤$ be an L-∞-algebra, dually characterized by its Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤)\in \mathrm{dgAlg}$.

Write $\exp (b𝔤)\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ for the object

$$U,[n]\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),{\Omega }_{\mathrm{si}}^{•}({\Delta }_{\mathrm{Diff}}^n)\otimes C^{\mathrm{\infty }}(U)).$$U,[n] \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet_{si}(\Delta^n_{Diff}) \otimes C^\infty(U)) \,.

For $n\in \mathbb{N}$ we say the Lie $n$-groupoid universally intergating $𝔤$ is the $n$-truncation of this object

$${\tau }_n\exp (𝔞).$$\tau_n \exp(\mathfrak{a}) \,.

Realized objectwise as the $(n+1)$-simplicial coskelton

$$U,[n]\mapsto {\mathrm{cosk}}_{n+1}{\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),{\Omega }_{\mathrm{si}}^{•}({\Delta }_{\mathrm{Diff}}^n)\otimes C^{\mathrm{\infty }}(U)).$$U,[n] \mapsto \mathbf{cosk}_{n+1} Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet_{si}(\Delta^n_{Diff}) \otimes C^\infty(U)) \,.

Remark

The generalization of this procedure from $\mathrm{\infty }$-Lie algebras $𝔤$ to ∞-Lie algebroids $𝔞$ is essentially straightforward, except for a slight technicality: instead of the presheaf construction $U\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),C^{\mathrm{\infty }}(U)\otimes {\Omega }^{•}({\Delta }_{\mathrm{Diff}}^n))$ one needs to consider the construction

$$\mathrm{conc}(U\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔞),{\Omega }^{•}(U\times {\Delta }_{\mathrm{Diff}}^n))),$$conc(U \mapsto Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(U \times \Delta^n_{Diff}))) \,,

where $\mathrm{conc}$ denotes the operation of passing to concrete presheaves.

Remark

The bare ∞-groupoid

$$\Gamma \exp (𝔞):[n]\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }_{\mathrm{Diff}}^n))$$\Gamma \exp(\mathfrak{a} ) : [n] \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(\Delta_{Diff}^{n}))

underlying the untruncated $\mathrm{\infty }$-Lie groupoid integrating $𝔤$ is essentiall the Sullivan construction on $\mathrm{CE}(𝔤)$, familiar from rational homotopy theory. The fact that this construction may be thought of in terms of Lie integration was amplified in

• Ezra Getzler, Lie theory for nilpotent $L_{\mathrm{\infty }}$-algebras (arXiv)

following Hinich.

A refinement of this construction to internal ∞-groupoids in Banach spaces was considered in

• Andre Henriques, Integrating $L_{\mathrm{\infty }}$-algebras (arXiv)

For more see Lie integration.

The evident refinement of the Sullivan construction to $\mathrm{\infty }$-Lie groupoids as considered here, parameterized over smooth test spaces, was mentioned around

• U.S., Differential nonabelian cohomology talk at Higher Structures in Math and Physics Lausanne (2008) (pdf)

and was considered around the same times also by Dmitry Roytenberg.

In

• Pavol Severa, …

is given a proposal for how to realize Lie differentiation in this context. Below we will see that, in analogy to and generalizing the above examples, the Lie differentiation of ${\tau }_n\exp (𝔤)$ is canonically induced by passing to its intrinsic de Rham coefficient object ${♭}_{\mathrm{dR}}{\tau }_n\exp (𝔞)$. Below in Infinitesimal differential coefficients we discuss how these are realized in terms of infinitesimal paths in the $\mathrm{\infty }$-Lie groupoid. This is at least in spirit close to Ševera’s construction. A more detailed discussion of the relation should be given somewhere, eventually.

The Lie-integrated universal principal $\mathrm{\infty }$-bundle

We describe a construction of the universal principal ∞-bundle of a Lie integrated $\mathrm{\infty }$-Lie group $G$.

Let $𝔤$ be a Lie n-algebra. The construction we consider works for choices $n\in \mathbb{N}$ such that the $\mathrm{\infty }$-Lie groupoid

$${\tau }_{\le n}\exp (𝔤)$$\tau_{\leq n}\exp(\mathfrak{g})

is equivalent to

$$BG:={\tau }_{\le n+1}\exp (𝔤),$$\mathbf{B}G := \tau_{\leq {n+1}} \exp(\mathfrak{g}) \,,

i.e. that we can truncate the Sullivan construction one degree “too high” without picking up a superfluous homotopy group (see Lie integration for details on this).

Example. This is for instance the case for $𝔤$ an ordinary Lie algebra, $n=1$: in that case ${\tau }_{\le 1}\exp (𝔤)$ is the one-object Lie groupoid $BG$ for $G$ the simply connected Lie group integrating it, and since for every Lie algebra we have ${\pi }_2(G)=0$ there is a trivial homotopy group in degree 2 and so (by the discussion at Lie integration) $BG\simeq {\tau }_{\le 2}\exp (𝔤)$.

The 1-morphism in ${\tau }_{\le 2}\exp (𝔤)$ are based paths in $G$, the 2-morphisms are homotopy-classes (rel boundary) of surfaces in $G$. The equivalence ${\exp }_{\le 2}\exp (𝔤)\to BG$ is obtained by sending a path to its endpoint. This situation is discussed in detail at The string Lie 2-group.

Definition

For $𝔤$ an L-∞-algebra write $\mathrm{inn}(𝔤)$ for the $L_{\mathrm{\infty }}$-algebra whose Chevalley-Eilenberg algebra is the Weil algebra $W(𝔤)$ of $𝔤$:

$$\mathrm{CE}(\mathrm{inn}(𝔤))=W(𝔤).$$CE(inn(\mathfrak{g})) = W(\mathfrak{g}) \,.

The $L_{\mathrm{\infty }}$-algebra $\mathrm{inn}(𝔤)$ is an $L_{\mathrm{\infty }}$-algebroidal model for the universal $𝔤$-principal bundle, in analogy to a groupal model for universal principal ∞-bundles.

Example

(Cartan model)

For $𝔤$ an ordinary Lie algebra, the Lie 2-algebra $\mathrm{inn}(𝔤)$ is the one coming from the differential crossed module $(𝔤\stackrel{\mathrm{Id}}{\to }𝔤)$. That this is a Lie 2-algebraic model for the Segal groupal model of the universal G-bundle is implicit in the Cartan model of equivariant cohomology?, which is a Lie algebroid-model of the Borel construction? $EG{\times }_{G}V$.

Definition

(integrated universal principal $\mathrm{\infty }$-bundle)

The Lie integration of $\mathrm{inn}(𝔤)$ we write

$$BEG:={\tau }_{\le n+1}\exp (\mathrm{inn}(𝔤))).$$\mathbf{B}\mathbf{E}G := \tau_{\leq {n+1}}\exp(inn(\mathfrak{g}))) \,.

The canonical inclusion $𝔤\to \mathrm{inn}(𝔤)$ induces a morphism of $\mathrm{\infty }$-Lie groupoids

$${\tau }_{n+1}\exp (𝔤\to \mathrm{inn}(𝔤)):BG\to BEG.$$\tau_{n+1}\exp(\mathfrak{g} \to inn(\mathfrak{g})) : \mathbf{B}G \to \mathbf{B}\mathbf{E}G \,.

This induces the corresponding morphism of ∞-Lie groups.

$$G\to EG.$$G \to \mathbf{E}G \,.

Observation

Since $W(𝔤)$ is contractible (has vanishing cochain cohomology) we have a weak equivalence

$$EG\simeq *.$$\mathbf{E}G \simeq * \,.

So the construction $EG={\tau }_{n+1}\exp (𝔤)$ is a groupal model for universal principal ∞-bundles.

(…)

Examples

We spell out some examples of Lie integration.

