Differential cohomology in an $(\infty,1)$-topos
differential cohomology
Every (∞,1)-topos comes with its intrinsic cohomology. For the special case of an cohesive (∞,1)-topos we define an intrinsic differential cohomology.
A cocycle $g : X \to A$ in the intrinsic cohomology of an (∞,1)-topos $\mathbf{H}$ classifies a principal ∞-bundle $P \to X$ in $\mathbf{H}$. A refinement of this cocycle to a cocycle in intrinsic differential cohomology corresponds to equipping this principal $\infty$-bundle with a connection.
The notion of differential cohomology makes sense when $\mathbf{H}$ is a cohesive (∞,1)-topos or at least an ∞-connected (∞,1)-topos: in this case it comes with an intrinsic fundamental path ∞-groupoid functor $\Pi : \mathbf{H} \to \infty Grpd$ whose internal reflection $(\mathbf{\Pi} \dashv \flat) : \mathbf{H} \stackrel{\to}{\leftarrow} \mathbf{H}$ allows to encode the behaviour of a bare cocycle $X \to A$ under parallel transport along the geometric paths defined by $\Pi$.
The intrinsic differential cohomology in $\mathbf{H}$ is the twisted cohomology induced by the obstruction problem of finding extensions $\nabla_{flat}$ in
If such an extension exists, we say that $\nabla_{flat}$ is a cocycle in intrinsic flat differential cohomology in $\mathbf{H}$, also known as a flat connection or a local system.
In general a given cocycle does not admit a flat differential refinement. But the obstructions to its existence arrange themselves neatly into curvature characteristic classes . The curvature-twisted cohomology induced by these classes we identify with intrinsic differential cohomology.
Throughout, we take $\mathbf{H} \simeq Sh_{(\infty,1)}(C)$ to be a ∞-connected (∞,1)-topos of (∞,1)-sheaves on some (∞,1)-site $C$, with global section essential geometric terminal morphism
and induced internal path ∞-groupoid adjunction
A list of examples is at Ambient (∞,1)-toposes.
For discussion of concrete representative of differential cocycles we make use of models for ∞-stack (∞,1)-toposes in terms of the model structure on simplicial presheaves $[C^{op}, sSet]_{proj,cov}$ obtained from the global model structure on simplicial presheaves on the sSet-site $C$ by left Bousfield localization? at Cech nerve projections.
Moreover, when dealing with fibrant objects in this model structure we make use of standard constructions in categories of fibrant objects, notably the factorization lemma .
In particular for $\mathbf{B} A$ a fibrant object with essentially unique point $* \to \mathbf{B}A$ – modelling the delooping of an ∞-group object $A$ – we write $\mathbf{E}A \to \mathbf{B}A$ for the fibration that is the $G$-universal principal ∞-bundle.
We conceive all of differential cohomology in $\mathbf{H}$ as the theory of extensions along the natural unit morphism $X \to \mathbf{\Pi}(X)$ that includes any object $X$ into its path ∞-groupoid. In the simplest case the extension simply exists and we are left with cocycles on $\mathbf{\Pi}(X)$. These we identify as cocycles in flat differential cohomology, which one may also think of as generalizations of the notion of local systems.
Of particular relevance is the case where the underlying cocycle of a flat differential cocycle is trivial. This we identify below with a cocycle in (nonabelian) de Rham cohomology intrinsic to $\mathbf{H}$.
Intrinsic de Rham cohomology serves as the home of curvature characteristic forms discussed further below in terms of which non-flat differential cohomology is defined. Passing to de Rham coefficients is also related to $\infty$-Lie differentiation in $\mathbf{H}$.
Given objects $X$ and $A$ in $\mathbf{H}$, a flat differential cocycle or $\infty$-local system on $X$ with coefficients in $A$ is a morphism $\mathbf{\Pi}(X) \to A$ in $\mathbf{H}$, equivalently a morphism $X \to \mathbf{\flat}(A)$.
We write
for the ∞-groupoid of flat differential $A$-cocycles on $X$ and accordingly
for the corresponding cohomology set. The constant path inclusion $X \to \mathbf{\Pi}(X)$ induces the projection
The image of a cocycle under this morphism we call the underlying cocycle or bare cocycle of a given flat differential cocycle.
