structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
∞-Lie theory (higher geometry)
This is a sub-entry of Smooth∞Grpd. See there for background.
We discuss the general abstract structures in a cohesive (∞,1)-topos realized in Smooth∞Grpd.
We discuss the intrinsic fundamental ∞-groupoid construction realized in $Smooth \infty Grpd$.
If $X \in Smooth\infty Grpd$ is presented by $X_\bullet \in SmoothMfd^{\Delta^{op}} \hookrightarrow [CartSp_{smooth}^{op}, sSet]$, then its image $i_!(X) \in$ ETop∞Grpd under the relative topological cohesion morphism is presented by the underlying simplicial topological space $X_\bullet \in TopMfd^{\Delta^{op}} \hookrightarrow [CartSp_{top}^{op}, sSet]$.
Let first $X \in SmoothMfd \hookrightarrow SmoothMfd^{\Delta^{op}}$ be simplicially constant. Then there is a differentiably good open cover $\{U_i \to X\}$ such that the Cech nerve projection
is a cofibrant resolution in $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ which is degreewise a coproduct of representables. That means that the left derived functor $\mathbb{L} Lan_i$ on $X$ is computed by the application of $Lan_i$ on this coend, which by the fact that this is defined to be the left Kan extension along $i$ is given degreewise by $i$, and since $i$ preserves pullbacks along covers, this is
The last step follows from observing that we have manifestly the Cech nerve as before, but now of the underlying topological spaces of the $\{U_i\}$ and of $X$.
The claim then follows for general simplicial spaces by observing that $X_\bullet = \int^{[k] \in \Delta} \Delta[k] \cdot X_k \in [CartSp_{smooth}^{op}, sSet]_{proj,loc}$ presents the (∞,1)-colimit over $X_\bullet : \Delta^{op} \to SmoothMfd \hookrightarrow Smooth \infty Grpd$ and the left adjoint (∞,1)-functor $i_!$ preserves these (∞,1)-colimits.
If $X \in Smooth\infty Grpd$ is presented by $X_\bullet \in SmoothMfd^{\Delta^{op}} \hookrightarrow [CartSp_{smooth}^{op}, sSet]$, then the image of $X$ under the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor
is equivalent to the geometric realization of (a Reedy cofibrant replacement of) the underlying simplicial topological space
In particular if $X$ is an ordinary smooth manifold then
is equivalent to the standard fundamental ∞-groupoid of $X$.
By a proposition above the functor $\Pi$ factors as $\Pi X \simeq \Pi_{ETop} i_! X$. By the above proposition this is $\Pi_{Etop}$ applied to the underlying simplicial topological space. The claim then follows with the corresponding proposition discussed at ETop∞Grpd.
The $\Pi : \mathrm{Smooth}\infty\mathrm{Grpd} \to \infty \mathrm{Grpd}$ preserves homotopy fibers of morphisms that are presented in $[\mathrm{CartSp}_{\mathrm{smooth}}^{\mathrm{op}}, \mathrm{sSet}]_{\mathrm{proj}}$ by morphisms of the form $X \to \bar W G$ with $X$ fibrant and $G$ a simplicial group in SmoothMfd.
By the above the functor factors as
and $i_!$ assigns the underlying topological spaces. If we can show that this preserves the homotopy fibers in question, then the claim follows with the analogous discussion at ETop∞Grpd.
We find this as in the proof of the latter proposition, by considering the pasting diagram of pullbacks of simplicial presheaves
Since the component maps of the right vertical morphisms are surjective, the degreewise pullbacks in $\mathrm{SmoothMfd}$ that define $P'$ are all along transversal maps, and thus the underlying objects in TopMfd are the pullbacks of the underlying topological manifolds. Therefore the degreewise forgetful functor $\mathrm{SmoothMfd} \to \mathrm{TopMfd}$ presents $i_!$ on the outer diagram and sends this homotopy pullback to a homotopy pullback.
We discuss the notion of geometric path ∞-groupoids realized in $Smooth\infty Grpd$.
For $X \in$ SmthMfd write
for the simplicial presheaf that sends a test space $U \in$ CartSp to the singular simplicial complex of smooth simplices smoothly parameterized over $U$:
The object $\mathbf{Sing} X \in [CartSp^{op}, sSet]$ presents the abstractly defined fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos $\mathbf{\Pi}(X)$.
Using the Steenrod-Wockel approximation theorem this comes down to the same argument as for Euclidean topological ∞-groupoids. See Presentation of the path ∞-groupoid there.
This section is at
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We discuss cohesive ∞-groups in $Smooth \infty Grpd$: smooth $\infty$-groups.
Let $G$ be a Lie group. Under the embedding $SmoothMfd \hookrightarrow Smooth \infty Grpd$ this is canonically identifed as a 0-truncated ∞-group object in $Smooth \infty Grpd$. Write $\mathbf{B}G \in Smooth \infty Grpd$ for the corresponding delooping object.
A fibrant presentation of the delooping object $\mathbf{B}G$ in the projective local model structure on simplicial presheaves $[CartSp^{op}_{smooth}, sSet]_{proj, loc}$ is given by the simplicial presheaf that is the nerve of the one-object Lie groupoid
regarded as a simplicial manifold and canonically embedded into simplicial presheaves:
The presheaf is clearly objectwise a Kan complex, being objectwise the nerve of a groupoid. It satisfies descent along good open covers $\{U_i \to \mathbb{R}^n\}$ of Cartesian spaces, because the descent $\infty$-groupoid $[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}G)$ is $\cdots \simeq G Bund(\mathbb{R}^n) \simeq G TrivBund(\mathbb{R}^n)$: an object is a Cech 1-cocycle with coefficients in $G$, a morphism a Cech coboundary, which yields the groupoid of $G$-principal bundles over $U$, which for the Cartesian space $U$ is however equivalent to the groupoid of trivial $G$-bundles over $U$.
To show that $\mathbf{B}G$ is indeed the delooping object of $G$ it is sufficient (by the discussion at model structure on simplicial presheaves – homotopy limits) to compute the (∞,1)-pullback $G \simeq * \times_{\mathbf{B}G} *$ in the global model structure $[CartSp^{op}, sSet]_{proj}$.
This is accomplished by the ordinary pullback of the fibrant replacement diagram
as discussed at universal principal ∞-bundle.
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 $\alpha : G_1 \to Aut(G_2)$.
A fibrant representative of $\mathbf{B}G$ in $[CartSp^{op}, sSet]_{proj,cov}$ is given by the crossed complex
As above for Lie groups.
Write equivalently
for the abelian Lie group called the circle group or 1-dimensional unitary group , regarded as a 0-truncated ∞-group object in $Smooth\infty Grpd$ under the embedding $SmoothMfd \hookrightarrow Smooth \infty Grpd$.
For $n \in \mathbb{N}$ the $n$-fold delooping $\mathbf{B}^n U(1) \in Smooth \infty Grpd$ we call the circle Lie (n+1)-group.
Write
for the chain complex of sheaves concentrated in degree $n$ on $U(1)$.
Recall the right Quillen functor $\Xi : [CartSp^{op}, Ch_\bullet^+] \to [CartSp^{op}, sSet]$ from above.
The simplicial presheaf $\Xi(U(1)[n]) \in [CartSp^{op}, sSet]$ is a fibrant representative in $[CartSp^{op},sSet]_{proj,loc}$ of the circle Lie $(n+1)$-group $\mathbf{B}^n U(1)$.
First notice that since $U(1)[n]$ is fibrant in $[CartSp_{smooth}^{op}, Ch_\bullet]_{proj}$ we have that $\Xi U(1)[n]$ is fibrant in $[CartSp^{op}, sSet]_{proj}$. We may compute the homotopy pullback that defines the loop space object in the global model structure (by the discussion at model structure on simplicial presheaves – homotopy (co)limits) we may check the statement about the delooping in the global model structure.
Consider the global fibration resolution of the point inclusion $* \to \Xi(U(1)[n])$ given by
The underlying morphism of chain complexes is clearly degreewise surjective, hence a projective fibration, hence its image under $\Xi$ is a projective fibration. Therefore the homotopy pullback in question is given by the ordinary pullback
computed in $[CartSp^{op}, Ch_\bullet]$ and then using that $\Xi$ is the right adjoint and hence preserves pullbacks. This shows that the loop object $\Omega \Xi(U(1)[n])$ is indeed presented by $\Xi (U(1)[n-1])$.
Now we discuss the fibrancy of $U(1)[n]$ in the local model structure $[CartSp^{op}, sSet]_{proj,loc}$. We need to check that for all good open covers $\{U_i \to U\}$ of a Cartesian space $U$ we have that the mophism
is an equivalence of Kan complexes, where $C(\{U_i\})$ is the Cech nerve of the cover. Observe that the Kan complex on the right is that whose vertices are cocycles in degree-$n$ Cech cohomology (see there for details) with coefficients in $U(1)$ and whose morphisms are coboundaries between these.
We proceed by induction on $n$. For $n = 0$ the condition is just that $C^\infty(-,U(1))$ is a sheaf, which clearly it is. For general $n$ we use that since $C(\{U_i\})$ is cofibrant, the above is the derived hom-space functor which commutes with homotopy pullbacks and hence with forming loop space objects, so that
by the above result on delooping. So we find that for all $0 \leq k \leq n$ that $\pi_k [CartSp^{op}, sSet](C(\{U_i\}), \Xi(U(1)[n]))$ is the Cech cohomology of $U$ with coefficients in $U(1)$ in degree $n-k$. By standard facts about Cech cohomology (using the short exact sequence of abelian groups $\mathbb{Z} \to U(1)\to \mathbb{R}$ and the fact that the cohomology with coefficients in $\mathbb{R}$ vanishes in positive degree, for instance by a partition of unity argument) we have that this is given by the integral cohomology groups
for $n \geq 1$. For the contractible Cartesian space all these cohomology groups vanish.
So we have that $\Xi(U(1)[n])(U)$ and $[CartSp^{op}, sSet](C(\{U_i\}), \Xi U(1)[n])$ both have homotopy groups concentrated in degree $n$ on $U(1)$. The above looping argument together with the fact that $U(1)$ is a sheaf also shows that the morphism in question is an isomorphism on this degree-$n$ homotopy group, hence is indeed a weak homotopy equivalence.
In the equivalent presentation of $Smooth\infty Grpd$ by simplicial presheaves on all of SmoothMfd the objects $\Xi U(1)[n]$ are far from being locally fibrant. Instead, their local fibrant replacements are given by the $n$-stacks of circle n-bundles.
Every connected object $X \in \infty Lie Grpd$ is – by definition – the delooping $X = \mathbf{B}G$ of a Lie ∞-group $G = \Omega X$, its loop space object formed in $\infty LieGrpd$. Since the discussion of group objects, loop space objects etc. involves only finite (∞,1)-limits and ∞-stackification preserves these, this may be discussed in the (∞,1)-category of (∞,1)-presheaves on CartSp. Since there $(\infty,1)$-limits are computed objectwise, an ∞-group object $G$ in $\infty LieGrpd$ is modeled by a (∞,1)-presheaf with values in ∞-groups in ∞Grpd.
By standard results on Models for group objects in ∞Grpd the latter may equivalently be modeled by simplicial groups. A simplicial group is possibly weak ∞-groupoid equipped with a strict group object structure. While strict ∞-groupoids with weak group object structure do not model all ∞-groups, weak $\infty$-groupoids with strict group structure do.
There is a good supply of standard results for and constructions with simplicial groups which makes this model useful for applications.
We discuss the intrinsic cohomology and pricipal ∞-bundles in $Smooth \infty Grpd$.
Let $A \in$ ∞Grpd be any discrete ∞-groupoid. Write $|A| \in$ Top for its geometric realization. For $X$ any topological space, the nonabelian cohomology of $X$ with coefficients in $A$ is the set of homotopy classes of maps $X \to |A|$
We say $Top(X,|A|)$ itself is the cocycle ∞-groupoid for $A$-valued nonabelian cohomology on $X$.
Similarly, for $X, \mathbf{A} \in Smooth \infty Grpd$ two smooth $\infty$-groupoids, write
for the intrinsic cohomology of $Smooth \infty Grpd$ on $X$ with coefficients in $\mathbf{A}$.
Let $A \in$ ∞Grpd, write $Disc A \in Smooth \infty Grpd$ for the corresponding discrete smooth ∞-groupoid. Let $X \in SmoothMfd \stackrel{i}{\hookrightarrow} Smooth \infty Grpd$ be a paracompact topological space regarded as a 0-truncated Euclidean-topological $\infty$-groupoid.
We have an isomorphism of cohomology sets
and in fact an equivalence of cocycle ∞-groupoids
More generally, for $X_\bullet \in SmoothMfd^{\Delta^{op}}$ presenting an object in $Smooth \infty Grpd$ we have
This follows from the $(\Pi \dashv Disc)$-adjunction and the above proposition asserting that $|\Pi(X_\bullet)| \simeq |X_\bullet|$ is the ordinary geometric realization of simplicial topological spaces.
Let $G$ be a Lie group, regarded as a group object in $Smooth \infty Grpd$ as in the above discussion.
The universal G-principal bundle is a replacement of the point inclusion $* \to \mathbf{B}G$ by a fibration $\mathbf{E}G \to \mathbf{B}G$.
For $G$ an ordinary group one model for this is given by the Lie groupoid
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 $G \to \mathbf{E}G$: it is an example of a groupal model for universal principal ∞-bundles.
To emphasize this group structure, we also write $INN(G)$ for this groupoid, following SchrRob. The corresponding crossed module is
Accordingly we write $\mathbf{B E}G$ or $\mathbf{B}INN(G)$ for the 2-groupoid given by the Lie crossed complex
The following proposition asserts that the general definition of principal ∞-bundles in an (∞,1)-topos $\mathbf{H}$ applied to the coefficient object $\mathbf{B}G$ in $\mathbf{H} = \infty LieGrpd$ for $G$ a Lie group does reprpduce the ordinary notion of $G$-principal bundles.