Integration of a Lie algebra

Proposition

Let $𝔤$ be an ordinary finite-dimensional Lie algebra and write $G$ for the corresponding simply connected Lie group.

We have

$$BG\simeq {\tau }_1\exp (b𝔤)$$\mathbf{B}G \simeq \tau_1 \exp(b \mathfrak{g})

Proof

We observe that $\exp (𝔤)$ has a single object. A k-morphism in $\exp (𝔤)$ is a flat $𝔤$-valued 1-form on ${\Delta }_{\mathrm{Diff}}^k$. Hence morphisms of ${\tau }_1\exp (𝔤)$ are smooth $𝔤$-valued 1-forms on the interval, two of them are connected by a contractibel space of higher cells if they may be interpolated by a flat $𝔤$-valued 1-form on the disk. That this collection of morphisms modulo this relation is indeed the simply connected Lie group $G$, with its standard smooth structure, is reviewed and discussed in detail in

• Marius Crainic, Rui Fernandes, Integrability of Lie brackest (arXiv)

Integration to line $n$-groups

Definition

Write $b^nℝ$ for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree $(n+1)$ and vanishing differential.

Write $e{b}^nℝ$ for the dg-algebra on one generator in degree $(n+1)$ and one in degree $(n+2)$ and the differential taking the former to the latter.

Lemma

For each $n\in \mathbb{N}$ the diagram

$$\begin{array}{ccc}\exp (b^nℝ)& \to & *\\ \downarrow & & \downarrow \\ \exp (e{b}^nℝ)& \to & \exp (b^{n+1}ℝ)\end{array}$$\array{ \exp(b^n \mathbb{R}) &\to& * \\ \downarrow && \downarrow \\ \exp(e b^n \mathbb{R}) &\to& \exp(b^{n+1}\mathbb{R}) }

is a pullback diagram in $\mathrm{sPSh}(\mathrm{CartSp})$. Moreover, there is a morphism of diagrams

$$\begin{array}{ccccc}\exp (b^nℝ)& \to & \exp (e{b}^nℝ)& \to & \exp (b^{n+1}ℝ)\\ \downarrow & & \downarrow & & \downarrow \\ B^nℝ& \to & E{B}^nℝ& \to & B^{n+1}ℝ\end{array}$$\array{ \exp(b^{n} \mathbb{R}) &\to& \exp(e b^{n} \mathbb{R}) &\to& \exp(b^{n+1} \mathbb{R}) \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}^n \mathbb{R} &\to& \mathbf{E}\mathbf{B}^n \mathbb{R} &\to& \mathbf{B}^{n+1} \mathbb{R} }

where the vertical morphisms are acyclic fibrations, induced by sending differential $n$-forms on the $n$-simplex to their integral over that $n$-simplex.

Proof

The key fact underlying this is that a closed smooth $n$-form on the $n$-sphere may be extended smoothly to a closed $n$-form on the $(n+1)$-ball precisely if its integral over the sphere vanishes.

From this it follows that also every closed $n$-form on the $k$-sphere for $k>n$ may be extended as a closed $n$-form to the $(n+1)$-ball. The same holds for smooth families of forms. This implies that $\exp (b^nℝ)\to B^nℝ$ is an acyclic fibration for all $n$.

Integration of $\mathrm{\infty }$-Lie algebra cocycles

We discuss how under Lie integration a cocycle in ∞-Lie algebra cohomology integrates to a cocycle on $\mathrm{\infty }$-Lie groupoids.

For $𝔤$ an ∞-Lie algebra, an $n$-cocycle on $𝔤$ (with “values in the trivial module $ℝ$”) is a morphism

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

Dually this is a dg-algebra-morphism of Chevalley-Eilenberg algebras

$$\mathrm{CE}(𝔤)\leftarrow \mathrm{CE}(b^{n-1}ℝ):\mu .$$CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R}) : \mu \,.

Since the dg-algebra on the right is semifree on a single generator in degree $(n+1)$ and with vanishing differential, this is the same as a closed element in the CE-algebra

$$\mu \in {\wedge }^n{𝔤}^{*}{d}_{𝔤}\mu =0.$$\mu \in \wedge^n \mathfrak{g}^* \;\;\; d_{\mathfrak{g}} \mu = 0 \,.

Applying the Lie integration functor $\exp (-)$ we obtain a morphism between the $\mathrm{\infty }$-Lie groupoid incarnations of $𝔤$ and $b^{n-1}ℝ$

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

Remember that the Lie integration proper of a Lie $k$-algebra $𝔤$ is the $k$-truncation

$$BG:={\tau }_k\exp (𝔤)$$\mathbf{B}G := \tau_k \exp(\mathfrak{g})

of $\exp (𝔤)$. Similarly, the Lie integration of $b^{n-1}ℝ$ is the $n$-truncation, which by Integration of abelian L-infinity algebras is

$${\tau }_n\exp (b^{n-1}ℝ)\simeq B^nℝ.$$\tau_n \exp(b^{n-1} \mathbb{R}) \simeq \mathbf{B}^n \mathbb{R} \,.

In general $k$ and $n$ will differ. The integration of the cocycle involves finding a discrete $\mathrm{\infty }$-group $K$ such that the morphism nevertheless lifst to

$$\begin{array}{ccc}{\tau }_n\exp (𝔤)& \stackrel{{\tau }_n\exp (\mu )}{\to }& {\tau }_n(b^{n-1}ℝ)\simeq B^nℝ\\ \downarrow & & \downarrow \\ {\tau }_k\exp (𝔤)& \to & B^nℝ/K\end{array}$$\array{ \tau_n \exp(\mathfrak{g}) &\stackrel{\tau_n \exp(\mu)}{\to}& \tau_n(b^{n-1} \mathbb{R}) \simeq \mathbf{B}^n \mathbb{R} \\ \downarrow && \downarrow \\ \tau_k \exp(\mathfrak{g}) &\to& \mathbf{B}^n \mathbb{R}/K }

Integration of the String 3-cocycle

For $𝔤$ a semisimple Lie algebra with binary invariant polynomial $⟨-,-⟩$, we have a canonical 3-cocycle

$$\mu =⟨-,[-,-]⟩\in \mathrm{CE}(𝔤).$$\mu = \langle -, [-,-]\rangle \in CE(\mathfrak{g}) \,.

Since this controls the string Lie 2-algebra, we call it the String-cocycle.

We have that ${\tau }_2\exp (𝔤)\simeq BG$ for $G$ the simply connected Lie group integrating $𝔤$. On the other hand, since ${\tau }_3(G)\simeq \mathbb{Z}$ we have the pushout

$$\begin{array}{ccc}{B}^3\mathbb{Z}& \to & {\tau }_3\exp (𝔤)\\ \downarrow & & \downarrow \\ E{B}^2\mathbb{Z}& \to & {\tau }_3\exp (𝔤)/\mathbb{Z}\end{array}$$\array{ \mathbf{B}^3 \mathbb{Z} &\to& \tau_3 \exp(\mathfrak{g}) \\ \downarrow && \downarrow \\ \mathbf{E} \mathbf{B}^2 \mathbb{Z} &\to& \tau_3 \exp(\mathfrak{g})/\mathbb{Z} }

and a weak equivalence ${\tau }_3\exp (𝔤)/\mathbb{Z}\to BG$.

Accordingly, the Lie algebra 3-cocycle integrates to a Lie group cocycle

$$\begin{array}{ccc}{\tau }_3\exp (𝔤)& \stackrel{\exp (\mu )}{\to }& B^3ℝ\\ \downarrow & & \downarrow \\ BG& \stackrel{{\int }_{\mu }}{\to }& B^3(ℝ/\mathbb{Z})\end{array}.$$\array{ \tau_3 \exp(\mathfrak{g}) &\stackrel{\exp(\mu)}{\to}& \mathbf{B}^3 \mathbb{R} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\int_\mu}{\to}& \mathbf{B}^3 (\mathbb{R}/\mathbb{Z}) } \,.