We therefore have an obstruction problem: given a bare underlying cocycle in $\mathbf{H}(X,A)$, what is the obstruction to equipping it with a flat connection and thus lifting it to $\mathbf{H}_{flat}(X,A)$? The next sections provide answers to this question.
Let $g : X \to \mathbf{B}G$ be a cocycle classifying a $G$-principal ∞-bundle $P \to X$. Above we defined a flat differential cocycle that refines $g$ as an extension $\nabla_P : \mathbf{\Pi}(X) \to A$ of $g$ along the constant path inclusion
It is sometimes useful to reformulate this extension problem as a lifting problem. We would like to have an object $At(P) \to \mathbf{\Pi}(X)$ in $\mathbf{H}$ such that a section $\hat \nabla_g$ of this projection
corresponds to a choice of flat differential cocycle $\nabla_g$ refining $g$.
For $G$ an ordinary Lie group and $P$ an ordinary $G$-principal bundle, the nature of the object $At(P)$ is well known: it is the Atiyah Lie groupoid of the principal bundle $P$, a Lie groupoid whose Lie algebroid is the Atiyah Lie algebroid of $P$.
The following definition is supposed to generalize this notion to the general context of a locally $\infty$-connected $(\infty,1)$-topos $\mathbf{H}$
For a cocycle $g : X \to A$, a morphism in $\mathbf{H}$, we say its Atiyah $\infty$-groupoid is the object
The universal property of the fiber product $X \times_A \mathbf{\flat}(A)$ says that sections $\sigma : X \to X \times_A \mathbf{\flat}(A)$
correspond precisely to lifts
But by the adjunction $(\Pi \dashv LConst \dashv \Gamma)$ we have (see essential geometric morphism) that this corresponds to extensions
Similarly by the adjunction property, we have that every such section also gives a lift
Of particular interest is the special case of Atiyah $\infty$-groupoids over the point . These we identify in the next section with coefficient objects for closed differential forms.
We describe now an abstract intrinsic notion of (abelian and nonabelian) de Rham cohomology in any locally ∞-connected (∞,1)-topos $\mathbf{H}$ as a constrained form of intrinsic flat differential cohomology in $\mathbf{H}$ as described above. We give intrinsic definitions for fundamental de Rham cohomological concepts such as the the Maurer-Cartan form-datum on an ∞-group object in $\mathbf{H}$ and show that in the case $\mathbf{H} =$ ∞LieGrpd these definitions reproduce ordinary smooth ∞-Lie algebroid valued differential forms.
Below in Differential cohomology with groupal coefficients we show that the obstructions to lifts from bare intrinsic cohomology of $\mathbf{H}$ to cocycles in intrinsic flat differential cohomology live precisely in this intrinsic de Rham cohomology: these are the intrinsic curvature characteristic forms.
(intrinsic de Rham coefficients)
Let $* \to A$ be a pointed object in $\mathbf{H}$. We write $A_{dR}$ or $\mathbf{\flat}_{dR} A$ for the Atiyah ∞-groupoid of the unique $A$-principal ∞-bundle over the point, i.e. for the (∞,1)-pullback
in $\mathbf{H}$.
Similarly for $X \in \mathbf{H}$ any object, we write $\mathbf{\Pi}(X)/X$ or $\mathbf{\Pi}_{dR}(X)$ for the pushout
In the model $[C^{op}, sSet]_{proj,cov}$ of $\mathbf{H}$ let $(\mathbf{\Pi}_R \dashv \mathbf{\flat}_R)$ be a Quillen adjunction that model the adjoint (∞,1)-functors $(\mathbf{\Pi} \dashv \mathbf{\flat})$ such that the counit components $\mathbf{\flat}_R(A) \to A$ are fibration (see path ∞-groupoid for onstructions of such models).
Then for $X$ given by a cofibrant object in $[C^{op},sSet]_{proj}$, the object $\mathbf{\Pi}_{dR}(A) \in \mathbf{H}$ is represented by the ordinary pushout
and is hence in particular itself cofibrant.
Similarly, for $A$ a fibrant representative the object $\mathbf{\flat}_{dR} A$ is constructed by the ordinary pullback
in $[C^{op}, sSet]$, and is in particular itself fibrant.