Let $X$ be a paracompact smooth manifold. The ordinary first nonabelian cohomology of $X$ with coefficients in $G$ coincided with the intrinsic cohomology of $\infty Lie Grpd$
and the $G$-principal bundle $P \to X$ corresponding to a cocycle $X \to \mathbf{B}G$ in $\infty LieGrpd$ is indeed the homotopy fiber of that cocycle.
By the discussion at model structure on simplicial presheaves we have that a cofibrant resolution for $X$ in the model $[CartSp^{op}, sSet]_{proj,cov}$ for $\infty LieGrpd$ is civen by the Cech nerve $C\{U_i\}$ of a good open cover $\{U_i \to X\}$. It follows that $\pi_0 \infty LieGrpd(X,\mathbf{B}G)$ is the Cech cohomology of $X$ with coefficients in $G$ (see there for details).
Concretely, a cocycle
is canonically identified with a transition function
satisfying on $U_i \cap U_j \cap U_k$ the cocycle condition $g_{i j} g_{j k} = g_{i k}$.
From this we can compute the homotopy fiber of $g : C(\{U_i\}) \to \mathbf{B}G$ by forming the ordinary pullback of the fibrant replacement $\mathbf{E}G \to \mathbf{B}G$ of the point inclusion $* \to \mathbf{B}G$, where $\mathb{E}G = \mathbf{B}G^{I} \times_{\mathbf{B}G} *$ is the smooth groupoid
From this we find the pullback $\hat P$ in
to be the smooth Lie groupoid
i.e.
The evident projection $\hat P \to P$ is objectwise a surjective and full and faithful functor.
We consider the cohomology in $\mathbf{H}$ of smooth delooping groupoids $\mathbf{B}G$ for $G$ an ordinary Lie group. This is a form of group cohomology for Lie groups.
We discuss how this relates to other definitions of Lie group cohomology in the literature.
(naive Lie group cohomology)
For $G$ a Lie group and $A$ an abelian Lie group and $n \in \mathbb{N}$ define $H^n_{rest}(G,A)$ to be the group of equivalence classes of cocycles given by smooth functions $G^\times_n \to A$ by coboundatries given by smooth functions $G^{\times_{n-1}} \to A$ subject to the usual relations.
Observe that with $\mathbf{B}G = G^{\times_\bullet}$ regarded as an object of $sPSh(CartSp)$, this is
Written this way it is evident that this definition misses to take into account any cofibrant replacement of $\mathbf{B}G$.
A more refined definition of cohomology of Lie groups has been given by (Segal), which was later rediscovered by (Brylinski), following (Blanc). A review is in section 4 of (Schommer-Pries).
(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 $\{U^{k}_{i} \to G^{\times_k}| i \in I_k\}$ such that
the index sets arrange themselves into a simplicial set $I : [k] \mapsto I_k$;
and for $d_j(U^k_i)$ and $s_j(U^k_i)$ the images of the face and degeneracy maps of $G^{\times\bullet}$ we have
and
For instance start with a good open cover $\{U^1_i \to G\}$ and define a good open cover $\{U^2_{i_0 i_1 i_2}\}$ of $G \times G$ by $U^2_{i_0 i_1 i_2} := d_0^* U^1_{i_0} \cap d_1^* U^1_{i_1} \cap d_2^* U^1_{i_2}$. And so on.
Then the differentiable Lie group cohomology $H^\bullet_{diffr}(G,A)$ of $G$ with coefficients in $A$ is the cohomology of the total complex of the Cech double complex $C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)$ whose differentials are the alternating sums of the face maps of $G^{\times_\bullet}$ and of the Cech nerves, respectively:
This is (Brylinski, definition 1.1).
As discussed there, this is equivalent to other definitions, notably to a definition given earlier in (Segal).
There is a natural map
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 $\mathbf{B}G$ in $sPSh(CartSp)_{proj,cov}$:
For $\{U^\bullet_{i_\bullet}\}$ a system of good open covers as above, we obtain a simplicial diagram of Cech nerves
which is degreewise a cofibrant resolution on $sPSh(CartSp)_{proj,cov}$ of $G^{\times_n}$. Its totalization coend is connected by a zig-zag of weak equivalences in $sPSh(CartSp)_{proj,cov}$ to $\mathbf{B}G$
and we have
The proof of this will also show the following
Write $H^\bullet(G,A) := \pi_0 Sh_{(\infty,1)}(CartSp)(\mathbf{B}G, \mathbf{B}^n A)$ for the intrinsic cohomology of $\mathbf{B}G$ regarded as an object of the $(\infty,1)$-topos of $\infty$-Lie groupoids.
There is a natural morphism
Since $\mathbf{B}^n A$ does satisfy descent with respect to good open covers of Cartesian spaces (every $(n-1)$ $A$-bundle gerbe over an $\mathbb{R}^n$ is trivializable), to compute the intrinsic cohomology we have to find a cofibrant replacement for $\mathbf{B}G$.
A cofibrant replacement of any paracompact manifold $X$ in $[CartSp^{op}, sSet]_{proj,cov}$ 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 $\{U^k_i \to G^{\times k}\}$ as in the definition of differentiable group cohomology above, we obtain cofibrant replacements
for each $k$, and arranging themselves into a simplicial simplicial presheaf
which is cofibrant in the injective model structure on functors $[\Delta^{op}, [CartSp^{op}, sSet]_{proj,cov}]_{inj}$.
Observe that the fact that this is inded a simplicial diagram in $sPSh(CartSp)$ is due to the extra compatibility condition in the above definition of differentiable Lie group cohomology.
Notice from the discussion at model structure on simplicial presheaves that we have canonically another cofibrant replacement $Q (G^{\times_n})$ of $G^{\times_n}$ in $[CartSp^{op}, sSet]_{proj}$ which is functorial and has the special property that the diagonal
is a cofibrant replacement of $\mathbf{B}G$. Notice that in $[\Delta^{op}, [CartSp^{op}, sSet]_{proj}]_{inj}$ we therefore have a zig-zag of weak equivalences
between cofibrant replacements of $G^{\times_\bullet}$.
Write $\mathbf{\Delta}^\bullet : \Delta \to sSet : [k] \mapsto N(\Delta/[k])$ for the standard Bousfield-Kan cofibrant replacement of the point, hence also of $\Delta$, in $[\Delta, sSet_{Quillen}]_{proj}$ (more discussion of this is at homotopy colimit).
Then by the fact that the coend
over the tensoring $\otimes sSet_{Quillen} \times [CartSp^{op}, sSet_{Quillen}]_{proj,cov} \to [CartSp^{op}, sSet_{Quillen}]_{proj,cov}$ is a left Quillen bifunctor (as discussed there), we have weak equivalences in $[CartSp^{op}, sSet]_{proj,cov}$
where
the first one is Dugger’s cofibrant replacement mentioned above, even a global weak equivalence;
the second one is objectwise the Bousfield-Kan map, hence also even a global weak equivalence;
the zig-zag is the image under the left Quillen functor $\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot (-)$ (using that $\mathf{\Delta}$ is cofibrant) of the above zig-zag of weak equivalences between cofibrant objects;
the last one is again the global weak equivalence coming objectwise from the Bousfield-Kan map.
This shows the first claim, that $\mathbf{B}G \stackrel{\simeq}{\leftarrow}\stackrel{\simeq}{\rightarrow} \int^{k} \Delta[k] \cdot C(\{U^k_i\})$.
Moreover, again by the left Quillen bifunctor property of $\int(-)\cdot(-)$ we have that $\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot C(\{U^k_i\}))$ is cofibrant. Therefore the intrinsic cohomology $H^n(G,A)$ is
By pullback along the Bousfield-Kan map $\mathbf{\Delta[k]} \to \Delta[k]$ we have a natural morphism
It remains to show that the set on the left is the differentiable Lie group cohomology.
For that observe that $\mathbf{B}^n A$ is in the image of the Dold-Kan correspondence $\Xi : [CartSp^{op}, Ch_\bullet] \to [CartSp^{op}, sAb] \stackrel{U}{\to} [CartSp^{op}, sSet]$ of the chain-complex valued presheaf $A[n] = A \otimes_\mathbb{Z} \mathbb{Z}[n]$ to reduce the computation of this simplicial hom-complex to that of a cochain complex:
Setting here $K_\bullet = \int^{[k]} \Delta[k] \cdot C(\{U^k_i\})$ and using the Eilenberg-Zilber theorem to identify the Moore complex of the diagonal to the total complex of the double complex of Moore complexes, the claim follows.
For $G$ a Lie group and $A$ either
the additive Lie group of real numbers $\mathbb{R}$;
the intrinsic cohomology of $G$ in $Smooth\infty Grpd$ coincides with the refined Lie group cohomology of (Segal/Brylinski)
In particular we have in general
and for $G$ compact also
The first statement is a special case of the above proposition about cohomology with constant coefficients.
The second statement is a special case of the more general statement that is proven at SynthDiff∞Grpd.
The last statement follows then from the observation (Brylinski) that $H^n_{diffr}(G,\mathbb{R}) \simeq H^n_{naive}(G,\mathbb{R})$ and the classical result (Blanc) that $H_{naive}^n(G,\mathbb{R}) = 0$ in positive degree, and using the fiber sequence induced from the short exact sequence of abelian groups $\mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z}$.
Let $G$ be a Lie group and $\mathbf{B}G \in \mathbf{H} := \infty LieGrpd$ its delooping as discussed above. Let $X$ be a paracompact smooth manifold. Then
is equivalent to the groupoid of smooth $G$-principal bundles on $X$. In particular
is the nonabelian smooth Cech cohomology of $X$ with coefficients in $G$.
Analogously, let $G$ be a strict Lie 2-group such that $\pi_0 G$ is a smooth manifold and $G_0 \to \pi_0 G$ is a submersion. Then we have that
is the 2-groupoid of $G$-principal 2-bundles, whose connected components are given by first smooth nonabelian Cech cohomology $H^1_{Smooth}(X,G)$.
The first case is a special case of the second, it is sufficient to consider that.
First we argue that $\mathbf{B}G$ is fibrant in $[CartSp^{op}, sSet]_{proj, loc}$, hence that for $\{U_i \to \mathbb{R}^n\}$ an open cover we have a weak equivalence
for $C(U)$ the Cech nerve of the good cover. Since the site CartSp with good open cover coverage is a Verdier site, it follows by the statements discussed at hypercover that every hypercover has a refinement by a split hypercover, which is a cofibrant resolution in $[CartSp^{op}, sSet]_{proj,loc}$. But also $C(U) \to X$ is a cofibrant resolution. Hence by the existence of the global model structure and using that $\mathbf{B}G$ is fibrant in $[CartSp^{op}, sSet]_{proj}$, it follows that
is nonabelian degree 1 Cech cohomology on $\mathbb{R}^n$ with coefficients in the 2-group $G$. By (NikolausWaldorf, prop. 4.1) this is the singleton set on the contractible $\mathbb{R}^n$. This shows that the descent morphism in quesion is an isomorphism on $\pi_0$.
Next, $\pi_1$ on the right is similarly the set of equivalence classes of $G_0//(G_0 \ltimes G_1)$-groupoid principal bundles. The underlying $G_0 \times G_1$-principal bundle on the contractible $\mathbb{R}^n$ is trivializable, hence $\pi_1([CartSp^{op}, sSet](C(U), \mathbf{B}^g)) \simeq C^\infty(\mathbb{R}^n , G_0)$, and so we have also an isomorphism on $\pi_1$. Finally the isomorphism on $\pi_2$ is clear.
Now for $X$ an arbitrary paracompact smooth manifold and $U \to X$ a good open cover, we have again that $C(U) \to X$ is a cofibrant resolution in the local model structure, hence hence by the above it follows that
By the same argument about hypercovers on Verdier sites as above, it follows that the connected components of this are
We discuss implementation in $Smooth \infty Grpd$ of the notion of twisted cohomology in a cohesive (∞,1)-topos.
The definition of differential cohomology below is an instance of twisted cohomology, with the twist being given by curvature characteristic forms.
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Given a Lie group $G$, we may regard it either as a topological group and form the classifying space $\mathcal{B}G \in$ Top $\simeq$ ∞Grpd, or we may form its delooping Lie groupoid $\mathbf{B}G \in \infty LieGrpd$, as discussed in the section on Lie groups above. By the discussion in the section on geometric realization we have that under the canonical geometric realization functor
we have
Analogous statements hold for $\infty$-Lie groups. For instance by the same result we have that the circle n-groupoid $\mathbf{B}^n U(1)$ maps to the Eilenberg-MacLane space
This means that given a cocycle
in integral cohomology (defining a characteristic class), it is of interest to ask of this morphism also lifts through $\Pi$ to a morphism
Classes of examples of such lifts have been described in the section on integration of ∞-Lie algebra cocycles.
By general reasoning, the extensions/principal ∞-bundles classified by such cocycles are their homotopy fiber
in $\infty LieGrpd$ and
in $Top \simeq \infty Grpd$. By the theorem discussed in the section on Geometric realization we have that these two homotopy fibers correspond to each other under $\Pi : \infty LieGrpd \to \infty Grpd$:
So this means with a smooth lift of a cocycle, we automatically obtain the corresponding smooth lift of the extension that it classifies (assuming that the simplicial topological group $G$ is well-sectioned). This is notably useful for finding smooth lifts of Whitehead towers. Examples of this we discuss in the following.
Let $O$ denote the orthogonal group, regarded as a Lie group. We discuss steps of the Whitehead tower of $O$ refined to $\infty LieGrpd$.
A lift of the first Stiefel-Whitney class $w_1 : \mathcal{B}O \to K(\mathbb{Z}_2 ,1) \in Top \simeq \infty Grpd$ to $\infty LieGrpd$ is given by the morphism
of Lie groupoids that is the delooping of the group homomorphism that sends $g \in O$ to its signature $\sigma(g) \in \mathbb{Z}_2$, where $\sigma(g) = 1$ if $g$ is in the connected component of $O$ (is orientation preserving as an orthogonal map) and $\sigma(g) = -1$ otherwise.