Unwinding the definitions, we see that

• ${\tau }_3\exp (𝔤)$ has as 1-morphisms based paths in $G$, as 2-morphisms based surfaces in $G$; as 3-morphisms unique morphisms between any pair of parallel 2-morphisms;

• the integrated cocycle ${\int }_{\mu }$ takes on 3-morphism the value obtained by choosing a 3-ball $\sigma :{\Sigma }_3\to G$ cobounding the boundary 2-morphisms-surfaces (which always exists since ${\pi }_2(G)=0$) and sending it to the 3-morphism labeled by ${\int }_{{\Sigma }_3}{\sigma }^{*}\mu \in ℝ/\mathbb{Z}$, which is well defined since $\mu$ has integral periods on $G$ (times some normalization constant).

We find that the string 2-group is the homotopy fiber of this integrated cocycle, i.e the (∞,1)-pullback

$$\begin{array}{ccc}B{\mathrm{String}}_{\mu }(G)& \to & *\\ \downarrow & & \downarrow \\ BG& \stackrel{\int \mu }{\to }& B^3ℝ/\mathbb{Z}\end{array}$$\array{ \mathbf{B}String_\mu(G) &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\int \mu}{\to}& \mathbf{B}^3 \mathbb{R}/\mathbb{Z} }

in $H$.

Parallel transport: integration of $\mathrm{\infty }$-Lie algebra valued differential forms

A special case of the general notion of cocycle in an ∞-Lie algebroid is of general interest: for $X$ a smooth manifold write $TX$ for the tangent Lie algebroid of $X$. For $𝔤$ any ∞-Lie algebra or ∞-Lie algebroid, a cocycle on $TX$ with coefficients in $𝔤$, i.e. a morphism

$$\omega :TX\to 𝔤$$\omega : T X \to \mathfrak{g}

is a collection of flat ∞-Lie algebroid valued differential forms. In the special case that $𝔤$ is an ordinary Lie algebra, this reduces to the standard notion of flat Lie-algebra valued 1-form with vanishing curvature 2-form.

The integration of $TX$ is (at least locally) the path ∞-groupoid. The integration of the cocycle $\omega$ is its parallel transport

$$\exp (\omega ):\Pi (X)\to \exp (𝔤).$$\exp(\omega) : \mathbf{\Pi}(X) \to \exp(\mathfrak{g}) \,.

(…)

Differential coefficients

Above we have defined for every ∞-Lie algebra $𝔤$ a tower of $\mathrm{\infty }$-Lie groupoids ${\tau }_n\exp (𝔤)$ integrating it. Now we consider the corresponding de Rham $\mathrm{\infty }$-Lie groupoids ${♭}_{\mathrm{dR}}{\tau }_n\exp (𝔤)$.

Recall that in the above examples we saw for $G$ a Lie $\mathrm{\infty }$-group that the underlying discrete Lie $\mathrm{\infty }$-groupoid $♭BG$ is resolved by the presheaf $G{\mathrm{TrivBund}}_{\mathrm{flat}}$ of trivial $G$-principal $\mathrm{\infty }$-bundles with flat connection. From this resolution the de Rham object ${♭}_{\mathrm{dR}}BG$ is obtained as an ordinary pullback of presheaves. These statements we now produce in the full generality of an $\mathrm{\infty }$-Lie group obtained from the integration of an $L_{\mathrm{\infty }}$-algebra.

First we produce a resolution of the underlying bare $\mathrm{\infty }$-groupoid

$$♭\exp (𝔤):(U,[n])\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }_{\mathrm{Diff}}^n)).$$\mathbf{\flat}\exp(\mathfrak{g}) : (U,[n]) \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(\Delta^n_{Diff})) \,.

Definition

Write $\exp (𝔤){\mathrm{Bund}}_{\mathrm{flat}}$ for the simplicial presheaf given by

$$\exp (𝔤){\mathrm{Bund}}_{\mathrm{flat}}:(U,[n])\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),{\Omega }^{•}(U\times {\Delta }_{\mathrm{Diff}}^n)).$$\exp(\mathfrak{g})Bund_{flat} : (U,[n]) \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(U \times \Delta^n_{Diff})) \,.

Here and in the following it is understood that diferential forms on a space that contains a ${\Delta }_{\mathrm{Diff}}^n$ as a factor have sitting instants : for every $k$ and every $k$-face of ${\Delta }_{\mathrm{Diff}}^n$ there is a neighbourhood of the boundary of that face on which the form is constant in the direction perpendicular to that boundary.

Lemma

The canonical morphism

$$♭\exp (𝔤)\to \exp (𝔤){\mathrm{Bund}}_{\mathrm{flat}}$$\mathbf{\flat}\exp(\mathfrak{g}) \to \exp(\mathfrak{g}) Bund_{flat}

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

Below we use this to factor the inclusion $♭\exp (𝔤)\to \exp (𝔤)$ as $♭\exp (𝔤)\stackrel{\simeq }{\to }\exp (𝔤){\mathrm{Bund}}_{\mathrm{flat}}\to \exp (𝔤)$ with the last morphism being a fibration.

Proof

The morphism of simplicial presheaves is on each object $U$ the morphism of simplicial sets

$$\alpha (U):\mathrm{Hom}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }^{•}))\to \mathrm{Hom}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }^{•}\times U))$$\alpha(U) : Hom(CE(\mathfrak{g}), \Omega^\bullet(\Delta^\bullet)) \to Hom(CE(\mathfrak{g}), \Omega^\bullet(\Delta^\bullet \times U))

which is given degreewise by postcomposition with the morphisms of dg-algebras ${\Omega }^{•}({\Delta }_{\mathrm{Diff}}^n)\to {\Omega }^{•}({\Delta }_{\mathrm{Diff}}^n)\otimes {\Omega }^{•}(U)$ that sends $\omega$ to $\omega \otimes 1$.

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

$$F_U:\mathrm{Hom}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }^{•}\times U))\to \mathrm{Hom}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }^{•}))$$F_U : Hom(CE(\mathfrak{g}), \Omega^\bullet(\Delta^\bullet \times U)) \to Hom(CE(\mathfrak{g}), \Omega^\bullet(\Delta^\bullet))

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

We take $F_U$ to be given by evaluation at $0\in U$, i.e. by postcomposition with the morphisms

$${\Omega }^{•}({\Delta }^n\times U)\stackrel{\mathrm{Id}\times 0^{*}}{\to }{\Omega }^{•}({\Delta }^n\times *)={\Omega }^{•}({\Delta }^n).$$\Omega^\bullet(\Delta^n \times U) \stackrel{Id \times 0^*}{\to} \Omega^\bullet(\Delta^n \times * ) = \Omega^\bullet(\Delta^n) \,.