An equivalent way to compute this is using the global model structure on functors $[I,sPSh(C)_{proj}^{loc}]_{proj}$, where $I = \Delta[1] = \{0 \to 1\}$ is the interval. Notice that a cofibration between cofibrant objects in $sPSh(C)_{proj,cov}$ is a cofibrant object in $[I,sPSh(C)_{proj,cov}]_{proj}$.
Since all representables $U$ are cofibrant in $sPSh(C)_{proj,cov}$ and since $U \to \mathbf{\Pi}_R(U)$ is a cofibration in $sPSh(C)_{proj,cov}$ it is a cofibrant object in the functor category. Therefore for $A$ fibrant in $\mathbf{H}$ we have that $\mathbf{\flat}_{dR}A$ is presented in $[I,sPSh(c)_{proj,loc}]_{proj}$ by
By the standard lore about computing homotopy pullbacks and homotopy pushouts.
Formation of loop space objects commutes with passing to de Rham coefficients, in that
For $* \to A$ pointed, we have a natural equivalence
… details …
(intrinsic de Rham cohomology)
We call
the intrinsic de Rham cohomology of $X$ with coefficients in $A$.
A cocycle in $\mathbf{H}_{dR}(X,A)$ is given by a diagram
in $\mathbf{H}$. However in terms of the model $[I,sPSh(C)]$ every such cocycle is represented, up to equivalence, by a strictly commuting diagram
for $A$ fibrant and $Y$ a cofibrant representative of $Y$. In practice one is often faced with flat differential cocycles modeled by morphisms $\nabla : \mathbf{\Pi}_R(Y) \to A$ whose underlying bare cocycle $g : Y \to \mathbf{\Pi}_R(Y) \to A$ is not yet trivialized, but only trivializable. Given any such trivialization $\eta : g \to *$ one would like to lift it to a coboundary $\hat \eta : \nabla \to \nabla'$ that identifies an equivalent flat differential cocycle $\nabla'$ whose underlying bare cocycle is exactly trivial.
This can always be achieved by solving a lifting problem. The given data specifies the following diagram, except for the diagonal morphism $\hat \eta$:
Here $A^I$ is the path object of $A$ used to present this right homotopy, and therefore $d_0 : A^I \to A$ is an acyclic fibration. Since moreover the constant path inclusion $Y \hookrightarrow \mathbf{\Pi}_R(Y)$ is a cofibration, a lift $\hat \eta$ exists as indicated.
The resulting diagram
is a cocycle representative in the model $[I,sPSh(C)]$ of the cocycle $\nabla$ in $\mathbf{H}_{dR}(Y,A)$ that we started with.
A k-morphism in $A_{dR}$(is modeled in $[I,sPSh(C)_{proj,loc}]_{proj}$ by a diagram
In particular this means that coboundaries given by $A_1$-valued functions on $U$ may appear in $A_{flat}(U)$ but not in $\mathbf{\flat}_{dR}A(U)$. For $A$ 1-truncated but not 1-connected all coboundaries in $\mathbf{H}_{flat}(X,A)$ disappear in $\mathbf{H}_{dR}(X,A)$, such that every cocycle in $\mathbf{H}_{dR}(X,A)$ represents its own cohomology class.
As we shall see, this maybe somewhat curious phenomenon is precisely correct for $\mathbf{\flat}_{dR} A$ to be the right coefficient object for general $\infty$-categorical curvatures: it implies that, in contrast to all higher degree form curvature?s, 1-form curvatures – those that arise in lowest degree for Lie algebroid-valued differential cocycles – are not to be taken up to de Rham coboundaries (doing which would make them all trivial). More on this in the section Example: Lie algebroid-valued connections.
The intrinsic flat differential forms that we described are in general cocycles in nonabelian de Rham cohomology: a generalization of Lie-algeba valued 1-forms? for general Lie algebras. We now describe in more detail how the intrinsic flat differential forms may be thought of as being ∞-Lie algebroid valued differential forms.