We may compute the principal bundle in $\infty LieGrpd$ classified by $\mathbf{w}_1$ as the ordinry pullback of $\mathbf{E} \mathbb{Z}_2 \to \mathbf{B}\mathbb{Z}_2$, where $\mathbf{E}\mathbb{Z}_2$ is the groupoid $\mathbb{Z}_2 \times \mathbb{Z}_2 \stackrel{\ovserset{\cdot}{\to}}{\underset{p_1}{\to} \mathbb{Z}_2}$.
The resulting pullback groupoid
has two objects, $s = +1,-1$ and morphisms of the form $s \stackrel{g}{\to} \sigma(g)\cdot s$ for $g \in O$. The smooth structure is that induced from $O$.
Evidently the canonical embedding $\mathbf{B} SO \to Q$ which sends a morphism $\bullet \stackrel{h}{\to} \bullet$ for $h \in SO$ an element of the special orthogonal group is an essentially surjective functor and a full and faithful functor and hence an equivalence of $\infty$-Lie groupoids.
This establishes the (∞,1)-pullback diagram
in $\infty LieGrpd$, covering under $\Pi$ the pullback
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Let $\mu_3 \in CE(\mathfrak{so}(n))$ be the Lie algebra 3-cocycle $\langle -,[-,-]\rangle$ normalized such its left-invariant continuation to a differential 3-form on $Spin(n)$ is the image in deRham cohomology of a generator of $H^3(Spin, \mathbb{Z})$.
The integration of this cocycle
is a smooth refinement of the first fractional Pontryagin class $\frac{1}{2} p_1 : \mathcal{B}Spin \to \mathcal{B}^4 \mathbb{Z}$.
By the discussion at Chern-Simons circle 3-bundle.
Therefore the homotopy fiber
is a smooth model of the string group.
Indeed, this homotopy fiber is given by the Lie string 2-group.
Compute the homotopy pullback as the ordinary limit over
By inspection, this gives the Lie 2-group whose
objects are based paths in $G$; the product is by concatenation of such paths;
morphisms are equivalence classes of based surfaces in $G$ labeled by some $c \in U(1)$; where two of these are equivalence if there is a ball cobounding them such that the integral of $\mu$ over this ball accounts for the difference in the two labels.
Here me may equivalently take thin homotopy-classes of paths and surfaces. This is indeed one of the three incarnations of the string 2-group as a strict 2-group.
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We discuss the intrinsic flat cohomology in $Smooth \infty Grpd$.
Let $X$ be a paracompact smooth manifold regarded as a 0-truncated smooth $\infty$-groupoid under the embedding $SmoothMfd \hookrightarrow Smooth \infty Grpd$. Let $A \in$ ∞Grpd be any ∞-groupoid and $Disc A \in Smooth \infty Grpd$ the coresponding discrete ∞-groupoid.
We have an equivalence of cocycle ∞-groupoids
and hence in particular an isomorphism on cohomology
This also means that
Same proof as of the analogous statement at ETop∞Grpd.
Let $\mathbf{B}^n U(1)$ be the circle $(n+1)$-Lie group as discussed above. Recall the notation and model category presentations as discussed there.
For $n \geq 1$ a fibration presentation in $[CartSp^{op}, sSet]_{proj}$ of the canonical morphism $\mathbf{\flat} \mathbf{B}^n U(1) \to \mathbf{B}^n U(1)$ in $Smooth \infty Grpd$ is given by the image under $\Xi : [CartSp^{op}, Ch_\bullet^+] \to [CartSp^{op}, sSet]$ of
where at the top we have the flat Deligne complex.
It is clear that the morphism of chain complexes is an objectwise surjection and hence maps to a projective fibration under $\Xi$. It remains to observe that the flat Deligne complex is a presentation of $\mathbf{\flat} \mathbf{B}^n U(1)$:
By the discussion at ∞-cohesive site we have that $\mathbf{\flat} = Disc \Gamma$ is given on fibrant objects by first evaluating on the point and then extending back to a constant simplicial presheaf. By the above discussion we have that $\Xi (U(1)[n])$ is indeed fibrant, and so a fibrant presentation of $\mathbf{\flat} \mathbf{B}^n U(1)$ is given by the constant presheaf $U(1)_{const} [n] : U \mapsto \Xi(U(1)[n])$.
The inclusion $U(1)_{const}[n] \to U(1)[n]$ is not yet a fibration. But by a basic fact of abelian sheaf cohomology – using the Poincare lemma – we have a global weak equivalence $U(1)[n]_{const} \stackrel{\simeq}{\to} [C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)]$ that factors this inclusion by the above fibration.
Let $G$ be a Lie group regarded as a 0-truncated ∞-group in $Smooth \infty Grpd$. Write $\mathfrak{g}$ for its Lie algebra. Write $\mathbf{B}G \in Smooth \infty Grpd$ for its delooping. Recall the fibrant presentation $\mathbf{B}G_c \in [CartSp_{smooth}^{op}, sSet]_{proj,loc}$ from above.
The object $\mathbf{\flat}\mathbf{B}G \in Smooth \infty Grpd$ has a fibrant presentation $\mathbf{\flat} \mathbf{B}G_{c} \in [CartSp^{op}, sSet]_{proj,loc}$ given by the groupoid of Lie-algebra valued forms
and this is such that the canonical morphism $\mathbf{\flat} \mathbf{B}G \to \mathbf{B}G$ is presented by the canonical morphism of simplicial presheaves $\mathbf{\flat}\mathbf{B}G_{c} \to \mathbf{B}G_{c}$ which is a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$.
This means that a $U$-parameterized family of objects of $\mathbf{\flat}\mathbf{B}G_{c}$ is given by a Lie-algebra valued 1-form $A \in \Omega^1(U)\otimes \mathfrak{g}$ whose curvature 2-form $F_A = d_{dR} A + [A ,\wedge A] = 0$ vanishes,
and a $U$-parameterized family of morphisms $g : A \to A'$ is given by a smooth function $g \in C^\infty(U,G)$ such that $A' = Ad_g A + g^{-1} d g$, where $Ad_g A = g^{-1} A g$ is the adjoint action of $G$ on its Lie algebra, and where $g^{-1} d g := g^* \theta$ is the pullback of the Maurer-Cartan form on $G$ along $g$.
By the discussion at ∞-cohesive site we have that $\mathbf{\flat} \mathbf{B}G$ is presented by the simplicial presheaf that is constant on the nerve of the one-object groupoid
for the discrete group underlying the Lie group $G$. The canonical morphism of that into $\mathbf{B}G_c$ is however not a fibration.
We claim that the canonical inclusion $N(G_{disc}\stackrel{\to}{\to}) \to \mathbf{\flat} \mathbf{B}G_{c}$ factors the inclusion into $\mathbf{B}G_c$ by a weak equivalence followed by a global fibration.
To see the weak equivalence, notice that it is objectwise an equivalence of groupoids: it is essentially surjective since every flat $\mathfrak{g}$-valued 1-form on the contractible $\mathbb{R}^n$ is of the form $g d g^{-1}$ for some function $g : \mathbb{R}^n \to G$ (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 transformation automorphism of the trivial $\mathfrak{g}$-valued 1-form are precisely given by the constant $G$-valued functions, this is also objectwise a full and faithful functor.
Similarly one sees that the map $\mathbf{\flat}\mathbf{B}G_c \to \mathbf{B}G$ is a fibration.
Finally we need to show that $\mathbf{\flat}\mathbf{B}G_c$ is fibrant in $[CartSp^{op}, sSet]_{proj,loc}$. 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 on a Cartesian space is equivalent to a trivial $G$-principal bundle whose connection is given by a globally defined $\mathfrak{g}$-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.
We discuss the intrinsic de Rham cohomology in $Smooth \infty Grpd$.
Let $\mathbf{B}^n U(1)$ be the circle Lie $(n+1)$-group, as discussed above. Recall the notation and model category presentations from the discussion there.
A fibrant representative in $[CartSp^{op}, sSet]_{proj,loc}$ of the de Rham coefficent object $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ as well as $\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}$ is given by the truncated de Rham complex of sheaves of abelian groups of differential forms
By definition and the fact that homotoppy pullbacks in the local structure may be computed in global structure (discussed here) the object $\mathbf{\flat}_{dR}\mathbf{B}^n U(1)_{chn}$ is given, up to equivalence, by the homotopy pullback in $[CartSp^{op}, Ch_\bullet]_{proj}$ of the inclusion $U(1)_{const}[n] \to U(1)[n]$ along the point inclusion $* \to U(1)[n]$. We may compute this as the ordinary pullback after passing to a resolution of this inclusion by a fibration. By the above discussion of flat cohomology with coefficients in $\mathbf{B}^n U(1)$ such a fibration replacement is given by the map from the flat Deligne complex. Using this we find the ordinary pullback diagram
Let $X$ be an orientable smooth manifold regarded under the embedding $SmoothMfd \hookrightarrow Smooth \infty Grpd$.
Write $H^n_{dR}(X)$ for the ordinary de Rham cohomology of $X$.
For $n \in \mathbb{N}$ we have isomorphisms
Let $\{U_i \to X\}$ be a good open cover. By the disucssion at model structure on simplicial presheaves the Cech nerve $C(\{U_i\}) \to X$ is a cofibrant resolution of $X$ in $[CartSp^{op}, sSet]_{proj,loc}$. Therefore we have for all $n$
The right hand is the $\infty$-groupoid of cocylces in the Cech hypercohomology of the truncated complex of sheaves of differential forms. A cocycle is given by a collection
of differential forms, with $C_i \in \Omega^n_{cl}(U_i)$, $B_{i j} \in \Omega^{n-1}(U_i \cap U_j)$, etc. , such that this collection is annihilated by the total differential $D = d_{dR} \pm \delta$, where $d_{dR}$ is the de Rham differential and $\delta$ the alternating sum of the pullbacks along the face maps of the Cech nerve.
It is a standard result of abelian sheaf cohomology that such cocycles represent classes in de Rham cohomology.
But for the record and since the details of this computation will show up again at some mildly subtle points in further discussion below, we spell this out in some detail.
For $n = 1$ and $n= 0$ our truncated de Rham complex degenerates to $\mathbf{\flat}_{dR}\mathbf{B}U(1)_{chn} = \Xi[\Omega^1_{\mathrm{cl}}(-)]$ and $\mathbf{\flat}_{dR}U(1)_{chn} = \Xi[0]$, respectively, which obviously has the cohomology as claimed above.
For $n \geq 2$ we can explicitly construct coboundaries connecting such a generic cocycle to one of the form
by using a partition of unity $(\rho_i \in C^\infty(X))$ subordinate to the cover $\{U_i \to X\}$, i.e. $x \in U_i \Rightarrow \rho_i(x) = 0$ and $\sum_i \rho_i = 1$.
For consider
where we use that from $(\delta Z)_{i_1, \cdots, i_{n+2}} = 0$ it follows that
where I am suppressing some evident signs…
By recurseively adding coboundaries this way, we can annihilate all the higher Cech-components of the original cocycle and arrive at a cocycle of the form $(F_i, 0, \cdots, 0)$.
Such a cocycle being $D$-closed says precisely that $F_i = F|_{U_i}$ for $F \in \Omega^n_{cl}(X)$ a globally defined closed differential form. Moreover, coboundaries between two cocycles both of this form
are necessarily themselves of the form $(\lambda_i, \lambda_{i j}, \cdots) = (\lambda_i, 0 ,\cdots, 0)$ with $\lambda_i = \lambda|_{U_i}$ for $\lambda \in \Omega^{n-1}(X)$ a globally defined differential $n$-form and $F = F' + d_{dR} \lambda$.
For $n \geq 1$ we have that the intrinsic de Rham cohomology of the circle n-groupoid $\mathbf{B}^n U(1)$ in $\mathbf{H} = \infty LieGrpd$ is concentrated in degree $(n+1)$, where it is $\mathbb{R}$:
By the discussion at circle n-group – differential coefficients above, we have that $\mathbf{\flat}_{dR} \mathbf{B}^k U(1) \simeq \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}$, reflecting the familiar fact that since $Lie(U(1)) \simeq Lie(\mathbb{R})$ we have that $Lie(U(1))$-valued forms are naturally identified with plain $\mathbb{R} = Lie(\mathbb{R})$-valued forms. Therefore the above may equivalently be restated as
Since both domain and codomain are abelian, it will be sufficient to demonstrate the statement for $n = 1$, $k$ arbitrary, and for $k = 1$, $n$ arbitrary. All other combinations of $n$ and $k$ are then implied by repeatedly using the delooping/looping (∞,1)-adjunction $(\Sigma \dashv \Omega)$ on abelian objects in
Here we use that since $\mathbf{\flat}_{dR}$ is a right adjoint, it commutes with forming loop space objects.
Using this, the main work in proving the theorem is involved in establishing the statement for $k = n+1 = 2$, which we now spell out. We note that this would be easy to prove if we did not have to pass to a cofibrant resolution of $\mathbf{B}^n U(1)$ for computing the derived hom-space in the projectve model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,cov}$. For then we would compute, using the above fibrant model for $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$
where $\Xi : [CartSp^{op}, Ch_\bullet] \to [CartSp^{op}, sSet]$ is the Dold-Kan correspondence map.
To see what this last set is, notice that if we forgot the abelian group structure, and looked at the last hom-set as one of sheaves, we find by the Yoneda lemma that is is the set $\Omega^1(U(1))$ of 1-forms. Among these the forms $\omega \in \Omega^1(U(1))$ that do respect the group structure are those such that for all $U \in CartSp$ and all $f,g : U \to U(1)$ we have
A little reflection shows that this is satisfied precisely by the constant forms, i.e. those that are a linear multiple of the Maurer-Cartan form on $U(1)$. Hence the above hom-complex is indeed just
But $\mathbf{B}^n U(1)$ is not cofibrant in $[CartSp^{op}, sSet]_{proj}$. We will pass now to a cofibrant resolution and discuss that computing the homs out of that does nevertheless reduce to the above computation.