Recall that the morphism $F_U$ will be an acyclic Kan fibration precisely if all diagrams of the form

$$\begin{array}{ccc}\partial \Delta [n]& \to & \mathrm{Hom}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }^{•}\times U))\\ \downarrow & & {\downarrow }^{F_U}\\ \Delta [n]& \to & \mathrm{Hom}(\mathrm{CE}(𝔤),{\Omega }^{•}({\Delta }^{•}))\end{array}$$\array{ \partial \Delta[n] &\to& Hom(CE(\mathfrak{g}), \Omega^\bullet(\Delta^\bullet \times U)) \\ \downarrow && \downarrow^\mathrlap{F_U} \\ \Delta[n] &\to& Hom(CE(\mathfrak{g}), \Omega^\bullet(\Delta^\bullet)) }

have a lift. Since the differential forms on the simplices have sitting instances, we may for convenience equivalently reformulate this in terms of spheres as follows:

for every morphism $\mathrm{CE}(𝔤)\to {\Omega }^{•}(D^n)$ and morphism $\mathrm{CE}(𝔤)\to {\Omega }^{•}(U\times S^{n-1})$ such that the diagram

$$\begin{array}{ccc}\mathrm{CE}(𝔤)& \to & {\Omega }^{•}(U\times S^{n-1})\\ \downarrow & & \downarrow \\ {\Omega }^{•}(D^n)& \to & {\Omega }^{•}(S^{n-1})\end{array}$$\array{ CE(\mathfrak{g}) &\to& \Omega^\bullet(U \times S^{n-1}) \\ \downarrow && \downarrow \\ \Omega^\bullet(D^n) &\to& \Omega^\bullet(S^{n-1}) }

commutes, this may be factored as

$$\begin{array}{c}\mathrm{CE}(𝔤)\\ & \searrow \\ & & {\Omega }^{•}(U\times D^n)& \to & {\Omega }^{•}(U\times S^{n-1})\\ & & \downarrow & & \downarrow \\ & & {\Omega }^{•}(D^n)& \to & {\Omega }^{•}(S^n)\end{array}.$$\array{ CE(\mathfrak{g}) \\ & \searrow \\ &&\Omega^\bullet(U \times D^n) &\to& \Omega^\bullet(U \times S^{n-1}) \\ &&\downarrow && \downarrow \\ &&\Omega^\bullet(D^n) &\to& \Omega^\bullet(S^n) } \,.

This factorization we now construct.

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

$$U\times [\mathrm{0,1}]\to U$$U \times [0,1] \to U

by

$$(u,\sigma )\mapsto u\cdot f(1-\sigma ),$$(u,\sigma) \mapsto u \cdot f(1-\sigma) \,,

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

Pullback of differential forms along this map produces a morphism

$${\Omega }^{•}(U\times S^{n-1})\to {\Omega }^{•}(U\times S^{n-1}\times [\mathrm{0,1}])$$\Omega^\bullet(U \times S^{n-1}) \to \Omega^\bullet(U \times S^{n-1} \times [0,1])

which is such a form $\omega$ is sent to a form which in a neighbourhood $(1-ϵ\mathrm{,1}]$ of $1\in [\mathrm{0,1}]$ is constant along $(1-ϵ\mathrm{,1}]\times U$ on the value $(0\otimes {\mathrm{Id}}_{S^{n-1}}{)}^{*}\omega$.

We can then glue to the morphism

$$\mathrm{CE}(𝔤)\to {\Omega }^{•}(U\times S^{n-1})\to {\Omega }^{•}(U\times [\mathrm{0,1}]\times S^{n-1})$$CE(\mathfrak{g}) \to \Omega^\bullet(U \times S^{n-1}) \to \Omega^\bullet(U \times [0,1] \times S^{n-1})

the morphism

$$\mathrm{CE}(𝔤)\to {\Omega }^{•}(D^n)\to {\Omega }^{•}(U\times \{1\}\times D^n)$$CE(\mathfrak{g}) \to \Omega^\bullet(D^n) \to \Omega^\bullet(U \times \{1\} \times D^n)

by smoothly identifying the union $[\mathrm{0,1}]\times S^{n-1}{\coprod }_{S^{n-1}}{D}^n$ with another copy of $D^n$ to obtain in total a morphism

$$\mathrm{CE}(𝔤)\to {\Omega }^{•}(U\times D^n)$$CE(\mathfrak{g}) \to \Omega^\bullet(U \times D^n)

with the desired properties.

Lemma

The canonical morphism

$$\exp (𝔤){\mathrm{Bund}}_{\mathrm{flat}}\to \exp (𝔤)$$\exp(\mathfrak{g}) Bund_{flat} \to \exp(\mathfrak{g})

is a fibration in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$.

Proof

Over each $U\in \mathrm{CartSp}$ the morphisms is induced from the morphism of dg-algebras

$${\Omega }^{•}(U)\to C^{\mathrm{\infty }}(U)$$\Omega^\bullet(U) \to C^\infty(U)

that discards all differential forms of non-vanishing degree.

It will be sufficient to show that for

$$\mathrm{CE}(𝔤)\to {\Omega }^{•}(D^n\times [\mathrm{0,1}])\otimes \prime C^{\mathrm{\infty }}(U)$$CE(\mathfrak{g}) \to \Omega^\bullet( D^n \times [0,1] ) \otimes' C^\infty(U)

a morphism and

$$\mathrm{CE}(𝔤)\to {\Omega }^{•}(D^n\times U)$$CE(\mathfrak{g}) \to \Omega^\bullet(D^n \times U)

a lift of its restriction to $\sigma =0\in [\mathrm{0,1}]$ we have a lift

$$\mathrm{CE}(𝔤)\to {\Omega }^{•}(D^n\times [\mathrm{0,1}]\times U)$$CE(\mathfrak{g}) \to \Omega^\bullet(D^n \times [0,1] \times U)

extending the lift. 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 $L_{\mathrm{\infty }}$-algebra valued forms.

1st case: $𝔤=𝔲(1)$

To warm up, consider the simplest case where $𝔤=𝔲(1)$.

Then a morphism $C(𝔤)\to {\Omega }^{•}(D^1\times U\times [\mathrm{0,1}])\otimes C^{\mathrm{\infty }}(U)$ is an element $A_v(u)dv+A_{\sigma }(u)d\sigma$ that satisfies

$${\partial }_v{A}_u={\partial }_u{A}_v$$\partial_v A_u = \partial_u A_v

and

$${\partial }_{\sigma }{A}_v={\partial }_v{A}_{\sigma }.$$\partial_\sigma A_v = \partial_v A_\sigma \,.

To lift this to a morphism $\mathrm{CE}(𝔤)\to {\Omega }^{•}(D^1\times U\times [\mathrm{0,1}])$ that restricts to the former for $\sigma =0$ we need to add a term $A_u(u,\sigma )du$. That satisfies the differential equations

$${\partial }_{\sigma }{A}_u={\partial }_u{A}_{\sigma }$$\partial_\sigma A_u = \partial_u A_\sigma

and

$${\partial }_v{A}_u={\partial }_u{A}_v.$$\partial_v A_u = \partial_u A_v \,.

The first one already uniquely defines and fixes $A_u$: since the value of $A_0$ at $\sigma =0$ is given, this differential equation has a unique solution along $\sigma \in [\mathrm{0,1}]$.

So it remains to check that this unique solution to the first equation also solves the second

$${\partial }_v{A}_u={\partial }_u{A}_v$$\partial_v A_u = \partial_u A_v

for all $\sigma$. It is true by assumption at $\sigma =0$, so it is sufficient to show that the $\sigma$-derivatives of both sides coincide.

On the left we have

$${\partial }_{\sigma }{\partial }_v{A}_u={\partial }_v{\partial }_{\sigma }{A}_u={\partial }_v{\partial }_u{A}_{\sigma }$$\partial_\sigma \partial_v A_u = \partial_v \partial_\sigma A_u = \partial_v \partial_u A_\sigma

by the above, and similarly on the right

$${\partial }_{\sigma }{\partial }_u{A}_v={\partial }_u{\partial }_{\sigma }{A}_v={\partial }_u{\partial }_v{A}_{\sigma }$$\partial_\sigma \partial_u A_v = \partial_u \partial_\sigma A_v = \partial_u \partial_v A_\sigma

so that indeed this is equal. This constitutes the required lift.