Above we defined a cocycle in the intrinsic de Rham cohomology in $\mathbf{H}$ of $X \in \mathbf{H}$ with coefficients in $A \in \mathbf{H}$ as a morphism
Being a morphism out of a left adjoint this factors through a universal morphism $\mathbf{\Pi}_{dR}\mathbf{flat}_{dR}A \to A$ as
We will now identify $\exp(\mathfrak{a}) := \exp(Lie(A)) := \Pi_{dR}\mathbf{\flat}_{dR} A$ with the intrinsic incarnation of the exponentiated ∞-Lie algebroid underlying $A \in \mathbf{H}$ and morphisms $\mathbf{\Pi}_{dR}(X) \to A$ accordingly as generalizations of ordinary flat Lie-algebra valued 1-forms: ∞-Lie algebroid valued differential forms.
(Lie differentiation to $\infty$-Lie algebroids)
We write
and for $A \in \mathbf{H}$ call $\exp(\mathfrak{a}) := \exp(Lie(A))$ the intrinsic exponentiated ∞-Lie algebroid of $A$.
For $G$ an ordinary Lie group regarded naturally as an ∞-group inside $\mathbf{H} =$ ∞LieGrpd we have that $\mathbf{\flat}_{dR} \mathbf{B}G$ is represented in the model by simplicial presheaves $[CartSp^{op}, sSet]_{proj,cov}$ by the sheaf of flat Lie-algebra valued 1-forms
This and further examples are discussed at ∞-Lie groupoid. See the section Lie group – differential coefficients.
If $\mathbf{H}$ is given as an infinitesimal thickening of an underlying (∞,1)-topos $\mathbf{H}_{red}$
then we may extract from such an intrinsic definition of $\infty$-Lie algebroids a more explicit (but still intrinsic) definition of differential forms along the lines of synthetic differential geometry. Moreover, such synthetic differential forms with values in $A$ will factor through the ∞-Lie algebroid $\mathfrak{a} \hookrightarrow A$ of $A$ and thus provide a generalization of ordinary Lie algebra valued differential forms to ∞-Lie algebroid valued differential forms.
Recall from the discussion at path ∞-groupoid that from the above infinitesimal thickening is induced the Infinitesimal path ∞-groupoid adjunction
Accordingly we get analogs of the above definitions of de Rham objects.
Write
and for pointed $* \to A$ in $\mathbf{H}$
The canonical natural inclusion
of infinitesimal paths into all paths accordingly induces natural inclusions
As before we have on pointed objects adjoint (∞,1)-functors
The study of this adjunction is effectively the study of Lie theory inside $\mathbf{H}$.
(Lie differentiation and $\infty$-Lie algebroids)
For $A \in \mathbf{H}$ we say that
is the ∞-Lie algebroid of $A$.
Notice that the counit of the adjunction gives a natural inclusion
(flat $\infty$-Lie algebroid valued differential forms)
By the characterization of adjunctions in terms of universal factorizations we have by a factorization
in $\mathbf{H}$
We call $\mathbf{\Pi}_{inf}(X) \to \mathfrak{a}$ here flat ∞-Lie algebroid valued differential forms whose integrated parallel transport is the given morphism denoted $P \exp(\int \omega) : \mathbf{\Pi}(X) \to A$.
By the general formula for right adjoints in presheaf categories we have the expression
in terms of an (∞,1)-colimit.
This means that $\mathfrak{a}$ is the universal object such that morphisms $\mathbf{\Pi}_{inf,dR}(U) \to A$ factor through it.
We have given above a definition of coefficient objects $\mathbf{\flat}_{dR}A$ for intrinsic nonabelian de Rham cohomology in terms of the homotopy fiber of the morphism $\mathbf{\flat} A \to A$. By the general logic of fiber sequences it follows that when $A = \mathbf{B}G$ is a delooping of an ∞-group $G$ we obtain a canonical morphism $G \to \mathbf{\flat}_{dR} \mathbf{B}G$. The following examples show that this is the general abstract generalization of the Maurer-Cartan form on a Lie group.
Let $G \in \mathbf{H}$ be an ∞-group object and write $\mathbf{B}G$ for its delooping object.
Consider the pasting of two (∞,1)-pullback diagrams
The object $\mathbf{\flat}_{dR}\mathbf{B}G$ is the lower pullback by definition, the object $G$ is the upper pullback since by the pasting law for (∞,1)-pullbacks it is also the pullback of the outer rectangle, which is $G$ by definition of the delooping $\mathbf{B}G$.