By Dugger’s construction of cofibrant replacements in the projective model structure we have that a cofibrant replacement for a simplciial presheaf $A$ is given by the diagonal of the bisimplicial presheaf which in degree $(n,k)$ has
where the coproduct ranges over all sequences of morphisms of representables $U_i \in CartSp$ into $A_k$. And face and degeneracy maps are the evident ones.
So we prove now that the computation of $\mathbf{H}(\mathbf{B}U(1), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1))$ does reduce to the above computation of morphisms out of $\mathbf{B}U(1)$. Write $Q(\mathbf{B}^1 U(1)) \stackrel{\simeq}{\to} \mathbf{B}^n U(1)$ for the above cofibrant replacement.
Then a morphism $Q(\mathbf{B} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1))$ is a collection of horizontal morphisms in the diagram
Explicitly, such a cocycle is
a collection $\{\lambda_{U} \in \Omega^2_{cl}(U) | U \in CartSp\}$;
a collection $\{\omega_{U_0 \to U_1 \to U(1)} \in \Omega^1(U_9)\}$
such that
for all $f : U_0 \to U_1$ we have $f^* \lambda \lambda_{U_1} = \lambda_{U_0} + \omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)}$;
for all $U_0 \stackrel{f}{\to} U_1 \to U_2 \to U(1)\times U(1)$ we have
for all $U$ we have $\omega_{U \stackrel{Id}{\to} U \to \stackrel{e}{\to} U(1)} = 0$.
A coboundary between two such cocycles is
such that
$\lambda'_U = \lambda_U + d \kappa_U$;
$\omega'_{U_0 \stackrel{f}{\to} U_1 \to U(1)} = \omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)} + f^* \kappa_{U_1} - \kappa_{U_0}$.
We claim that any such cocycle $(\lambda,\omega)$ is coboundant to one such that
for all $U$ we have $\lambda_U = 0$;
for all $U_0 \stackrel{f}{\to} U(1)$ we have $\omega_{U_0 \stackrel{f}{\to} U_1 \stackrel{e}{\to} U(1)} = 0$.
The coboundary establishing this is given by setting
This follows from the cocycle law for for maps of the form
which asserts that
Moreover we claim that such cocycles with $\lambda_U = 0$ for all $U$ and $\omega_{U_0 \to U_1 \stackrel{e}{\to} U(1)}$ for all $U_0 \stackrel{f}{\to} U_1$ have no nontrivial morphisms between them, which means that these do constitute unique representatives of their cohomology classes. This is because such a morphism would be given by a collection $\kappa_U \in \Omega^1_{cl}(U)$ of closed 1-forms for each $U \in CartSp$, such that in particular for all $U_0 \stackrel{f}{\to} U_1$ we’d have $\kappa_{U_0} = f^* \kappa_{U_1}$. Clearly for any choice of $\kappa_U$s, one can find $f$ such that this is not satisfied. For instance simply take $f : U \to *$, which imposes the relation $\kappa_U = 0$ explicitly.
This shows that the cohomology in question is the set of cocycles as above, satisfying the two extra constraints. Now using the cocycle law for the diagram
we find
and using it for
we find
In view of the above gauge condition this is equivalently
This says that $\omega_{U_0 \stackrel{f}{\to} U_1 \to U(1)} =: \omega_{U_0 \to U(1)}$ in fact only depends on the domain of $f$, hence only on the map $h : U_0 \to U(1)$ and that as such these forms constitute the components of a morphism of sheaves
But this means that the cocycle $Q(\mathbf{B}U(1)) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)$ indeed factors through $Q(\mathbf{B}U(1)) \to \mathbf{B}U(1)$ and therefore the naive computation at the beginning indeed applies and shows that these cocycle are in bijection with multiples of the standard Maurer-Cartan form on $U(1)$, hence with $\mathbb{R}$, so that
The same kind of argument as above shows that for $k \geq 2$ cocycles $Q(\mathbf{B}U(1)) \to \mathbf{\flat}_{dR}\mathbf{B}^k U(1)$ have underlying them sheaf morphisms $C^\infty(-,U(1)) \to \Omega^{k-1}(-)$. But all these are necessarily trivial, due to the fact that $U(1)$ is 1-dimensional. This establishes the theorem for $n=1$ and arbitrary $k$.
Finally the same argument that above showed that no nontrivial automorphisms of certain cocycles exist shows the theorem for $k=1$ and arbitrary $n \geq 1$, namely that no nontrivial morphisms $Q(\mathbf{B}^n U(1)) \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$ exist: such are given by a collection of 1-forms $\lamba_U \in \Omega^1(U)$ for all $U$, satisfyine for all possible maps $f : U_0 \to U_1$ the relation $\lambda_{U_0} = f^* \lambda_{U_1}$. If $\lambda_{U}$ does not vanish for all $U$, one can always find some $f$ for which this is not satisfied.
Let $G$ be a Lie group. Write $\mathfrak{g}$ for its Lie algebra.
The object $\mathbf{\flat}_{dR}\mathbf{B}G \in Smooth \infty Grpd$ has a fibrant presentation in $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ by the sheaf $\mathbf{\flat}\mathbf{B}G_c = \Omega^1_{flat}(-, \mathfrak{g})$ of flat Lie-algebra valued forms
By a proposition above we have a fibration $\mathbf{\flat}\mathbf{B}G_c \to \mathbf{B}G_c$ in $[CartSp_{smooth}^{op}, sSet]_{proj}$ modeling the canonical inclusion $\mathbf{\flat}\mathbf{B}G \to \mathbf{B}G$. Therefore we may get a presentation for the defining (∞,1)-pullback
in $Smooth \infty Grpd$ by the ordinary pullback
in $[CartSp^{op}, sSet]$.
The resulting simplicial presheaf is fibrant in $[CartSp^{op}, sSet]_{proj,loc}$ because it is a sheaf.
Writing $T U$ for the tangent Lie algebroid of $U$ we may equivalently write
where on the right we have the set of morphisms of Lie algebroids. Equivalently in terms of Chevalley-Eilenberg algebras this is
For $X \in$ SmoothMfd $\hookrightarrow Smooth \infty Grpd$ we find the intrinsic de Rham cohomology set
is the set of smooth flat $\mathfrak{g}$-valued differential forms on $X$.
We work out, following the general definition the coefficient object for Lie 2-algabra valued forms $\mathbf{\flat}_{dR} \mathbf{B}[G_2\to G_1]$ for $(G_2 \to G_1)$ a Lie crossed module.
Let $\Xi : CrsdCplx \to KanCplx$ now denote the inclusion of crossed complexes into all Kan complexes/∞-groupoids.
Write $[\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1]$ for the corresponding differential crossed module with action $\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2)$ corresponding to the Lie strict 2-group crossed module $(G_2 \stackrel{\delta}{\to} G_1)$ with action $\alpha : G_1 \to Aut(G_2)$.
The Lie 2-groupoid $\mathbf{\flat} \mathbf{B}[G_2 \stackrel{\delta}{\to} G_1]$ is represented in $[CartSp^{op}, sSet]$ by the Lie 2-groupoid which on $U \in CartSp$ s the following 2-groupoid:
An object is a pair
such that
and
A 1-morphism $(g,a) : (A,B) \to (A',B')$ is a pair
such that
and
The composite of two 1-morphisms
is given by the pair
a 2-morphism $f : (g,a) \to (g', a')$ is a function
such that
and
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.
The 2-groupoid $\mathbf{\flat}_{dR} \mathbf{B}[G_2 \to G_1]$ is as the one above, discarding the piece $C^\infty(-,G_1)$ in the 1-morphisms and the piece $C^\infty(-,G_2)$ in the 2-morphismms.
Form the defining pullback as before. (…)
We discuss here the intrinsic exponentiated ∞-Lie algebras in $Smooth \infty Grpd$.
Write dgAlg for the category of dg-algebras over the real numbers $\mathbb{R}$.
Write
for the full subcategory of the opposite category on the graded-commutative semi-free dgas of finite type on generators in positive degree. This is the category of L-∞ algebras identified by their dual Chevalley-Eilenberg algebras.
We describe a presentation of the exponentiation an L-∞ algebra to a smooth $\infty$-group. The following somewhat technical definition serves to control the smooth structure on these exponentiated objects.
For $k \in \mathbb{N}$ regard the $k$-simplex $\Delta^k$ as a smooth manifold with corners in the standard way. We think of this embedded into the Cartesian space $\mathbb{R}^k$ in the standard way with maximal rotation symmetry about the center of the simplex, and equip $\Delta^k$ with the metric space structure induced this way.
A smooth differential form $\omega$ on $\Delta^k$ is said to have sitting instants along the boundary if, for every $(r \lt k)$-face $F$ of $\Delta^k$ there is an open neighbourhood $U_F$ of $F$ in $\Delta^k$ such that $\omega$ restricted to $U$ is constant in the directions perpendicular to the $r$-face on its value restricted to that face.
More generally, for any $U \in$ CartSp a smooth differential form $\omega$ on $U \times\Delta^k$ is said to have sitting instants if there is $0 \lt \epsilon \in \mathbb{R}$ such that for all points $u : * \to U$ the pullback along $(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k$ is a form with sitting instants on $\epsilon$-neighbourhoods of faces.
Smooth forms with sitting instants form a sub-dg-algebra of all smooth forms. We write $\Omega^\bullet_{si}(U \times \Delta^k)$ for this sub-dg-algebra.
We write $\Omega_{si,vert}^\bullet(U \times \Delta^k)$ for the further sub-dg-algebra of vertical differential forms with respect to the projection $p : U \times \Delta^k \to U$, hence the coequalizer
For $\mathfrak{g} \in L_\infty$ write $\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet]$ for the simplicial presheaf defined over $U \in$ CartSp and $n \in \mathbb{N}$ by
with the evident structure maps given by pullback of differential forms.
For references related to this definition see Lie integration .
The objects $\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet]$ are
Kan complexes over each $U \in$ CartSp.
That $\exp(\mathfrak{g})_0 = *$ follows from degree-counting: $\Omega^\bullet_{si,vert}(U \times \Delta^0) = C^\infty(U)$ is entirely in degree 0 and $CE(\mathfrak{g})$ is in degree 0 the ground field $\mathbb{R}$.
To see that $\exp(\mathfrak{g})$ has all horn-fillers over each $U \in CartSp$ observe that the standard continuous horn retracts $f : \Delta^k \to \Lambda^k_i$ are smooth away from the preimages of the $(r \lt k)$-faces of $\Lambda[k]^i$.
For $\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i)$ a differential form with sitting instants on $\epsilon$-neighbourhoods, let therefore $K \subset \partial \Delta^k$ be the set of points of distance $\leq \epsilon$ from any subface. Then we have a smooth function
The pullback $f^* \omega \in \Omega^\bullet(\Delta^k \setminus K)$ may be extended constantly back to a form with sitting instants on all of $\Delta^k$.
The resulting assignment
We say that the loop space object $\Omega \exp(\mathfrak{g})$ is the smooth $\infty$-group exponentiating $\mathfrak{g}$.
The objects $\exp(\mathfrak{g}) \in Smooth \infty Grpd$ are geometrically contractible:
Observe that every simplicial presheaf $X$ is the homotopy colimit over its component presheaves $X_n \in [CartSp_{smooth}^{op}, Set] \hookrightarrow [CartSp_{smooth}^{op}, sSet]_{loc}$
(Use for instance the injective model structure for which $X_\bullet$ is cofibrant in the Reedy model structure $[\Delta^{op},[CartSp_{smooth}^{op}, sSet]_{inj,loc}]_{Reedy}$ ).
Therefore it is sufficient to show that in each degree $n$ the 0-truncated object $\exp(\mathfrak{g})_{n}$ is geometrically contractible. To exhibit a geometric contraction, choose for each $n \in \mathbb{N}$, a smooth retraction
of the $n$-simplex: a smooth map such that $\eta_n(-,1) = Id$ and $\eta_n(-,0)$ factors through the point.
We claim that this induces a diagram of presheaves
where over $U \in CartSp$ the middle morphism is given by
where
$\alpha : CE(\mathfrak{g}) \to \Omega^\bullet_{vert}(U \times \Delta^n)$ is an element of the set $\exp(\mathfrak{g})_n(U)$,
$f$ is an element of $[0,1](U)$;
$(f,\eta_n)$ is the composite morphism
$(f,\eta)^* \alpha$ is the postcomposition of $\alpha$ with the image of $(f,\eta_n)$ under $\Omega^\bullet_{vert}(-)$.
Here the last item is well defined given the coequalizer definition of $\Omega^\bullet_{vert}$ because $(f,\eta_n)$ is a morphism of bundles over $U$
Similarly, for $h : K \to U$ any morphism in CartSp${}_{smooth}$ the naturality condition for a morphism of presheaves follows from the fact that the composites of bundle morphisms
and
coincide.
Moreover, notice that the lower morphism in our diagram of presheaves indeed factors through the point as indicated, becase for an L-∞ algebra $\mathfrak{g}$ we have that the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ is in degree 0 the ground field algebra algebra $\mathbb{R}$, so that there is a unique morphism $CE(\mathfrak{g}) \to \Omega^\bullet_{vert}(U \times \Delta^0) \simeq C^\infty(U)$ in $dgAlg$.
Finally, since $[0,1]$ is a contractible paracompact manifold, we have that $\Pi([0,1]) \simeq *$ by the above proposition. Therefore the above diagram of presheaves presents a geometric homotopy in $Smooth \infty Grpd$ from the identity map to a map that factors through the point.
By the propositon about geometric homotopy in a cohesive (∞,1)-topos it follows that $\Pi(\exp(\mathfrak{g})_n) \simeq *$ for all $n \in \mathbb{N}$. And since $\Pi$ preserves the homotopy colimit $\exp(\mathfrak{g}) \simeq \mathbb{L} {\lim_\to}_n \exp(\mathfrak{g})_n$ we have that $\Pi(\exp(\mathfrak{g})) \simeq *$, too.