2nd case: $𝔤\mathrm{an}\mathrm{arbitrary}\mathrm{Lie}\mathrm{algebra}$

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

the original element is a collection of 1-forms $A_v^{a}dv+A_{\sigma }^{a}d\sigma$ satisfying

$${\partial }_{\sigma }{A}_v^a={\partial }_v{A}_{\sigma }-C^a{}_{bc}{A}_v^b{A}_{\sigma }^c.$$\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 $A_u^{a}du$ that is uniquely fixed by the condition that it solves the differential equation

$${\partial }_{\sigma }{A}_u={\partial }_u{A}_{\sigma }-C^a{}_{bc}{A}_{\sigma }^b{A}_u^c$$\partial_\sigma A_u = \partial_u A_\sigma - C^a{}_{b c} A_\sigma^b A_u^c

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

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

$$F_{vu}^a:={\partial }_v{A}_u^a-{\partial }_u{A}_v^a+C^a{}_{bc}{A}_v^b{A}_u^c=0.$$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 $\sigma =0$. We now show that the $\sigma$-derivative of this expression satisfies the Binachi-type equation

$${\partial }_{\sigma }{F}_{vu}^a=C^a{}_{bc}{A}_{\sigma }^b{F}_{vu}^c.$$\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 $F_{vu}^a=0$. Since this solution is guaranteed to be unique, we will have shown our claim.

Now compute:

$$\begin{array}{rl}{\partial }_{\sigma }{F}_{uv}^a& :={\partial }_{\sigma }{\partial }_u{A}_v^a-{\partial }_{\sigma }{\partial }_v{A}_u^a+{\partial }_{\sigma }{C}^a{}_{bc}{A}_u^b{A}_v^c\\ & ={\partial }_u{\partial }_{\sigma }{A}_v^a-{\partial }_v{\partial }_{\sigma }{A}_u^a+{\partial }_{\sigma }{C}^a{}_{bc}{A}_u^b\wedge A_v^c\\ & ={\partial }_u({\partial }_v{A}_{\sigma }^a-C^a{}_{bc}{A}_{\sigma }^b{A}_v^c)-{\partial }_v({\partial }_u{A}_{\sigma }^a-C^a{}_{bc}{A}_{\sigma }^b{A}_u^c)+{\partial }_{\sigma }{C}^a{}_{bc}{A}_u^b{A}_v^c\\ & ={\partial }_u{C}^a{}_{bc}{A}_{\sigma }^b{A}_v^c-{\partial }_v{C}^a{}_{bc}{A}_{\sigma }^b{A}_u^c+{\partial }_{\sigma }{C}^a{}_{bc}{A}_u^b{A}_v^c\\ & =C^a{}_{bc}{A}_{\sigma }^b({\partial }_u{A}_v^c-{\partial }_v{A}_u^c)+C^a{}_{bc}({\partial }_{\sigma }{A}_u^b-C^b{}_{de}{A}_u^d{A}_{\sigma }^e)A_v^c-C^a{}_{bc}({\partial }_{\sigma }{A}_{\nu }^b-C^b{}_{de}{A}_v^d{A}_{\sigma }^e)A_u^b+{\partial }_{\sigma }{C}^a{}_{bc}{A}_v^b{A}_u^c\\ & =C^a{}_{bc}{A}_{\sigma }^c{F}_{uv}^b\end{array}.$$\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

$$C^a{}_{bc}{C}^b{}_{de}+C^a{}_{bd}{C}^b{}_{ec}+C^a{}_{be}{C}^b{}_{cd}=0.$$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 $𝔤$ a general $L_{\mathrm{\infty }}$-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 $u$-, $v$-, $\sigma$- direction that matter, all other indices just run along.

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

With $♭\exp (𝔤)\to \exp (𝔤)$ realized as a fibration between fibrant objects, we now obtain the de Rham coefficient object ${♭}_{\mathrm{dR}}\exp (𝔤)$ as an ordinary pullback, as in the above discussions.

Corollary

For $𝔤$ an $L_{\mathrm{\infty }}$-algebra, a representive in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ of the object ${♭}_{\mathrm{dR}}\exp (𝔤)$ is the presheaf

$$(U,[n])\mapsto {\mathrm{Hom}}_{\mathrm{dgAlg}}(\mathrm{CE}(𝔤),{\Omega }^{•\ge 1,•}(U\times {\Delta }_{\mathrm{Diff}}^n)),$$(U,[n]) \mapsto Hom_{dgAlg}( CE(\mathfrak{g}), \Omega^{\bullet\geq 1,\bullet}(U \times \Delta^n_{Diff}) ) \,,

where the notation on the right denotes the dg-algebra of differential forms on $U\times {\Delta }_{\mathrm{Diff}}^n$ that (apart from having setting instants on the faces of ${\Delta }_{\mathrm{Diff}}^n$) are along $U$ of non-vanishing degree.

Compare this to the more explicit examples that we had discussed above.

Corollary

All statements go through verbatim for the $n$-truncation ${\tau }_n\exp (𝔤)$.

(…)

Remark

Observe that $\exp (𝔤)$ is the concretization (in the sense of concrete presheaf) of $G{\mathrm{TrivBund}}_{\mathrm{flat}}\simeq ♭BG$. And ${♭}_{\mathrm{dR}}BG$, being the kernel of the concretization map, is in a sense the maximally non-concrete sub-presheaf of $G{\mathrm{TrivBund}}_{\mathrm{flat}}$.

For line $n$-groups

For the circle n-groupoid $B^{n}U(1)$ we have now obtained two different models for its de Rham coefficient object ${♭}_{\mathrm{dR}}{B}^{n}U(1)$:

1. The image under the Dold-Kan map $\Xi :[{\mathrm{CartSp}}^{\mathrm{op}},{\mathrm{Ch}}_{•}]\to [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ of the complex of sheaves of forms

   $${♭}_{\mathrm{dR}}^{\mathrm{chn}}{B}^nℝ:=\Xi ({\Omega }^1(-)\stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^2(-)\stackrel{d_{\mathrm{dR}}}{\to }\cdots {\Omega }_{\mathrm{cl}}^n(-)).$$\mathbf{\flat}_{dR}^{chn}\mathbf{B}^n \mathbb{R} := \Xi (\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \Omega^n_{cl}(-)) \,.

   (This is discussed here).

2. The $(n+1)$-coskeleton of the simplicial presheaf

   $${♭}_{\mathrm{dR}}^{\mathrm{simp}}{B}^nℝ:U,[k]\mapsto {\tilde{\Omega }}_{\mathrm{cl}}^n{}^n(U\times {\Delta }^k),$$\mathbf{\flat}^{simp}_{dR}\mathbf{B}^n \mathbb{R} : U,[k] \mapsto \tilde \Omega^n_{cl}^n(U \times \Delta^k) \,,

   where the tilde indicates the subset of all those forms with at least one leg along $U$.

   (This is the result of the Lie integration algorithm.)

There is an evident degreewise map

$${\int }_{{\Delta }^k}:{♭}_{\mathrm{dR}}^{\mathrm{simp}}{B}^nℝ\to {♭}_{\mathrm{dR}}^{\mathrm{chn}}{B}^nℝ$$\int_{\Delta^k} : \mathbf{\flat}_{dR}^{simp} \mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{dR}^{chn} \mathbf{B}^n \mathbb{R}

that sends a closed $n$-form $\omega \in {\Omega }_{\mathrm{cl}}^n(U\times {\Delta }^n)$ to its fiber integration ${\int }_{{\Delta }^k}\omega$.

Proposition

This map yields a morphism of simplicial presheaves

$$\int :{♭}_{\mathrm{dR}}^{\mathrm{simp}}{B}^nℝ\to {♭}_{\mathrm{dR}}^{\mathrm{chn}}{B}^nℝ$$\int : \mathbf{\flat}_{dR}^{simp} \mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{dR}^{chn} \mathbf{B}^n \mathbb{R}

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

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 $\omega \in {\Omega }_{\mathrm{cl}}^n(U\times {\Delta }^k)$ 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 $\omega \mapsto {\int }_{\partial {\Delta }^k}\omega$.