The canonical morphism
defined this way, which by the above corresponds to a flat $\mathfrak{g}$-valued differential form datum, we call the $\infty$-Maurer-Cartan form on $G$.
For $G$ an ordinary Lie group regarded naturally as a group object in $\mathbf{H} =$ ∞LieGrpd, $\theta$ is the ordinarty Maurer-Cartan form $\theta = g^{-1} d g$ in that the morphism $G \to \mathbf{\flat}_{dR}\mathbf{B}G$ is represented in the model $[CartSp^{op}, sSet]_{proj,cov}$ by the morphism of sheaves
given over $U \in CartSp$ by
where $\Omega^1_{flat}(-,\mathfrak{g})$ is the sheaf of Lie-algebra valued 1-forms whose curvature 2-form vanishes.
This is discussed at ∞-Lie groupoid in the section Canonical form on a Lie group.
We have given above a definition of the canonical $\mathfrak{g}$-valued flat differential Maurer-Cartan form on an ∞-group $G$. The construction has a straightforward generalization to canonical form on the fibers of a $G$-principal ∞-bundle $P$, generalizing the vertical $\mathfrak{g}$-valued forms appearing in Ehresmann connections.
For $P \to X$ the $G$-principal ∞-bundle classified by a cocycle $X \to \mathbf{B}G$, for each point $x : * \to X$ the pasting diagram of (∞,1)-pullback squares
exhibits the canonical flat $\mathfrak{g}$-valued vertical 1-form
on the fiber $P_x$ of $P$ over $x$.
For $G$ an ordinary Lie group, $P \to X$ an ordinary smooth $G$-principal bundle, all regarded naturally as objects in $\mathbf{H} =$ ∞LieGrpd this defintiion reproduces the ordinary notion of flat vertical $\mathfrak{g}$-valued differential form.
For $G$ an ordinary Lie group, and $p : P \to X$ an ordinary $G$-principal bundle, a basic form on $P$ is a differential form $\omega \in \Omega^\bullet(P)$ that is invariant under the $G$-action on $P$. A standard fact is that basic forms are precisely the image of the forms on $X$ under pullback along $p$. The following is the analog of this statement in an $(\infty,1)$-topos. This is just a formal tautology, which is however sometimes useful to make explicit.
(basic forms on a $G$-principal $\infty$-bundle)
For $G$ an ∞-group in $\mathbf{H}$ and $P \to X$ a $G$-principal ∞-bundle, an intrinsic differential form $\omega \in \mathbf{H}_{dR}(P, A)$ on $P$ for any coefficient object $A$ is called basic precisely if it is homotopy $G$-invariant in that for its representing morphism $\omega : P \to \mathbf{\flat}_{dR} A$ there exists an extension to a cocone
under the action-groupoid object of $G$ acting on $P$ (see principal ∞-bundle for the details). We write
for the ∞-groupoid of basic $A$-valued forms on $P$, being the $\infty$-groupoid of cocones in $\mathbf{H}$ under the action groupoid object $P\times G^{\times \bullet}$ with tip $\mathbf{\flat}_{dR} A$.
Pullback of forms along the bundle projection $p : P \to X$ establishes an equivalence between forms on $X$ and basic forms on $P$:
Up to notation and terminology, this is just a restatement of the fact that $X$ is the $(\infty,1)$-colimit over the action groupoid object
as discussed at principal ∞-bundle.
Objects in $\mathbf{H}$ of the form $LConst S$ for $S \in$ ∞Grpd have the interpretation of being discrete ∞-Lie groupoids in the sense of discrete Lie groups. The latter have trivial Lie algebras and accordingly the former are to be expected to have trivial ∞-Lie algebroids.
That this is indeed the case follows immediately from the above definitions.
For $\mathbf{H}$ a ∞-connected (∞,1)-topos and for $* \to S \in$ ∞Grpd a pointed ∞-groupoid, we have
in $\mathbf{H}$.
By the assumption that $\mathbf{H}$ is $\infty$-connected we have that $LConst : \infty Grpd \to \mathbf{H}$ is a full and faithful (∞,1)-functor and hence that $\Gamma \circ LConst \simeq Id$.