We may think of $\Omega \exp(\mathfrak{g})$ as the smooth ∞-simply connected Lie integration of $\mathfrak{g}$.
Notice however that $\Omega \exp(\mathfrak{g}) \in Smooth \infty$ in general has nontrivial and interesting categorical homotopy groups. The above statement says that its geometric homotopy groups vanish .
Let $\mathfrak{g} \in L_\infty$ be an ordinary (finite dimensional) Lie algebra. Standard Lie theory (see Lie's three theorems) provides a simply connected Lie group $G$ integrating $\mathfrak{g}$.
With $G$ regarded as a smooth ∞-group write $\mathbf{B}G \in Smooth\infty Grpd$ for its delooping. Recall from above the standard presentation of this by a simplicial presheaf $\mathbf{B}G_c \in [CartSp_{smooth}^{op}, sSet]$.
The operation of parallel transport $P \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G$ yields a weak equivalence (in $[CartSp^{op}, sSet]_{proj}$)
The proof is spelled out at Lie integration. In the section Integrating Lie algebras to Lie groups.
Let $n \in \mathbb{N}$, $n \geq 1$.
Write
for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree $n$ and vanishing differential. We call this the line Lie n-algebra
The discrete ∞-groupoid underlying $\exp(b^{n-1} \mathbb{R})$ is given by the Kan complex that in degree $k$ has the set of closed differential $n$-forms (with sitting instants) on the $k$-simplex
We write equivalently
We have that $\mathbf{B}^n \mathbb{R}_{smp}$ is a presentation of the smooth line n-group $\mathbf{B}^{n} \mathbb{R}$.
Concretely, with $\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet]$ the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree $n$ on $C^\infty(-, \mathbb{R})$ the equivalence is induced by the fiber integration of differential $n$-forms over the $n$-simplex:
The proof of this is spelled out at Lie integration in the section Integration to line n-groups.
We consider now the flat coefficient objects $\mathbf{\flat} \exp(\mathfrak{g})$ of exponentiated ∞-Lie algebras $\exp(\mathfrak{g})$.
Write $\mathbf{\flat}\exp(\mathfrak{g})_{smp}$ for the simplicial presheaf given by
The canonical morphism $\mathbf{\flat} \mathbf{B}^n \mathbb{R} \to \mathbf{B}^n \mathbb{R}$ in $Smooth \infty Grpd$ is presented in $[CartSp_{smooth}^{op}, sSet]$ by the composite
where the first morphism is a weak equivalence and the second a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$.
We discuss the two morphisms in the composite separately in two lemmas.
The canonical inclusion
is a weak equivalence in $[CartSp^{op}, sSet]_{proj}$.
The morphism in question is on each object $U \in CartSp$ the morphism of simplicial sets
which is given by pullback of differential forms along the projection $U \times \Delta^k \to \Delta^k$.
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
and show that this is an acyclic fibration of simplicial sets. The statement then follows by the 2-out-of-3-property of weak equivalences.
We take $F_U$ to be given by evaluation at $0: * \to U$, i.e. by postcomposition with the morphisms
(This of course is not natural in $U$ and hence does not extend to a morphism of simplicial presheaves. But for our argument here it need not.)
The morphism $F_U$ is an acyclic Kan fibration precisely if all diagrams of the form
have a lift. Using the Yoneda lemma over the simplex category and since the differential forms on the simplices have sitting instants, we may, as above, equivalently reformulate this in terms of spheres as follows:
for every morphism $CE(\mathfrak{g}) \to \Omega^\bullet_{si}(D^n)$ and morphism $CE(\mathfrak{g}) \to \Omega^\bullet_{si}(U \times S^{n-1})$ such that the diagram
commutes, this may be factored as
(Here the subscript “${}_{si}$” denotes differential forms on the disk that are radially constant in a neighbourhood of the boundary.)
This factorization we now construct.
Let first $f : [0,1] \to [0,1]$ be any smoothing function, i.e. a smooth function which is surjective, non-decreasing, and constant in a neighbourhood of the boundary. Define a smooth map
by
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
which is such that a form $\omega$ is sent to a form which in a neighbourhood $(1-\epsilon,1]$ of $1 \in [0,1]$ is constant along $(1-\epsilon,1] \times U$ on the value $(0 , Id_{S^{n-1}})^* \omega$.
(Notice that this step does not respect vertical forms. This is the crucial difference between $\{\Omega^\bullet_{si}(U \times \Delta ^k) \leftarrow CE(\mathfrak{g})\}$ and $\{\Omega^\bullet_{si,vert}(U \times \Delta ^k) \leftarrow CE(\mathfrak{g})\}$).
Let now $0 \lt \epsilon \in \mathbb{R}$ some value such that the given forms $CE(\mathfrak{g}) \to \Omega^\bullet_{si}(D^k)$ are constant a distance $d \leq \epsilon$ from the boundary of the disk. Let $q : [0,\epsilon/2] \to [0,1]$ be given by multiplication by $1/(\epsilon/2)$ and $h : D_{1-\epsilon/2}^k \to D_1^n$ the injection of the $n$-disk of radius $1-\epsilon/2$ into the unit $n$-disk.
We can then glue to the morphism
to the morphism
by smoothly identifying the union $[0,\epsilon/2] \times S^{n-1} \coprod_{S^{n-1}} D^n_{1-\epsilon/2}$ with $D^n$ (we glue a disk into an annulus to obtain a new disk) to obtain in total a morphism
with the desired properties: at $u = 0$ the homotopy that we constructed is constant and the above construction hence restricts the forms to radius $\leq 1-\epsilon/2$ and then extends back to radius $\leq 1$ by the constant value that they had before. Away from 0 the homotopy in the rmaining $\epsilon/2$ bit smoothly interpolates to the boundary value.
The canonical morphism
is a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$.
Over each $U \in CartSp$ the morphism is induced from the morphism of dg-algebras
that discards all differential forms of non-vanishing degree.
It is sufficient to show that for
a morphism and
a lift of its restriction to $\sigma = 0 \in [0,1]$ we have an extension to a 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 ∞-Lie algebroid valued differential forms.
1st case: $\mathfrak{g} = b^{n-1} \mathbb{R}$
We condsider the case $\mathfrak{g} = b^{n-1} \mathbb{R}$.
A morphism $CE(b^{n-1} \mathbb{R}) \to \Omega_{si,vert}^\bullet( U \times (D^k \times [0,1]))$ is a $U$-parameterized family of $n$-forms $A = A_{D^k} + (d \sigma) \wedge \lambda$ on $D^k \times [0,1]$ satisfying
and
Consider a lift of the restriction to $\sigma = 0$ by a term $A_{D^k} + A_{U \times D^k}$ being a morphism $CE(b^{n-1}\mathbb{R}) \to \Omega^\bullet_{si}(U \times D_k )$.
We can extend this to a morphism $CE(b^{n-1}\mathbb{R}) \to \Omega^\bullet_{si}(U \times D_k \times [0,1])$ by lifting $\lambda$ without adding any terms to it and solving the differential equation
equivalently for the new term
with boundary condition the given value at $\sigma = 0$.
For this unique solution to be a lift we need in addition that $(d_U + d_{D^k}) (A_{D^k} + A_{U \times D^k}) = 0$. Since this is the case at $\sigma = 0$ by assumption of the lift at that end, we need to check that
But by the above this follows from the Bianchi identity, which for the special abelian case that we are considering is just the nilpotency of the de Rham differential
2nd case: $\mathfrak{g} an arbitrary Lie algebra$
Now let $\mathfrak{g}$ be an ordinary Lie algebra. Choose a dual basis $\{t^a\}$ and structue constants $C^a{}_{b c}$. We get a discussion analogous to the above with structure constant terms thrown in:
the original element is a collection of 1-forms $A^a_v d v + A^a_\sigma d \sigma$ satisfying
We lift by adding a term $A_u^a d u$ that is uniquely fixed by the condition that it solves the differential equation
for given boundary value at $\sigma = 0$.
We need to show that the lift found this way also satisfies the equation
By assumption, this is true at $\sigma = 0$. We now show that the $\sigma$-derivative of this expression satisfies the Binachi-type equation
A solution to this differential equation with initial value 0 is $F^a_{v u} = 0$. Since this solution is guaranteed to be unique, we will have shown our claim.
Now compute:
Here in the last step we use the Jacobi identity
general case
For $\mathfrak{g}$ a general $L_\infty$-algebra, the computation is essentially as above for the Lie algebra case only that all indices become multi-indices in a suitable sense.
For instance the structure constants now have components of arbitrary arity. But for the discussion of the lift it is still always just the components with two legs along the $u$-, $v$-, $\sigma$- direction that matter, all other indices just run along.
I’ll try to think of a convenient notation to express this.
We have discussed now two different presentations for the flat coefficient object $\mathbf{\flat}\mathbf{B}^n \mathbb{R}$:
$\mathbf{\flat} \mathbf{B}^n \mathbb{R}_{chn}$ – discussed here;
$\mathbf{\flat} \mathbf{B}^n \mathbb{R}_{smp}$ – discusse here;
There is an evident degreewise map
that sends a closed $n$-form $\omega \in \Omega^n_{cl}(U \times \Delta^k)$ to $(-1)^{k+1}$ times its fiber integration $\int_{\Delta^k} \omega$.
This map yields a morphism of simplicial presheaves
which is a weak equivalence in $[CartSp^{op}, sSet]_{proj}$.
First we check that we have a morphism of simplicial sets over each $U \in CartSp$. Since both objects are abelian simplicial groups we may, by the Dold-Kan correspondence, check the statement for sheaves of normalized chain complexes.
Notice that the chain complex differential on the forms $\omega \in \Omega^n_{cl}(U \times \Delta^k)$ on simplices sends a form to the alternating sum of its restriction to the faces of the simplex. Postcomposed with the integration map this is the operation $\omega \mapsto \int_{\partial \Delta^k} \omega$ of integration over the boundary.
Conversely, first integrating over the simplex and then applying the de Rham differential on $U$ yields
where we first used that $\omega$ is closed, so that $d_{dR} \omega = (d_U + d_{\Delta^k}) \omega = 0$, and then used Stokes' theorem. Therefore we have indeed objectwise a chain map.
By the discussion of the two objects we already know that both present the homotopy type of $\mathbf{\flat} \mathbf{B}^n \mathbb{R}$. Therefore it suffices to show that the integration map is over each $U \in CartSp$ an isomorphism on the simplicial homotopy group in degree $n$.
Clearly the morphism
is surjective on degree $n$ homotopy groups: for $f : U \to * \to \mathbb{R}$ constant, a preimage is $f \cdot vol_{\Delta^n}$, the normalized volume form of the $n$-simplex times $f$.
Moreover, these preimages clearly span the whole homotopy group $\pi_n (\mathbf{\flat} \mathbf{B}^n \mathbb{R}) \simeq \mathb{R}_{disc}$ (they are in fact the images of the weak equivalence $const \Gamma \exp(b^{n-1}\mathbb{R}) \to \mathbf{\flat} \mathbf{B}^n \mathbb{R}_{smp}$ ) and the integration map is injective on them. Therefore it is an isomorphism on the homotopy groups in degree $n$.
We now consider the de Rham coefficient object $\mathbf{\flat}_{dR} \exp(\mathfrak{g})$ of exponentiated L-∞ algebras $\exp(\mathfrak{g})$.
For $\mathfrak{g} \in L_\infty$ a representive in $[CartSp^{op}, sSet]_{proj}$ of the object de Rham coefficient object $\mathbf{\flat}_{dR} \exp(\mathfrak{g})$ is the presheaf
where the notation on the right denotes the dg-algebra of differential forms on $U \times\Delta^n$ that (apart from having sitting instants on the faces of $\Delta^n$) are along $U$ of non-vanishing degree.
By the above proposition we may present the defining (∞,1)-pullback $\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R} := * \times_{\mathbf{B}^n \mathbb{R}} \mathbf{\flat} \mathbf{B}^n \mathbb{R}$ in $Smooth \infty Grpd$ by the ordinary pullback
in $[CartSp_{smooth}^{op}, sSet]$.
We have discussed now two different presentations for the de Rham coefficient object $\mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}$:
$\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{chn}$ – discussed here;
$\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}_{smp}$ – discussed here;
There is an evident degreewise map
that sends a closed $n$-form $\omega \in \Omega^n_{cl}(U \times \Delta^k)$ to $(-1)^{k+1}$ times its fiber integration $\int_{\Delta^k} \omega$.
This map yields a morphism of simplicial presheaves
which is a weak equivalence in $[CartSp^{op}, sSet]_{proj}$.
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 $\omega \in \Omega^n_{cl}(U \times \Delta^k)$ on simplices sends a form to the alternating sum of its restriction to the faces of the simplex. Postcomposed with the integration map this is the operation $\omega \mapsto \int_{\partial \Delta^k} \omega$.
Conversely, first integrating over the simplex and then applying the de Rham differential on $U$ yields
where we first used that $\omega$ is closed, so that $d_{dR} \omega = (d_U + d_{\Delta^k}) \omega = 0$, and then Stokes' theorem.
Therefore we have indeed objectwise a chain map.
To see that it gives a weak equivalence, notice that this morphism is the morphism on pullbacks induced from the weak equivalence of diagrams
Since both of these pullbacks are homotopy pullbacks by the above discussion, the induced morphism between the pullbacks is also a weak equivalence.
We discuss the intrinsic Maurer-Cartan and curvature characteristic forms defined in any cohesive $(\infty,1)$-topos realized in $Smooth \infty Grpd$.
Let $G$ be a Lie group. Write $\mathfrak{g}$ for its Lie algebra.
Under the identification
from the above proposition, for $X \in$ SmoothMfd, we have that the canonical morphism
in $Smooth \infty Grpd$ corresponds to the ordinary Maurer-Cartan form on $G$.