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

$$\begin{array}{rl}\omega \mapsto d_U{\int }_{{\Delta }^k}\omega & ={\int }_{{\Delta }^k}{d}_U\omega \\ & =-{\int }_{{\Delta }^k}{d}_{{\Delta }^k}\omega \\ & =-{\int }_{\partial {\Delta }^k}\omega \end{array}$$\begin{aligned} \omega \mapsto 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}

using that $\omega$ is closed, so that $d_{\mathrm{dR}}\omega =(d_U+d_{{\Delta }^k})\omega =0$.

Therefore we have indeed objectwise a chain map.

To see that it gives a weak equivalence we need to check that this chain map is a quasi-isomorphism.

From the construction of both objects we know that they have the same cohomology, and that is concentrated in degree $n$, where it is ${\Omega }_{\mathrm{cl}}^1(U)$. Therefore it is in fact sufficient to check that the integration map is onto in degree $n$.

That amounts to observing that every 1-form $\alpha \in {\Omega }^1(U)$ may be obtained by integration of a closed $n$-form on $U\times {\Delta }^k$. This is clearly the case, for instance take $\frac{1}{c}\alpha \wedge d{x}^1\wedge \cdots \wedge d{x}^k-\frac{1}{c}(d_U\alpha )\wedge x^1\wedge d{x}^2\wedge \cdots \wedge d{x}^k$.

$\mathrm{\infty }$-Chern-Weil homomorphism

… under construction …

..

The ordinary Chern-Weil homomorphism constructs curvature characteristic forms for $G$-principal bundles from invariant polynomials of Lie algebras. The notion of invariant polynomial generalizes straightforwardly from Lie algebras to ∞-Lie algebras. We discuss a generalization of the Chern-Weil homomorphism for $\mathrm{\infty }$-Lie groups in the image of the Lie integration map applied to an $\mathrm{\infty }$-Lie algebra $𝔤$.

We discuss how to construct for each $\mathrm{\infty }$-Lie algebra $𝔤$ with Lie integration $BG:={\tau }_n\exp (𝔤)$ from each ∞-Lie n-cocycle $\mu :𝔤\to b^{n-1}ℝ$ in transgression with an invariant polynomial $P$ a morphism

$$P:BG\to {♭}_{\mathrm{dR}}{B}^nℝ/K$$P : \mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}/K

that represents a class in the intrinsic de Rham cohomology

$$P\in H_{\mathrm{dR}}(BG,B^nℝ).$$P \in \mathbf{H}_{dR}(\mathbf{B}G, \mathbf{B}^n \mathbb{R}) \,.

Definition

(differential resolution)

For $𝔤$ an ∞-Lie algebra with Lie integration $BG:={\tau }_n\exp (𝔤)\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$, write $B_{\mathrm{diff}}G\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ for the $n$-truncation of the simplicial presheaf given by

$$U,n\mapsto \left\{\begin{array}{ccc}{C}^{\mathrm{\infty }}(U)\otimes \prime {\Omega }^{•}({\Delta }_{\mathrm{diff}}^n)& \leftarrow & \mathrm{CE}(𝔤)\\ \uparrow & & \uparrow \\ {\Omega }^{•}(U\times {\Delta }_{\mathrm{Diff}}^n)& \leftarrow & W(𝔤)\end{array}\right\}.$$U,n \mapsto \left\{ \array{ C^\infty(U) \otimes' \Omega^\bullet(\Delta^n_{diff}) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^n_{Diff}) &\leftarrow& W(\mathfrak{g}) } \right\} \,.

Proposition

The evident projection

$$BG\stackrel{\simeq }{\leftarrow }{B}_{\mathrm{diff}}G$$\mathbf{B}G \stackrel{\simeq}{\leftarrow} \mathbf{B}_{diff} G

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

Proof

The projection without the truncation is a weak equivalence by the freeness of the Weil algebra $W(𝔤)$: morphisms of dg-algebras ${\Omega }^{•}(U\times {\Delta }^n)\leftarrow W(𝔤)$ are fixed by and uniquely corespond to their underlying morphisms of graded vector spaces ${\Omega }^{(}U\times {\Delta }^n)\leftarrow {\wedge }^1{𝔤}^{*}$. This implies that every diagram

$$\begin{array}{ccc}\partial \Delta [n]& \to & B_{\mathrm{diff}}G\\ \downarrow & & \downarrow \\ \Delta [n]& \to & BG\end{array}$$\array{ \partial\Delta[n] &\to& \mathbf{B}_{diff}G \\ \downarrow && \downarrow \\ \Delta[n] &\to& \mathbf{B}G }

has a lift over each $U\in \mathrm{CartSp}$, hence that the morphism on the right is over each $U$ an aacyclic Kan fibration.

Let $\mu :𝔤\to b^{n-1}ℝ$ be an ∞-Lie algebra cocycle which is in transgression with an invariant polynomial $P$, where the transgression is induced from a Chern-Simons element ${\mathrm{cs}}_P$. This data is a diagram

$$\begin{array}{ccccc}𝔤& \stackrel{\mu }{\to }& b^{n-1}ℝ& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ \mathrm{inn}(𝔤)& \stackrel{({\mathrm{cs}}_P,P)}{\to }& \mathrm{inn}(b^{n-1}ℝ)& \stackrel{P}{\to }& b^nℝ\end{array}$$\array{ \mathfrak{g} &\stackrel{\mu}{\to}& b^{n-1}\mathbb{R} &\to& * \\ \downarrow && \downarrow && \downarrow \\ inn(\mathfrak{g}) &\stackrel{(cs_P, P)}{\to}& inn(b^{n-1}\mathbb{R}) &\stackrel{P}{\to}& b^{n}\mathbb{R} }

of $\mathrm{\infty }$-Lie algebras, or dually a dg-algebra diagram

$$\begin{array}{ccccc}\mathrm{CE}(𝔤)& \stackrel{\mu }{\leftarrow }& \mathrm{CE}(b^{n-1}ℝ)& \leftarrow & 0\\ \uparrow & & \uparrow & & \uparrow \\ W(𝔤)& \stackrel{({\mathrm{cs}}_P,P)}{\leftarrow }& W(b^{n-1}ℝ)& \leftarrow & \mathrm{CE}(b^nℝ)\end{array}.$$\array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R}) &\leftarrow& 0 \\ \uparrow && \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs_P,P)}{\leftarrow}& W(b^{n-1}\mathbb{R}) &\leftarrow& CE(b^n \mathbb{R}) } \,.

This integrates to a morphism

$${\hat{B}}_{\mathrm{diff}}G\stackrel{\int \mu }{\to }{B}_{\mathrm{diff}}^nℝ/K{♭}_{\mathrm{dR}}{B}^{n+1}/K$$\hat \mathbf{B}_{diff}G \stackrel{\int \mu}{\to} \mathbf{B}_{diff}^{n} \mathbb{R}/K \mathbf{\flat}_{dR} \mathbf{B}^{n+1}/K

where $\hat{B}G\stackrel{\simeq }{\leftarrow }{B}_{\mathrm{diff}}G$ and $B^nℝ/K\stackrel{\simeq }{\leftarrow }{B}_{\mathrm{diff}}^nℝ/K$ are the resolutions.

The following proposition asserts that this definition does indeed capture the ordinary Chern-Weil homomorphism.

Let $X$ be a paracompact smooth manifold, $G$ a Lie group and $P\to X$ a $G$-principal bundle classified by a morphisms $X\to BG$ in $\mathrm{\infty }\mathrm{LieGrpd}$, hence a Cech cocycle given by a span

$$X\stackrel{\simeq }{\leftarrow }Y\to BG$$X \stackrel{\simeq}{\leftarrow} Y \to \mathbf{B}G

in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$.