It follows that $\mathbf{\flat}_{dR} LConst S$ is the pullback
and hence itself equivalent to the point. Then so is $Lie LConst S = \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} LConst S$.
While this is almost tautological, it still has noteworthy applications when applied to objects $\mathbf{\Pi}(X) = LConst \Pi(X)$ in the image of the homotopy ∞-groupoid? functor in view of its realization as the path ∞-groupoid.
We first record the following evident statements.
If $\mathbf{H}$ is ∞-connected, then the geometric realization / homotopy $\infty$-groupoid of $\mathbf{\Pi}_{dR}(X)$ is trivial:
Moreover we have for all $X \in \mathbf{H}$
$\mathbf{\Pi}_{dR}\mathbf{\Pi}(X) \simeq *$;
$\mathbf{\flat}_{dR} \mathbf{\Pi}(X) \simeq *$
Since $\Pi$ is left adjoint it preserves the pushout that defines $\mathbf{\Pi}_{dR}(X)$ so that the geometric realization is itself the pushout in
Now using the assumption that $\mathbf{H}$ is $\infty$-connected we have $\Pi \circ LConst \simeq Id$ and hence $\Pi(*) \simeq *$ and $\Pi \mathbf{\Pi}(X) = \Pi LCont \Pi(X) \simeq \Pi(X)$. Therefore the above pushout is equivalent to
The right vertical morphism is therefore a pushout of an equivalence and hence itself an equivalence.
Also by preservation of pushout we have
This now implies $\mathbf{\Pi}_{dR}(\mathbf{\Pi}(X)) \simeq *$, which is also immediately seen directly by using again from the ∞-connectedness that $\mathbf{\Pi} \mathbf{\Pi}(X) \simeq \mathbf{\Pi}(X)$.
Analogously we have $\mathbf{\flat} \mathbf{\Pi}(X) := LConst \Gamma LConst \Pi X \simeq LConst \Pi(X) = \mathbf{\Pi}(X)$ and hence the pullback
yields $\mathbf{\flat}_{dR} \mathbf{\Pi}(X) \simeq *$.
For $* \to A \in \mathbf{H}$ pointed, we have that the tangent $\infty$-Lie algebroid
This is a version of the following familiar fact: for $\mathfrak{g}$ an L-∞-algebra dual to its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$,
the corresponding Weil algebra $W(\mathfrak{g})$, which may be regarded as the Chevalley-Eilenberg algebra of the tangent $L_\infty$-algebra $T \mathfrak{g}$, has trivial cohomology.
…
The ordinary de Rham theorem asserts an equivalence between de Rham cohomology? of a smooth manifold and its “ordinary cohomology” (meaning: singular cohomology or anything equivalent to it) with coefficients the additive discrete group $\mathbb{R}$ of real numbers.
The natural generalization of this statement holds for the intrinsic de Rham cohomology and intrinsic cohomology in an $\infty$-connected $(\infty,1)$-topos.
(de Rham theorem)
Write $R$ for the intrinsic line object in $\mathbf{H}$, with $\Pi(R) \simeq *$. Let $X \in \mathbf{H}$ be such that $\mathbb{R}$-valued cohomology on $X$ is trivial. (For instance in $\mathbf{H} =$ ∞-LieGrpd? for $X$ a connected smooth manifold, but not for instance the path ∞-groupoid of a manifold.)
Write $|X|$ for the geometric realization of $X$, the image of $X$ under $\mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{\simeq}{\to} Top$. Notice that by adjunction
where on the left we have odinary real cohomology of $|X|$ (real singular cohomology) and on the right its reflection in $\mathbf{H}$.
Then: there is a natural isomorphism of cohomology sets
This even comes from a natural equivalence of cocycle $\infty$-groupoids
By the definition of intrinsic de Rham cohomology, this is essentially a tautology, since by the assumption that $\mathbf{H}(X,\mathbf{B}^n R) \simeq *$ we have
While this is a trivial formal consequence of the abstract definition of intrinsic de Rham cohomology, the point is that this abstract defrinition indeed captures ordinaty de Rham cohomology.