We compute the defining double (∞,1)-pullback
in $Smooth \infty Grpd$ as a homotopy pullback in $[CartSp_{smooth}^{op}, sSet]_{proj}$
In the above discussion of differential coefficients we already modeled the lower $(\infty,1)$-pullback square by the ordinary pullback
A standard fibration replacement of the point inclusion $* \to \mathbf{\flat}\mathbf{B}G$ is (as discussed at universal principal ∞-bundle) given by replacing the point by the presheaf that assigns groupoids of the form
where on the right the commuting triangle is in $(\mathbf{\flat}_{dR}\mathbf{B}G_c)(U)$ and here regarded as a morphism from $(g_1,A_1)$ to $(g_2,A_2)$. And the fibration $Q \to \mathbf{\flat}\mathbf{B}G_c$ is given by projecting out the base of these triangles.
The pullback of this along $\mathbf{\flat}_{dR}\mathbf{B}G_c \to \mathbf{\flat}\mathbf{B}G_c$ is over each $U$ the restriction of the groupoid $Q(U)$ to its set of objects, hence is the sheaf
equipped with the projection
given by
Under the Yoneda lemma (over SmoothMfd) this identifies the morphism $t$ with the Maurer-Cartan form $\theta \in \Omega^1_{flat}(G,\mathfrak{g})$.
We discuss presentations of universal curvature characteristics $\mathbf{B}^n U(1)\to \mthbf{\flat}_{dR}\mathbf{B}^{n+1} U(1)$ and $\mathbf{B}^n \mathbb{R}\to \mthbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}$ in $Smooth \infty Grpd$ by constructions in $[CartSp_{smooth}^{op}, sSet]$.
Recall the discussion of $\mathbf{B}^n U(1)$ and of $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ from above.
For $n \in \mathbb{N}$ define the simplicial presheaf
The evident projection
is a weak equivalence in $[CartSp^{op}, sSet]_{proj}$. Moreover, the universal curvature characteristic
in $Smooth \infty Grpd$ is presented in $[CartSp^{op}, sSet]_{proj,loc}$ by a span
where the horizontal morphism is the evident projection
We need to present the defining (∞,1)-pullback
by a homotopy pullback in $[CartSp^{op}, sSet]_{proj}$ (since (∞,1)-sheafification preserves finite (∞,1)-pullbacks it is sufficient to present the $(\infty,1)$-pullback in (∞,1)-presheaves).
We claim that we have a commuting diagram
in $[CartSp^{op},sSet]_{proj}$ where
the objects are fibrant models for the corresponding objects in the above $(\infty,1)$-pullback diagram;
the two right vertical morphisms are fibrations;
the two squares are pullback squares.
Therefore this is a homotopy pullback in $[CartSp^{op}, sSet]_{proj}$ that realizes the (∞,1)-pullback in question.
For the lower square we had discussed this already above. For the upper square the same type of reasoning applies. The main point is to find the chain complex in the top right such that it is a resolution of the point and maps by a fibration onto our model for $\mathbf{\flat}\mathbf{B}^n U(1)$. The top right complex is
and the vertical map out of it into $C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-) \stackrel{d_{dR}}{\to} \Omega^{n+1}_{cl}(-)$ is in positive degree the projection onto the lower row and in degree 0 the de Rham differential. This is manifestly surjective (by the Poincare lemma applied to each object $U \in$ CartSp) hence this is a fibration.
The pullback object in the top left is in this notation
and in turn the top left vertical morphism $curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$ is in positive degree the projection on the lower row and in degree 0 the de Rham differential.
Notice that the evident forgetful morphism $\mathbf{B}^n U(1) \stackrel{}{\leftarrow} \mathbf{B}^n_{diff} U(1)$ is indeed a weak equivalence.
In the section on de Rham coefficients for exponentiated Lie algebras we had discussed an equivalent presentation of most of the objects above. We now formulate the curvature characteristic in this alternative form.
We may write the simplicial presheaf $\mathbf{\flat}_{dR}\mathbf{B}^{n+1} \mathbb{R}_{smp}$ from above equivalently as follows
where on the right we have the set of commuting diagrams in dgAlg of the given form, with the vertical morphisms being the canonical projections.
Write $W(b^{n-1}\mathbb{R}) \in$ dgAlg for the Weil algebra of the line Lie n-algebra, defined to be the free commutative dg-algebra on a single generator in degree $n$, hence the graded commutative algebra on a generator in degree $n$ and a generator in degree $(n+1)$ equipped with the differential that takes the former to the latter.
We have the following properties of $\mathrm{W}(b^{n-1}\mathbb{R})$
There is a canonical natural isomorphism
between dg-algebra homomorphisms $A : W(b^{n-1}\mathbb{R}) \to \Omega^\bullet(X)$ from the Weil algebra of $b^{n-1}\mathbb{R}$ to the de Rham complex and degree-$n$ differential forms, not necessarily closed.
There is a canonical dg-algebra homomorphism $W(b^{n-1}\mathbb{R}) \to CE(b^{n-1}\mathbb{R})$ and the differential $n$-form corresponding to $A$ factors through this morphism precisely if the curvature $d_{dR} A$ of $A$ vanishes.
The image under $\exp(-)$
of the canonical dg-algebra morphism $\mathrm{W}(b^{n-1}\mathbb{R}) \leftarrow \mathrm{CE}(b^n \mathbb{R})$ is a fibration in $[\mathrm{CartSp}_{\mathrm{smooth}}^{\mathrm{op}}, \mathrm{sSet}]_{\mathrm{proj}}$ that presents the point inclusion ${*} \to \mathbf{B}^{n+1}\mathbb{R}$ in $\mathrm{Smooth}\infty \mathrm{Grpd}$.
Let $\mathbf{B}^n \mathbb{R}_{diff,smp} \in [CartSp_{smooth}^{op}, sSet]$ be the simplicial presheaf defined by
where on the right we have the set of commuting diagrams in dgAlg as indicated.
This means that an element of $\mathbf{B}^n \mathbb{R}_{diff,smp}(U)[k]$ is a smooth $n$-form $A$ (with sitting instants) on $U \times \Delta^k$ such that its curvature $(n+1)$-form $d A$ vanishes when restricted in all arguments to vector fields tangent to $\Delta^k$. We may write this condition as $d A \in \Omega^{\bullet \geq 1, \bullet}_{si}(U \times \Delta^k)$.
There are canonical morphisms
in $[CartSp_{smooth}^{op}, sSet]$, where the vertical map is given by remembering only the top horizontal morphism in the above square diagram, and the horizontal morphism is given by forming the pasting composite
This span is a presentation in $[CartSp_{smooth}^{op}, sSet]$ of the universal curvature characteristics $curv : \mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}$ in $Smooth \infty Grpd$.
We need to produce a fibration resolution of the point inclusion $* \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}_{smp}$ in $[CartSp_{smooth}^{op}, sSet]_{proj}$ and then show that the above is the ordinary pullback of this along $\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{smp} \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}_{smp}$.
We claim that this is achieved by the morphism
Here the simplicial presheaf on the left is that which assigns the set of arbitrary $n$-forms (with sitting instants but not necessarily closed) on $U \times \Delta^k$ and the map is simply given by sending such an $n$-form $A$ to the $(n+1)$-form $d_{dR} A$.
It is evident that the simplicial presheaf on the left resolves the point: since there is no condition on the forms every form on $U \times \Delta^k$ is in the image of the map of the normalized chain complex of a form on $U \times \Delta^{k+1}$: such is given by any form that is, up to a sign, equal to the given form on one $n$-face and 0 on all the other faces. Clearly such forms exist.
Moreover, this morphism is a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$, for instanxce because its image under the normalized chains complex functor is a degreewise surjection, by the Poincare lemma.
Now we observe that we have over each $(U,[k])$ a double pullback diagram in Set
hence a coresponding pullback diagram of simplicial presheaves, that we claim is a presentation for the defining double (∞,1)-pullback
for $curv$.
The bottom square is the one we already discussed for the de Rham coefficients. Since the top right vertical morphism is a fibration, also the top square is a homotopy pullback and hence exhibits the defining $(\infty,1)$-pullback for curv.
The degreewise map
that sends an $n$-form $A \in \Omega^n(U \times \Delta^k)$ and its curvature $d A$ to $(-1)^{k+1}$ times its fiber integration $(\int_{\Delta^k} A, \int_{\Delta^k} d A)$ is a weak equivalence in $[CartSp_{smooth}^{op}, sSet]_{proj}$.
Since under homotopy pullbacks a weak equivalence of diagrams is sent to a weak equivalence. See the analagous argument above.
Above we discussed the canonical differential form on smooth $\infty$-groups $G$ for the special cases where a) $G$ is a Lie group and b) where $G$ is a circle Lie n-group. These are both in turn special cases of the situation where $G$ is a Lie simplicial group. This we discuss now.
For $G$ a Lie simplicial group, the flat de Rham coefficient object $\mathbf{\flat}_{dR}\mathbf{B}G$ is presented by the simplicial presheaf which in degree $k$ is given by $\Omega^1_{flat}(-, \mathfrak{g}_k)$, where $\mathfrak{g}_k = Lie(G_k)$ is the Lie algebra of $G_k$.
Let
be the presheaf of simplicial groupoids which in degree $k$ is the groupoid of Lie-algebra valued forms with values in $G_k$ from above. As in the above discussion there we have that under the degreewise nerve this is a degreewise fibrant resolution of presheaves of bisimplicial sets
of the standard presentation of the delooping of the discrete group underlying $G$. By the discussion at bisimplicial set we know that under taking the diagonal
the object on the right is a presentation for $\mathbf{\flat}_{dR} \mathbf{B}G$, because
Now observe that the morphism
is a global fibration. This is in fact true for every morphism of the form
for $S_\bullet//G_\bullet \to *//G_\bullet$ a simlicial action groupoid projection with $G$ a simplicial group acting on a Kan complex $S$: we have that
On the second factor the horn filling condition is simply that of the identity map $diag N B G \to diag N B G$ which is evidently solvable, whereas on the first factor it amounts to $S \to *$ being a Kan fibration, hence to $S$ being Kan fibrant.
But the simplicial presheaf $\Omega^1_{flat}(-,\mathfrak{g}_\bullet)$ is indeed Kan fibrant: for a given $U \in CartSp$ we may use parallel transport to (non-canonically) identify
where on the right we have smooth functions that send the origin of $U$ to the neutral element. But since $G_\bullet$ is Kan fibrant and has smooth global fillers (by the discussion at simplicial group one can give algebraic formulas for the fillers, which translate into smooth manps) als $SmoothMfd_*(U,G_\bullet)$ is Kan fibrant.
In summary this means that the defining homotopy pullback
is presented by the ordinary pullback of simplicial presheaves
For $G$ a simplicial Lie group the canonical differential form
is presented in terms of the above presentation for $\mathbf{\flat}_{dR} \mathbf{B}G$ by the morphisms of simplicial presheaves
which is the presheaf-incarnation of the Maurer-Cartan form of the ordinary Lie group $G_k$.
Continuing with the strategy of the previous proof we find a resolution of $* \to \mathbf{\flat} \mathbf{B}G$ by applying the construction of The canonical form on a Lie group degreewise and then applying $diag N$.
The defining homotopy pullback
for $\theta$ is this way presented by the ordinary pullback
of simplicial presheaves, where $\Omega^1_{flat}(-,\mathfrak{g}_\k)$ is the set of flat $\mathfrak{g}$-valued forms $A$ equipped with a gauge transformation $0 \stackrel{g}{\to} A$. As in the above proof one finds that the right vertical morphism is a fibration, hence indeed a resolution of the point inclusion. The pullback is degreewise that from the case of ordinary Lie groups and thus the result follows.
We can now give a simplicial description of the canonical curvature form $\theta : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$ that above we obtained by a chain complex model:
The canonical form on the circle Lie n-group
is presented by the simplicial map
which is simply the Maurer-Cartan form on $U(1)$ in degree $n$.
The equivalence to the model we obtained before is given by noticing the equivalence in hypercohomology of chain complexes of abelian sheaves
on CartSp.
We discuss the realization of the general abstract concept of flat Ehresmann infinity-connections realized in $Smooth\infty Grpd$. We show that when applied to an ordinary Lie group this reproduces the traditional notion of Ehresmann connection.
(…)
We discuss the intrinsic differential cohomology in $Smooth \infty Grpd$
We first expose the simple special case of ordinary $U(1)$-principal bundles with connection in more detail. Then we turn to the general case.
Before discussing the full theorem, it is instructive to start by looking at the special case $n=1$ in some detail, which is about ordinary $U(1)$-principal bundles with connection.
This contains in it already all the relevant structure of the general case, but the low categorical degree is more transparently written out and will allow us to pause to highlight some maybe noteworthy aspects of the situation, such as the phenomenon of pseudo-connections below.
In terms of the Dold-Kan correspondence the object $\mathbf{B}U(1) \in \mathbf{H}$ is modeled in $[CartSp^{op}, sSet]$ by
Accordingly we have for the double delooping the model
and for the universal principal 2-bundle
In this notation we have also the constant presheaf
Above we already found the model
In order to compute the differential cohomology $\mathbf{H}_{diff}(-,\mathbf{B}U(1))$ by an ordinary pullback in sSet we also want to resolve the curvature characteristic morphism $\mathbf{B}U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)$ by a fibration. We claim that this may be obtained by choosing the resolution $\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B} U(1)_{diff,chn}$ given by
with the morphism $curv : \mathbf{B}_{diff}U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)$ given by
By the Poincare lemma applied to each Cartesian space, this is indeed a fibration.
In the next section we give the proof of this (simple) claim. Here in the warmup phase we instead want to discuss the geometric interpretation of this resolution, along the lines of the section curvature characteristics of 1-bundles in the survey-part.
We have the following geometric interpretation of the above models:
and
And in this presentation the morphism $curv : \mathbf{B}_{diff}U(1) \to \mathbf{B}^2 U(1)$ is given over $U \in CartSp$ by forming the pasting composite
and picking the lowest horizontal morphism.