Let $⟨\cdots ⟩$ be an $n$-ary invariant polynomial on $𝔤$

Proposition

The composite

$$X\to BG\to {♭}_{\mathrm{dR}}{B}^{2n}U(1)$$X \to \mathbf{B}G \to \mathbf{\flat}_{dR}\mathbf{B}^{2n} U(1)

in $\mathrm{\infty }\mathrm{LieGrpd}$, i.e. the morphism given by a zig zag

$$X\stackrel{\simeq }{\leftarrow }Y\stackrel{}{\to }BG\stackrel{\simeq }{\leftarrow }{B}_{\mathrm{diff}}G\to {♭}_{\mathrm{dR}}{B}^{2n}U(1)$$X \stackrel{\simeq}{\leftarrow} Y \stackrel{}{\to} \mathbf{B}G \stackrel{\simeq}{\leftarrow} \mathbf{B}_{diff}G \to \mathbf{\flat}_{dR}\mathbf{B}^{2n} U(1)

in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$ represents the de Rham cohomology class of the curvature characteristic form

$$[⟨F_{\nabla }\wedge \cdots \wedge F_{\nabla }⟩]\in H^{2n}(X)$$[\langle F_\nabla \wedge \cdots \wedge F_\nabla\rangle] \in H^{2n}(X)

of any connection $\nabla$ on $P$.

Proof

… sketch, am being interrupted …

By possibly refining the cover $Y$, we may lift the given cocycle $BG$-cocycle to a $B_{\mathrm{diff}}G$ cocycle which on patches $U_i$ assigns the local curvature forms $A_i$ of $\nabla$.

To a double intersection this cocycle assigns a based path ${\gamma }_{ij}$ in $G$ with endpoint $g_{ij}$. By the discussion at Chern-Simons form we find that the corresponding image of $⟨...⟩$ is the Chern-Simons form $\mathrm{CS}(A_i,A_j)$ for this path of gauge transformations. Since $A_i$ and $A_j$ are components of a genuine connection (which always exsist), this form is closed.

Similarly for higher paths. It follows that the cocycle in ${♭}_{\mathrm{dR}}{B}^{2n}U(1)=({\Omega }^1(-)\stackrel{d_{\mathrm{dR}}}{\to }\cdots \stackrel{d_{\mathrm{dR}}}{\to }{\Omega }_{\mathrm{closed}}^{2n}(-))$ that we obtain looks like

$$(⟨F_{A_i}\wedge \cdots \wedge F_{A_i}⟩,\mathrm{CS}(A_i,A_j),{\lambda }_{ijk},\cdots ).$$(\langle F_{A_i} \wedge \cdots \wedge F_{A_i}\rangle, CS(A_i,A_j), \lambda_{i j k}, \cdots) \,.

Using a partition of unity this is coboundant to a cocycle of the form

$$(⟨F_{A_i}\wedge \cdots \wedge F_{A_i}⟩+d\mathrm{something}\mathrm{,0},0,\cdots ).$$(\langle F_{A_i} \wedge \cdots \wedge F_{A_i}\rangle + d something,0, 0, \cdots) \,.

This represents a globally defined form which differs from $⟨F_{\nabla }\wedge \cdots F_{\nabla }⟩$ by an exact form.

Simplicial manifolds / simplicial diffeological space

Every smooth simplicial manifold, and more generally every simplicial object in diffeological spaces, naturally represents a simplicial presheaf $X_{•}\in [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$, and as such naturally represents an $\mathrm{\infty }$-Lie groupoid.

Simplicial de Rham complex

For $X_{•}$ a simplicial manifold, there are two main models for the simplicial de Rham complex of $X$.

1. the total complex of the double complex ${\Omega }^{•}(X_{•})$;

2. The complex whose elements in degree $n$ are collections

   $$\{{\omega }_p\in {\Omega }^n(X_p\times {\Delta }^p){\}}_{p=0}^{\mathrm{\infty }}$$\{\omega_p \in \Omega^n(X_p \times \Delta^p)\}_{p = 0}^\infty

   subject to the conditions

   $$(\mathrm{id}\times ({\Delta }^{p-1}\stackrel{{\sigma }_i}{\to }{\Delta }^p){)}^{*}{\omega }_p=((X_p\stackrel{{\delta }_i}{\to }{X}_{p-1})\times \mathrm{id}{)}^{*}{\omega }_{p-1}$$(id \times (\Delta^{p-1} \stackrel{\sigma_i}{\to} \Delta^p))^* \omega_p = ((X_p \stackrel{\delta_i}{\to} X_{p-1}) \times id)^* \omega_{p-1}

   for all $p,i$, and whose differential is degreewise the ordinary de Rham differential.

There is a quasi-isomorphism from the latter to the former, given by the fiber integration of forms over simplices.

Above we had obtained two different simplicial presheaves representing the intrinsic de Rham coefficient object ${♭}_{\mathrm{dR}}{B}^nℝ$, which we denoted ${♭}_{\mathrm{dR}}^{\mathrm{simp}}{B}^nℝ$ and ${♭}_{\mathrm{dR}}^{\mathrm{chn}}{B}^nℝ$, and a weak equivalence

$$\int :{♭}_{\mathrm{dR}}^{\mathrm{simp}}{B}^nℝ\to {♭}_{\mathrm{dR}}^{\mathrm{chn}}{B}^nℝ.$$\int \; : \; \mathbf{\flat}_{dR}^{simp}\mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{dR}^{chn}\mathbf{B}^n \mathbb{R} \,.

We want to claim now that

1. the cocycle $\mathrm{\infty }$-groupoids of the two simplicial de Rham complexes are the hom-complexes

   $$[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}](X_{•},{♭}_{\mathrm{dR}}^{\mathrm{simp}}{B}^nℝ)$$[CartSp^{op}, sSet](X_\bullet, \mathbf{\flat}_{dR}^{simp}\mathbf{B}^n \mathbb{R})

   and

   $$[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}](X_{•},{♭}_{\mathrm{dR}}^{\mathrm{chn}}{B}^nℝ),$$[CartSp^{op}, sSet](X_\bullet, \mathbf{\flat}_{dR}^{chn}\mathbf{B}^n \mathbb{R}) \,,

   respectively;

2. the quasi-isomorphism relating them is the morphism induced by the weak equivalence of these coeffient objects

   $$[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}](X_{•},\int ):[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}](X_{•},{♭}_{\mathrm{dR}}^{\mathrm{simp}},B^nℝ)\to [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}](X_{•},{♭}_{\mathrm{dR}}^{\mathrm{chn}},B^nℝ).$$[CartSp^{op}, sSet](X_\bullet, \int ) : [CartSp^{op}, sSet](X_\bullet, \mathbf{\flat}_{dR}^{simp}, \mathbf{B}^n \mathbb{R}) \to [CartSp^{op}, sSet](X_\bullet, \mathbf{\flat}_{dR}^{chn}, \mathbf{B}^n \mathbb{R}) \,.

To see this, notice for instance for the second version of the simplicial de Rham complex that its $n$-cocycles $\{{\omega }_p\in {\Omega }_{\mathrm{cl}}^n(X_p\times {\Delta }^p)\}$, are diagrams of sheaves on $\mathrm{CartSp}$ of the form

$$\begin{array}{ccc}⋮& & ⋮\\ X_2& \stackrel{{\omega }_2}{\to }& {\Omega }_{\mathrm{cl}}^n(-\times {\Delta }^2)\\ \downarrow \downarrow \downarrow & & \downarrow \downarrow \downarrow \\ X_1& \stackrel{{\omega }_1}{\to }& {\Omega }_{\mathrm{cl}}^n(-\times {\Delta }^1)\\ \downarrow \downarrow & & \downarrow \downarrow \\ X_0& \stackrel{{\omega }_0}{\to }& {\Omega }_{\mathrm{cl}}^n(-)\end{array}.$$\array{ \vdots && \vdots \\ X_2 &\stackrel{\omega_2}{\to}& \Omega^n_{cl}(- \times \Delta^2) \\ \downarrow \downarrow \downarrow && \downarrow \downarrow \downarrow \\ X_1 &\stackrel{\omega_1}{\to}& \Omega^n_{cl}(- \times \Delta^1) \\ \downarrow \downarrow && \downarrow \downarrow \\ X_0 &\stackrel{\omega_0}{\to}& \Omega^n_{cl}(-) } \,.