We define and study now differential cohomology in $\mathbf{H}$ for the case that the coefficient object $A \in \mathbf{H}$ is a group object, so that its delooping object $\mathbf{B}A$ exists. In this case there is a characteristic class $curv_A : A \to \mathbf{\flat}_{dR}\mathbf{B}A$ with values in the intrinsic de Rham cohomology with values in $\mathbf{B}A$ that measures curvature? of $A$-cocycles (the canonical Maurer-Cartan form on $A$), and we define differential $A$-cohomology to be the twisted cohomology induced by this class.
It turns out that this twisted cohomology is an exact obstruction theory: an $A$-cocycle $g : X \to A$ has an extension to flat differential cohomology if and only if its curvature class $curv_A(g)$ is trivial in cohomology.
The same is not true if $A$ is not groupal, i.e. if we are looking at differentuial refinements of fully nonabelian cohomology. Below in the section Differential cohomology with non-groupal coefficients we consider a more general but less exact curvature characteristic class that applies to arbitrary coefficients – hence also to fully nonabelian cohomology and its differential refinements – and which is in some sense the universal best approximation to the curvature class that we consider here.
We produce now a characteristic class, naturally defined on groupal objects $A \in \mathbf{H}$, that measures the obstructions to lifts of $A$-cocycles through $\mathbf{\flat}(A) \to A$ to flat differential $A$-cocycles.
(differential fibration sequence)
If $A$ is a group object with delooping $\mathbf{B}A$, then we have a fiber sequence
For groupal $A$, we call the morphism
from the above diagram the curvature characteristic class of $A$.
This is just the observation of the Maurer-Cartan form repeated – but one degree up: typically we have $A = \mathbf{B}G$ for $G$ an ∞-group so that $A$-cohomology classifies $G$-principal ∞-bundles. Then $\mathbf{B}A = \mathbf{B}^2 G$ is the 2-fold delooping of $G$ and the curvature characteristic form of $G$ is
which is the Maurer-Cartan form of the $\infty$-group $\mathbf{B}G$.
While formally trivial, this definition has the following important consequence.
By the fact that the $(\infty,1)$-hom-functor preserves (∞,1)-limits, it follows that for every $X \in \mathbf{H}$ we have a fiber sequence of cocycle ∞-groupoids
and hence a long exact sequence of pointed cohomology sets
This says exactly that for $g : X \to A$ an $A$-cocycle, the obstruction to lifting it to a flat differential $A$ cocycle through $H_{flat}(X,A) \to H(X,A)$ is the nontriviality of its image under $H(X,A) \to H_{dR}(X,\mathbf{B}A)$.
Notice that if $A = \mathbf{B}^n K$ is an Eilenberg-MacLane object for an abelian goup object $K$, the above long exact sequence may be written in possibly more familiar form equivalently as
By the intrinsic de Rham theorem, we may think of the morphism $H^n(X,K) \to H_{dR}^{n+1}(X,K)$ here as being the generalization of producing an image of an integral cohomology class in real cohomology. From classical theory we expect that this image is measured by curvature data, and that curvature is the obstruction to flatness. This is the situation that the above fiber sequence formalizes. For a detailed discussion of how this connects to standard theory see the examples.
Above we defined for an object $A \in \mathbf{H}$ with delooping $\mathbf{B}A$ the curvature class $curv : A \to \mathbf{\flat}_{dR}\mathbf{B}A$. This we now use to define differential cohomology with coefficients in $A$.
(differential cohomology with groupal coefficients)
For $A \in \mathbf{H}$ a group object write ${H}_{diff}(-,A)$ for the twisted cohomology induced by the curvature characteristic class $A \to \mathbf{\flat}_{dR}\mathbf{B}A$, i.e. for the connected components $H_{diff}(X,A) := \pi_0 \mathbf{H}_{diff}(X,A)$ of the homotopy pullback
For $c \in \mathbf{H}_{diff}(X,A)$ a cocycle, we call
$F(c) \in H_{dR}(X,\mathbf{B}A)$ the curvature class of $c$;
$[\eta(c)] \in H(X,A)$ the underlying class in $A$-cohomology.