Here the terms mean the following:
$INN(U(1))$ is the 2-group $\Xi(U(1) \to U(1))$, which is a groupal model for the universal U(1)-principal bundle $\mathbf{E}U(1)$;
$\mathbf{\Pi}_2(U)$ is the path 2-groupoid with homotopy class of 2-dimensional paths as 2-morphisms
the groupoids of diagrams in braces have as objects commuting diagrams in $[CartSp^{op}, sSet]$ as indicated, and horizontal 2-morphisms fitting into such diagrams as morphisms.
Using the discussion at 2-groupoid of Lie 2-algebra valued forms (SchrWalII) we have the following:
For $X$ a smooth manifold, morphisms in $[CartSp^{op}, 2Grpd]$ of the form $tra_A : \Pi_2(X) \to \mathbf{E}\mathbf{B}U(1)$ are in bijection with smooth 1-forms $A \in \Omega^1(X)$: the 2-functor sends a path in $X$ to the the parallel transport of $A$ along that path, and sends a surface in $X$ to the exponentiated integral of the curvature 2-form $F_A = d A$ over that surface. The Bianchi identity $d F_A = 0$ says precisely that this assignment indeed descends to homotopy classes of surfaces, which are the 2-morphisms in $\Pi_2(X)$.
Moreover 2-morphisms of the form $(\lambda,\alpha) : tra_A \to \tra_{A'}$ in $[CartSp^{op}, 2Grpd]$ are in bijection with pairs consisting of a $\lambda \in C^\infty(X,U(1))$ and a 1-form $\alpha \in \Omega^1(X)$ such that $A' = A + d_{dR} \lambda - \alpha$.
And finally 3-morphisms $h : (\lambda, \alpha) \to (\lambda', \alpha')$ are in bijection with $h \in C^\infty(X,U(1))$ such that $\lambda' = \lambda \cdot h$ and $\alpha' = \alpha + d_{dR} h$.
By the same reasoning we find that the coefficient object for flat $\mathbf{B}^2 U(1)$-valued differential cohomology is
So by the above definition of differential cohomology in $\mathbf{H}$ we find that $\mathbf{B}U(1)$-differential cohomology of a paracompact smooth manifold $X$ is given by choosing any good open cover $\{U_i \to X\}$, taking $C(\{U_i\})$ to be the Cech nerve, which is then a cofibrant replacement of $X$ in $[CartSp^{op}, sSet]_{proj,cov}$ and forming the ordinary pullback
(because the bottom vertical morphism is a fibration, by the fact that our model for $\mathbf{B}_{diff} U(1) \to \flat_{dR}\mathbf{B}^2 U(1)$ is a fibration, that $C(\{U_i\})$ is cofibrant and using the axioms of the sSet-enriched model category $[CartSp^{op}, sSet]_{proj}$).
A cocycle in $[CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1))$ is
a collection of functions
satsifying $g_{i j} g_{j k} = g_{i k}$ on $U_i \cap U_j \cap U_k$;
a collection of 1-forms
a collection of 1-forms
such that
on $U_i \cap U_j$ and
on $U_i \cap U_j \cap U_k$.
The curvature-morphism takes such a cocycle to the cocycle
in the above model $[CartSp^{op},sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1))$ for intrinsic de Rham cohomology.
Every cocycle with nonvanishing $(a_{i j})$ is in $[C(\{U_i\}), \mathbf{B}_{diff}U(1)]$ coboundant to one with vanishing $(a_{i j})$
The first statements are effectively the definition and the construction of the above models. The last statement is as in the above discussion of our model for ordinary de Rham cohomology: given a cocycle with non-vanishing closed $a_{i j}$, pick a partition of unity $(\rho_i \in C^\infty(X))$ subordinate to the chosen cover and the coboundary given by $(\sum_{i_0} \rho_{i_0} a_{i_0 i})$. This connects $(A_i,a_{i j}, g_{i j})$ with the cocycle $(A'_i, a'_{i j}, g_{i j})$ where
and
So in total we have found the following story:
In order to compute the curvature characteristic form of a Cech cohomology cocycle $g : C(\{U_i\}) \to \mathbf{B}U(1)$ of a $U(1)$-principal bundle, we first lift it
to an equivalent $\mathbf{B}_{diff}U(1)$-cocycle, and this amounts to putting (the Cech-representatitve of) a pseudo-connection on the $U(1)$-principal bundle.
From that lift the desired curvature characteristic is simply projected out
and we find that it lives in the sheaf hypercohomology that models ordinary de Rham cohomology.
Therefore we find that in each cohomology class of curvatures, there is at least one representative which is an ordinary globally defined 2-form. Moreover, the pseudo-connections that map to such a representative are precisely the genuine connections, those for which the $(a_{i j})$-part of the cocycle vaishes.
So we see that ordinary connections on ordinary circle bundles are a means to model the homotopy pullback
in a 2-step process: first the choice of a pseudo-connection realizes the bottom horizontal morphism as an anafunctor, and then second the restriction imposed by forming the ordinary pullback chooses from all pseudo-connections precisely the genuine connections.
The general version of this story is discussed in detail at differential cohomology in an (∞,1)-topos – Local (pseudo-)connections.
In the above discussion of extracting ordinary connections on ordinary $U(1)$-principal bundles from the abstract topos-theoretic definition of differential cohomology, we argued that a certain homotopy pullback may be computed by choosing in the Cech-hypercohomology of the complex of sheaves $(\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-))$ over a manifold $X$ those cohomology representatives that happen to be represented by globally defined 2-forms on $X$. We saw that the homotopy fiber of pseudo-connections over these 2-forms happened to have connected components indexed by genuine connections.
But by the general abstract theory, up to isomorphism the differential cohomology computed this way is guaranteed to be independent of all such choices, which only help us to compute things.
To get a feeling for what is going on, it may therefore be useful to re-tell the analgous story with pseudo-connections that are not genuine connections.
By the very fact that $\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B}_{diff}U(1)$ is a weak equivalence, it follows that every pseudo-connection is equivalent to an ordinary connection as cocoycles in $[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}_{diff}(G))$.
If we choose a partition of unity $(\rho_i \in C^\infty(X,\mathbb{R}))$ subordinate to the cover $\{U_i \to X\}$, then we can construct the corresponding coboundary explicitly:
let $(A_i g_{ij}, a_{i j})$ be an arbitrary pseudo-connection cocycle. Consider the Cech-hypercohomology coboundary given by $(\sum_{i_0} \rho_{i_0} a_{i_0 i}, 0)$. This lands in the shifted cocycle
and we can find the new pseudo-components $a'_{i j}$ by
Using the computation
we find that these indeed vanish.
The most drastic example for this is a lift $\nabla$ of a cocycle $g = (g_{i j})$ in
is one which takes all the ordinary curvature forms to vanish identically
This fixes the pseudo-components to be $a_{i j} = - d g_{i j}$. By the above discussion, this pseudo-connection with vanishing connection 1-forms is equivalent, as a pseudo-connection, to the ordinary connection cocycle with connection forms $(A_i := \sum_{i_0} \rho_{i_0} d g_{i_0 i})$. This is a standard formula for equipping $U(1)$-principal bundles with Cech cocycle $(g_{i j})$ with a connection.
We saw above that the intrinsic coefficient object $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ yields ordinary de Rham cohomology in degree $n \gt 1$. For $n = 1$ we have that $\mathbf{\flat}_{dR} \mathbf{B}U(1)$ is given simply by the 0-truncated sheaf of 1-forms, $\Omega^1(-) : CartSp^{op} \to Set \hookrightarrow sSet$. Accordingly we have for $X$ a paracompact smooth manifold
instead of $H^1_{dR}(X)$.
There is a good reason for this discrepancy: for $n \geq 1$ the object $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is the recipient of the intrinsic curvature characteristic morphism
For $X \to \mathbf{B}^{n-1} U(1)$ a cocycle (an $(n-2)$-gerbe without connection), the cohomology class of the composite $X \to \mathbf{B}^{n-1} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is precisely the obstruction to the existence of a flat extension $X \to \mathbf{\flat} \mathbf{B}^{n-1} U(1) \to \mathbf{B}^{n-1} U(1)$ for the original cocycle.
For $n = 2$ this is the usual curvature 2-form of a line bundle, for $n = 3$ it is curvature 3-form of a bundle gerbe, etc. But for $n = 1$ we have that the original cocycle is just a map of spaces
This can be understoody as a cocycle for a groupoid principal bundle, for the 0-truncated groupoid with $U(1)$ as its space of objects. Such a cocycle extends to a flat cocycle precisely if $f$ is constant as a function. The corresponding curvature 1-form is $d_{dR} f$ and this is precisely the obstruction to constancy of $f$ already, in that $f$ is constant if and only if $d_{dR} f$ vanishes. Not (necessarily) if it vanishes in de Rham cohomology .
This is the simplest example of a general statement about curvatures of higher bundles: the curvature 1-form is not subject to gauge transformations.
Recall the definition of the intrinsic differential cohomology on $X \in Smooth \infty Grpd$ with coefficients in $U(1)$ as the (∞,1)-pullback
in ∞Grpd, where the morphism on the right picks one base point in each connected component.
For $X \in SmoothMfd \hookrightarrow Smooth \infty Grpd$ a paracompact smooth manifold we have
Here on the right we have the subset of Deligne cocycles that picks for each integral de Rham cohomology class of $X$ only one curvature form representative.
If we make use of the explicit presentation of $Smooth \infty Grpd$ by the model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,loc}$ and the explicit presentation $\mathbf{\flat}_{\mathrm{dR}} \mathbf{B}^{n+1}U(1)_{chn}$ for $\mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$ by ordinary differential forms, as above we may replace the right morphism in this pullback by $\Omega^{n+1}_{cl}(X) \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}U(1))$ and consider the (∞,1)-pullback
For $X \in SmoothMfd \hookrightarrow Smooth \infty Grpd$ a paracompact smooth manifold we have that
is the ordinary Deligne cohomology of $X$ in degree $n+1$.
Choose a differentiably good open cover $\{U_i \to X\}$ and let $C(\{U_i\}) \to X$ in $[CartSp^{op}, sSet]_{proj}$ be the corresponding Cech nerve projection, a cofibrant resolution of $X$.
Since the above model $curv_{chn} : \mathbf{B}_{diff}^n U(1)_{chn} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{chn}$ for the intrinsic $curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$ is a fibration and $C(\{U_i\})$ is cofibrant, also
is a Kan fibration by the fact that $[CartSp^{op}, sSet]_{proj}$ is an simplicial model category. Therefore the homotopy pullback is computed as an ordinary pullback.
By the above discussion of de Rham cohomology we have that we can assume the morphism $H_{dR}^{n+1}(X) \to [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^{n+1}_{chn})$ picks only cocycles represented by globally defined closed differential forms $F \in \Omega^{n+1}_{cl}(X)$.
By the nature of the chain complexes $curv_{chn} : \mathbf{B}_{diff}^n U(1)_{chn} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{chn}$, we see that the elements in the fiber over such a globally defined $(n+1)$-form $F$ are precisely the cocycles with values only in the “upper row complex” of $\mathbf{B}_{diff}^{n}U(1)_{chn}$
such that $F$ is the differential of the last term.
This is the complex of sheaves that defines Deligne cohomology in degree $(n+1)$.
We discuss how the general abstract definition of differential cohomology in $Smooth \infty Grpd$ reproduces on general grounds the abstract properties of the traditional definition of ordinary differential cohomology.
For $X$ a smooth manifold, the cohomology classes $H'_{diff}(X, \mathbf{B}^n U(1))$ of the cocycle $\infty$-groupoid $\mathbf{H}'_{diff}(X, \mathbf{B}^n U(1))$ defined by the homotopy pullback
fit, with their canonical abelian group structure, into the two characteristic short exact sequences of ordinary differential cohomology:
$0 \to Omega^{n}(X)/\Omega^{n}_{cl}(X) \to H'_{diff}(X, \mathbf{B}^n U(1)) \to H^n(X, \mathbb{Z}) \to 0$.
$0 \to H^{n-1}(X, U(1)) \to H'_{diff}(X, \mathbf{B}^n U(1)) \to \Omega^{n+1}_{cl, int}(X) \to 0$
For the first sequence, the characteristic class exact sequence, use this general proposition from the discussion at cohesive (∞,1)-topos . This says that over the vanishing curvature form $0 \in \Omega_{cl}^{n+1}(X)$ we have the short exact sequence
where we used that $H_{smooth}(X, \mathbf{B}^n U(1)) \simeq H^{n+1}(X, \mathbb{Z})$ on the paracompact space $X$ and that $H_{dR}^n(X)/\Omega^n_{cl,int}(X) = \Omega^n_cl(X)/\Omega^n_{cl, int}(X)$ because exact forms are in particular closed integral forms (with periods $0 \in \mathbb{N}$). Looking at $H'_{diff}$ instead of $H_{diff}$, we get in the fibers one such contribution per closed $(n+1)$-form $\Lambda$ trivial in de Rham cohomology, hence per arbitrary $n$-form $\omega$ with $d \omega = \Lambda$
But this is the fiber sequence in question.
For the second sequence, the curvature exact sequence, this general proposition from the discussion at cohesive (∞,1)-topos , which implies that we have a short exact sequence
Then use prop. 1, which says that $H_{flat}^n(X, \mathbf{B}^n U(1)) \simeq H^{n}(X, U(1)_{disc})$.
(…)
The morphism given by fiber integration of differential forms over the simplex factor fits into a diagram
where the vertical morphisms are weak equivalences.
Fiber integration induces a weak equivalence
Observe that $\mathbf{B}^n \mathbb{R}_{diff,simp}$ is the pullback of $\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{smp} \to \mathbf{\flat}\mathbf{B}^{n+1} \mathbb{R}_{smp}$ along the evident forgetful morphism from
This forgetful morphism is evidently a fibration (because it is a degreewise surjection under Dold-Kan), hence this pullback models the homotopy fiber of $\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}$. Since by the above fiber integration gives a weak equivalence of pulback diagrams the claim follows.