But these are exactly the coend diagrams that encode the morphisms of simplicial presheaves $\omega :X_{•}\to {♭}_{\mathrm{dR}}^{\mathrm{simp}}{B}^nℝ$.

(…)

Simplicial Lie and diffeological groups

Every connected object $X\in \mathrm{\infty }\mathrm{Lie}\mathrm{Grpd}$ is – by definition – the delooping $X=BG$ of a Lie ∞-group $G=\Omega X$, its loop space object formed in $\mathrm{\infty }\mathrm{LieGrpd}$. 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 $(\mathrm{\infty }\mathrm{,1})$-limits are computed objectwise, an ∞-group object $G$ in $\mathrm{\infty }\mathrm{LieGrpd}$ 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 $\mathrm{\infty }$-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.

Delooping

For the moment see simplicial group - delooping.

Simplicial principal bundles

For the moment see the discussion about geometric realization further above.

Cohomology

As every (∞,1)-topos, $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ comes with its intrinsic cohomology.

With constant coefficients

We discus the intrinsic cohomology of $H$ with constant coefficients, i.e. coefficients that are constant (∞,1)-sheaves on CartSp.

Theorem

Let $X$ be a paracompact smooth manifold and $A\in$ ∞Grpd. Then we have an equivalence of cocycle ∞-groupoids

$${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})(X,\mathrm{LConst}A)\simeq \mathrm{Top}(X,\mid A\mid )$$Sh_{(\infty,1)}(CartSp)(X, LConst A) \simeq Top(X, |A|)

and hence in particular an isomorphism on cohomology

$$H(X,A)\simeq {\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})(X,\mathrm{LConst}A)$$H(X,A) \simeq \pi_0 Sh_{(\infty,1)}(CartSp)(X, LConst A)

Proof

The key point is that for paracompact $X$, the nerve theorem asserts that $\Pi (X)$ is weak homotopy equivalent to $\mathrm{Sing}X$, the standard fundamental ∞-groupoid of $X$. This is discussed in detail in the section geometric realization at path ∞-groupoid.

Using this, the statement follows by the (∞,1)-adjunction $(\Pi ⊣\mathrm{LConst})$, that is discussed in detail at Unstructured homotopy ∞-groupoid.

Cohomology of Lie groups

We consider the cohomology in $H$ of smooth delooping groupoids $BG$ for $G$ an ordinary Lie group. This is a form of group cohomology for Lie groups.

$$H^n(G,A):={\pi }_{0}H(BG,B^{n}A).$$H^n(G,A) := \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A) \,.

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

Definition

(naive Lie group cohomology)

For $G$ a Lie group and $A$ an abelian Lie group and $n\in \mathbb{N}$ define $H_{\mathrm{rest}}^n(G,A)$ to be the group of equivalence classes of cocycles given by smooth functions $G_n^{\times }\to A$ by coboundatries given by smooth functions $G^{{\times }_{n-1}}\to A$ subject to the usual relations.

Observe that with $BG=G^{{\times }_{•}}$ regarded as an object of $\mathrm{sPSh}(\mathrm{CartSp})$, this is

$$H_{\mathrm{rest}}^n(G,A)={\pi }_0\mathrm{sPSh}(BG,B^{n}A).$$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 $BG$.

A more refined definition of cohomology of Lie groups has been given by Segal, which was later rediscovered by Jean-Luc Brylinski, following Blanc.

See section 4 of

• Chris Schommer-Pries, A finite-dimensional String 2-group (arXiv:0911.2483)

for a review and applications.

Definition

(differential Lie group cohomology)

Let $G$ be a paracompact Lie group and $A$ an abelian Lie group.

For eack $k\in \mathbb{N}$ we can pick a good open cover $\{U_i^k\to G^{{\times }_k}\mid i\in I_k\}$ such that

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

• and for $d_j(U_i^k)$ and $s_j(U_i^k)$ the images of the face and degeneracy maps of $G^{\times •}$ we have

  $$d_j(U_i^k)\subset U_{d_j(i)}^{k-1}$$d_j(U^k_i) \subset U^{k-1}_{d_j(i)}

  and

  $$s_j(U_i^k)\subset U_{s_j(i)}^{k+1}.$$s_j(U^k_i) \subset U^{k+1}_{s_j(i)} \,.

For instance start with a good open cover $\{U_i^1\to G\}$ and define a good open cover $\{U_{i_0{i}_1{i}_2}^2\}$ of $G\times G$ by $U_{i_0{i}_1{i}_2}^2:=d_0^{*}{U}_{i_0}^1\cap d_1^{*}{U}_{i_1}^1\cap d_2^{*}{U}_{i_2}^1$. And so on.

Then the differentiable Lie group cohomology $H_{\mathrm{diffr}}^{•}(G,A)$ of $G$ with coefficients in $A$ is the cohomology of the total complex of the Cech double complex $C^{\mathrm{\infty }}(U_{i_0,\cdots ,i_{•}}^{•},A)$ whose differentials are the alternating sums of the face maps of $G^{{\times }_{•}}$ and of the Cech nerves, respectively:

$$H_{\mathrm{diff}}^n(G,A):=H^n\mathrm{Tot}{C}^{\mathrm{\infty }}(U_{i_0,\cdots ,i_{•}}^{•},A)$$H^n_{diff}(G,A) := H^n Tot C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)

This is definition 1.1 in

• Jean-Luc Brylinski, Differentiable cohomology of gauge groups (pdf).

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

Observation

There is a natural map

$$H_{\mathrm{restr}}^n(G,A)\to H_{\mathrm{diffr}}^n(G,A)$$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 $BG$ in $\mathrm{sPSh}(\mathrm{CartSp}{)}_{\mathrm{proj},\mathrm{cov}}$:

Proposition

For $\{U_{i_{•}}^{•}\}$ a system of good open covers as above, we obtain a simplicial diagram of Cech nerves

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

which is degreewise a cofibrant resolution on $\mathrm{sPSh}(\mathrm{CartSp}{)}_{\mathrm{proj},\mathrm{cov}}$ of $G^{{\times }_n}$. Its totalization coend is connected by a zig-zag of weak equivalences in $\mathrm{sPSh}(\mathrm{CartSp}{)}_{\mathrm{proj},\mathrm{cov}}$ to $BG$

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

and we have

$$H_{\mathrm{diffr}}^n(G,A)={\pi }_0\mathrm{sPSh}(\mathrm{CartSp})({\int }^{[k]}\Delta [k]\cdot C(\{U_i^k\}),B^{n}A).$$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 $H^{•}(G,A):={\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})(BG,B^{n}A)$ for the intrinsic cohomology of $BG$ regarded as an object of the $(\mathrm{\infty }\mathrm{,1})$-topos of $\mathrm{\infty }$-Lie groupoids.

There is a natural morphism

$$H_{\mathrm{diffr}}^n(G,A)\to H^n(G,A).$$H^n_{diffr}(G,A) \to H^n(G,A) \,.

Proof

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

A cofibrant replacement of any paracompact manifold $X$ in $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{cov}}$ is given by the Cech nerve $C(\{U_i\})\stackrel{\simeq }{\to }X$ of a good open cover $\{U_i\to X\}$, 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 $\{U_i^k\to G^{\times k}\}$ as in the definition of differentiable group cohomology above, we obtain cofibrant replacements

$$C(\{U_i^k\})\stackrel{\simeq }{\to }{G}^{{\times }_k}$$