Write $\mathbf{B}^n U(1)_{conn,simp} \hookrightarrow \mathbf{B}^n U(1)_{diff,simp}$ for the sub-presheaf which over $(U,[k])$ is the set of those forms $\omega$ on $U \times \Delta^k$ such that the curvature $d \omega$ has no leg along $\Delta^k$.
Under fiber integration over simplices, $\mathbf{B}^n U(1)_{conn,simp}$ is quasi-isomorphic to the Deligne cohomology-complex.
In summary this gives us the following alternative perspective on connections on $\mathbf{B}^{n-1}U(1)$-principal ∞-bundles: such a connection is a cocycle with values in the $\mathbf{B}^n \mathbb{Z}$-quotient of the $(n+1)$-coskeleton of the simplicial presheaf which over $(U,[k])$ is the set of diagrams of dg-algebras
where the restriction to the top morphism is the underlying cocycle and the restriction to the bottom morphism the curvature form.
The generalization to such diagram cocycles from $b^{n-1}\mathbb{R}$ to general ∞-Lie algebras $\mathfrak{g}$ we discuss below in ∞-Lie algebra valued connections.
The higher holonomy of a circle $n$-bundle with connection is well defined only over oriented smooth manifolds. In the unorientable or even unoriented case, extra structure is needed to define it.
See orientifold for more.
We discuss the general abstract notion of Chern-Weil homomorphism and ∞-connections realized in $Smooth \infty Grpd$.
Recall that for $A \in Smooth \infty Grpd$ a smooth $\infty$-groupoid regarded as a coefficient object for cohomology, for instance the delooping $A = \mathbf{B}G$ of an ∞-group $G$ we have abstractly that
a characteristic class on $A$ with coefficients in the circle Lie n-group is represented by a morphism
the (unrefined) Chern-Weil homomorphism induced from this is the differential characteristic class given by the composite
with the universal curvature characteristic on $\mathbf{B}^n U(1)$, or rather: is the morphism on cohomology
induced by this.
the ∞-connections with coefficients in $A$ are the cocycles in the ∞-groupoid $Smooth \infty Grpd(X,A)_{conn}$ that universally lifts these differential classs in de Rham cohomology to full differential cohomology, in that it universally fills the diagrams
Above we have discussed a presentation of the universal curvature class $\mathbf{B}^n \mathbb{R} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}$ by a span
in the model structure on simplicial presheaves $[CartSp_{smooth}^{op}, sSet]_{proj}$, given by maps of smooth families of differential forms.
We now insert this in the above general abstract definition of the $\infty$-Chern-Weil homomorphism to deduce a presentation of that in terms of smooth families of ∞-Lie algebroid valued differential forms.
The main step is the construction of a well-suited composite of two spans of morphisms of simplicial presheaves (of two ∞-anafunctors): we consider presentations of characteristic classes $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ in the image of the ∞-Lie integration map and presented by trunactions and quotients of morphisms of simplicial presheaves of the form
Then, using the above, the composite differential characteristic class $\mathbf{c}_{dR}$ is presented by the zig-zag
of simplicial presheaves. In order to efficiently compute which morphism in $Smooth \infty Grpd$ this presents we need to construct – preferably naturally in the L-∞ algebra $\mathfrak{g}$ – a simplicial presheaf $\exp(\mathfrak{g})_{diff}$ that fills this diagram as follows:
Given this, $\exp(\mathfrak{g})_{diff,smp}$ serves as a new resolution of $\exp(\mathfrak{g})$ for which the composite differential characteristic class is presented by the ordinary composite of morphisms of simplicial presheaves $curv_{smp}\circ \exp(\mu, cs)$.
This object $\exp(\mathfrak{g})_{diff}$ we shall see may be interpreted as the coefficient for pseudo ∞-connections with values in $\mathfrak{g}$.
There is however still room to adjust this presentation such as to yield in each cohomology class special nice cocycle representatives. This we will achieve by finding naturally a subobject $\exp(\mathfrak{g})_{conn} \hookrightarrow \exp(\mathfrak{g})_{diff}$ whose inclusion is an isomorphism on connected components and restricted to which the morphism $curv_{smp} \circ \exp(\mu,cs)$ yields nice representatives in the de Rham hypercohomology encoded by $\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{smp}$, namely globally defined differential forms. On this object the differential characteristic classes we will show factors naturally through the refinements to differential cohomology, and hence $\exp(\mathfrak{g})_{conn}$ is finally identified as a presentation for the the coefficient object for ∞-connections with values in $\mathfrak{g}$.
We discuss presentations for the differential characteristic classes in the image of ∞-Lie integration.
Let $\mathfrak{g} \in L_\infty \stackrel{CE}{\hookrightarrow} dgAlg^{op}$ be an L-∞ algebra.
A L-∞ algebra cocycle on $\mathfrak{g}$ in degree $n$ is a morphism
to the line Lie n-algebra.
Every $L_\infty$-algebra cocycle induces canonically a morphism of simplicial presheaves of exponentiated L-∞-algebras
given componentwise by postcomposition with the image of $\mu$ under $CE(-)$
The Weil algebra $W(\mathfrak{g}) \in dgAlg$ is the unique representative of the free dg-algebra on the chain complex $\mathfrak{g}_\bullet^*[1]$ underlying $\mathfrak{g}$ such that the canonical projection $\mathfrak{g}_\bullet^*[1] \oplus \mathfrak{g}_\bullet^*[2] \to \mathfrak{g}_\bullet^*[1]$ extends to a dg-algebra homomorphism
For $\mathfrak{g} \in L_\infty$ define the simplicial presheaf $\exp(\mathfrak{g})_{diff} \in [CartSp_{smooth}^{op}, sSet]$ by
where on the left we have the set of commuting diagrams in dgAlg as indicated, with the vertical morphisms being the canonical projections.
For $\mathfrak{g} = b^{n-1}\mathbb{R}$ the line Lie n-algebra, this subsumes the previous definition.
The canonical projection
is an acyclic fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$.
Moreover, for every $L_\infty$-algebra cocycle it fits into a commuting diagram
for some morphism $exp(\mu)_{diff}$.
Both claims follow from the free property of the Weil algebra.
For the first, we need to show that for all $U \in$ CartSp we have lifts in all diagrams of the form
The bottom morphism is a collection of differential forms on $U \times \Delta^k$ (with sitting instants), satisfying a flatness condition for their $d_{\Delta^k}$-differentials. The top morphism is a collection of forms on $U \times \partial \Delta^k$ with no flatness constraint except that those with no component along $U$ coincide with the restriction of those of the bottom morphism to the boundary $\partial \Delta^k$. These latter extend to a unique lift to the interior of the simplex. The remaining forms may be smoothly interpolated for instance to 0 in the interior of the simplex, while keeping at least one leg along $U$. Since for the top morphism there is no condition on the differentials, any choice will do.
For the second claim, let $U(CE(\mu))$ be the underlying morphism on chain complexes of $\mu$. Then we have the free dg-algebra homomorphism $F U (CE(\mu)) : W(b^{n-1}) \to W(\mathfrak{g})$ fitting into the commutative diagram
Pasting-precomposition with this diagram yields a morhism $\exp(\mu)_{diff}$ as desired.
Let $G \in Smooth \infty Grpd$ be an n-group given by Lie integration of an L-∞ algebra $\mathfrak{g}$, in that the delooping object $\mathbf{B}G$ is presented by the $(n+1)$-coskeleton simplicial presheaf $\mathbf{cosk}_{n+1}\exp(\mathfrak{g})$.
Then for $X \in [CartSp_{smooth}, sSet]_{proj}$ any object and $\hat X$ a cofibrant resolution, we say that
is the Kan complex of pseudo n-connections on $G$-principal ∞-bundles.
We discuss presentations in $[CartSp_{smooth}^{op}, sSet]$ of the the intrinsic notion of ∞-connections in $Smooth \infty Grpd$.
Let still $\mathfrak{g} \in _\infty \stackrel{CE(-)}{\hookrightarrow} dgAlg^{op}$ be an L-∞ algebra.
An invariant polynomial on $\mathfrak{g}$ is an element $\langle - \rangle \in W(\mathfrak{g})$ such that both $\langle - \rangle \in \wedge^\bullet$ as well as $d_{W(\mathfrak{g})}\; \langle - \rangle$ are elements in the graded subalgebra generated by the shifted generators $\mathfrak{g}^*[1] \hookrightarrow W(\mathfrak{g})$;
Write $inv(\mathfrak{g}) \hookrightarrow W(\mathfrak{g})$ for the sub-dg-algebra of invariant polynomials.
For the line Lie n-algebra we have
This allows us to identify an invariant polynomial $\langle - \rangle$ of degree $n+1$ with a morphism
in dgAlg.
We say an invariant polynomial $\langle - \rangle$ on $\mathfrak{g}$ is in transgression with an L-∞ algebra cocycle $\mu : \mathfrak{g} \to b^{n-1} \mathbb{R}$ if there is a morphism $cs : W(b^{n-1}\mathbb{R}) \to W(\mathfrak{g})$ such that we have a commuting diagram
We say that $cs$ is a Chern-Simons element exhibiting the transgression between $\mu$ and $\langle - \rangle$.
We say that an $L_\infty$-algebra cocycle is transgressive if it is in transgression with some invariant polynomial.
We have
There is a transgressive cocycle for every invariant polynomial.
Any two $L_\infty$-algebra cocycles in transgression with the same invariant polynomial are cohomologous.
Every decomposable invariant polynomial (the wedge product of two non-vanishing invariant polynomials) transgresses to a cocycle cohomologous to 0.
By the fact that the Weil algebra is free, its cochain cohomology vanishes and hence the definition property $d_{W(\mathfrak{g})} \langle -\rangle = 0$ implies that there is some element $cs \in W(\mathfrak{g})$ such that $d_{W(\mathfrak{g})} cs = \langle - \rangle$. Then the image of $cs$ along the canonical dg-algebra homomorphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$ is $d_{CE(\mathfrak{g})}$-closed hence is a cocycle on $\mathfrak{g}$. This is by construction in transgression with $\langle - \rangle$.
Let $cs_1$ and $cs_2$ be Chern-Simons elements for the to given $L_\infty$-algebra cocycles. Then by assumption $d_{(\mathfrak{g})} (cs_1 - cs_2) = 0$. By the acyclicity of $W(\mathfrak{g})$ there is then $\lambda \in W(\mathfrak{g})$ such that $cs_1 = cs_2 + d_{W(\mathfrak{g})} \lambda$.
Since $W(\mathfrak{g}) \to CE(\mathfrak{g})$ is a dg-algebra homomorphism this implies that also $\mu_1 = \mu_2 + d_{CE(\mathfrak{g})} \lambda|_{CE(\mathfrak{g})}$.
Given two nontrivial invariant polynomials $\langle - \rangle_1$ and $\langle - \rangle_2$ let $cs_1 \in W(\mathfrak{g})$ be any element such that $d_{W(\mathfrak{g})}cs_1 = \langle - \rangle_1$. Then $cs_{1,2} := cs_1 \wedge \langle -\rangle_2$ satisfies $d_{W(\mathfrak{g})} cs_{1,2} = \langle - \rangle_1 \wedge \langle -\rangle_2$. By the first observation the restriction of $cs_{1,2}$ to $CE(\mathfrak{g})$ is therefore a cocycle in transgression with $\langle - \rangle_1 \wedge \langle -\rangle_2$. But by the definition of invariant polynomials the restriction of $\langle - \rangle_2$ vanishes, and hence so does that of $cs_{1,2}$. The claim the follows with the second point above.
Define the simplicial presheaf $\exp(\mathfrak{g})_{ChW} \in [CartSp_{smooth}^{op}, sSet]$ by the assignment
where on the right we have the set of horizontal morphisms in dgAlg making commuting diagrams with the canonical vertical morphisms as indicated.
We call $\langle F_A \rangle$ the curvature characteristic forms of $A$.
Let
be the presentation, as above, of the product of all differential refinements of characteristic classes on $\exp(\mathfrak{g})$ induced from Lie integration of transgressive L-∞ algebra cocycles.
We have that $\exp(\mathfrak{g})_{ChW}$ is the pullback in $[CartSp_{smooth}^{op}, sSet]$ of the globally defined closed forms along the curvature characteristics induced by all transgressive $L_\infty$-algebra cocycles:
By the above proposition we have that the bottom horizontal morphims sends over each $(U,[k])$ and for each $i$ an element
of $\exp(\mathfrak{g})(U)_k$ to the composite
regarded as an element in $\mathbf{\flat}_{dR} \mathbf{B}^{n_i+1}_{smp}(U)_k$. The right vertical morphism $\Omega^{n_i + 1}(U) \to \mathbf{\flat}_{dR}\mathbf{B}^{n_i+1}\mathbb{R}_{smp}(U)$ from the constant simplicial set of closed $(n_i+1)$-forms on $U$ picks precisely those of these elements for which $\langle F_A\rangle$ is a basic form on the $U \times \Delta^k$-bundle in that it is in the image of the pullback $\Omega^\bullet(U) \to \Omega^\bullet_{si}(U \times \Delta^k)$.
This shows that $\exp(\mathfrak{g})_{ChW}$ serves as a convenient object on which the differential characteristic classes of $\exp(\mathfrak{g})$ are supported.
For more see connection on a smooth principal ∞-bundle.
For standard references on differential geometry and Lie groupoids see there.
The $(\infty,1)$-topos $Smooth \infty Grpd$ is discussed in section 3.3 of
A discussion of smooth $\infty$-groupoids as $(\infty,1)$-sheaves on $CartSp$ and the presentaton of the $\infty$-Chern-Weil homomorphism on these is in
For references on Chern-Weil theory in Smooth∞Grpd and connection on a smooth principal ∞-bundle, see there.
The results on differentiable Lie group cohomology used above are in
and
which parallels
A review is in section 4 of
Classification of topological principal 2-bundles is discussed in
and the generalization to classification of smooth principal 2-bundles is in