Recall that $H_{dR}(X,A) := H(X, \mathbf{\flat}_{dR} A)$ denotes the intrinsic de Rham cohomology of $\mathbf{H}$, as discussed in the section on intrinsic de Rham cohomology.
(differential fiber sequence)
Differential cohomology fits into a fiber sequence
This is a general statement about the definition of twisted cohomology: consider the diagram
The square on the right is a pullback by definition of twisted cohomology in general and our special case of differential cohomology in particular. Take the left square to be the pullback of the middle vertical morphism to the point and deduce the top left object from that: by the pasting law for (∞,1)-pullbacks this top left object is the pullback of the total diagram. But by the definition of $H(X,\mathbf{\flat}_{dR}\mathbf{B}A)$ as the set of connected components of $\mathbf{H}(X,\mathbf{\flat}_{dR}\mathbf{B}A)$ it follows that the pullback of the outer diagram is
Finally using that (as discussed at cohomology and at fiber sequence) $\Omega \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}A) \simeq \mathbf{H}(X,\Omega \mathbf{\flat}_{dR} \mathbf{B}A)$ and then using the above observation that $\Omega \mathbf{\flat}_{dR} \mathbf{B}A \simeq \mathbf{\flat}_{dR} \Omega \mathbf{B}A$ and finally the defining equivalence $\Omega \mathbf{B}A \simeq A$ the claim follows.
Let $\mathbf{B}^n K$ be an Eilenberg-MacLane object in $\mathbf{H}$, then differential cohomology in $\mathbf{H}$ fits into a short exact sequence
The above fiber sequence yields (as recalled there) a long exact sequence of pointed cohomology sets
If $A = \mathbf{B}^n K$ is an Eilenberg-MacLane object on an abelian group object $K$, then this reads
Moreover observing that by construction the last morphism $H_{diff}^n(X,K) \to H^n(X,K)$ is surjective (because in the defining $(\infty,1)$-pullback for $\mathbf{H}_{diff}$ the right vertical morphism is evidently surjective on connected components) this yields the short exact sequence as claimed.
Warning. This is essentially verbatim the expected short exact sequence familiar from ordinary generalized differential cohomology only up to the following slight nuances in notation:
The cohomology groups of the short exact sequence above denote the groups obtained in the given (∞,1)-topos $\mathbf{H}$, not in Top. Notably for $\mathbf{H} =$ ∞LieGrpd and $|X| \in Top$ the geometric realization of a paracompact manifold $X$, we have that $H^1(X,U(1))$ above is $H^2_{sing}(|X|,\mathbb{Z})$ and not $H^1_{sing}(|X|,U(1))$.
The fact that on the left of the short exact sequence we find the intrinsic de Rham cohomology set $H_{dR}^n(X,A)$ instead of something like the set of all flat forms as familiar from discussions of ordinary generalized differential cohomology is due to the fact that, following the general abstract procedure of twisted cohomology, we defined $\mathbf{H}_{diff}(X,A)$ as the $(\infty,1)$-pullback of the inclusion $H(X,\mathbf{\flat}_{dR}A ) \to \mathbf{H}(X,\mathbf{\flat}_{dR} A)$ of the set of connected components of curvature classes into the cocycle ∞-groupoid. This is the only natural thing to do in a fully natural (∞,1)-categorical setup. However, in applications we typically have a concrete model for the ∞-groupoid $\mathbf{H}(X,\mathbf{\flat}_{dR} A)$ in mind, and then can consider the inclusion $\mathbf{H}(X,\mathbf{\flat}_{dR} A)_0 \hookrightarrow \mathbf{H}(X,\mathbf{\flat}_{dR} A)$ of all of its objects. While this does not make intrinsic sense – it is a bit evil – , this is what is effectively done in ordinary generalized differential cohomology, and doing so in our definition changes the above short exact sequence slightly to become exactly the familiar sequence, for the suitable special cases.
A detailed discussion of how ordinary differential cohomology is reproduced from the above definition is in the examples.
Above we have given a general abstract definition of differential cocycles in terms of certain homotopy fibers. It is useful to concretely model these cocyclea in terms of intrinsic differential form data on the underlying underlying base $X$ space or the underlying principal ∞-bundle $P \to X$. Such data is called a connection on $P$.
We describe now a way to encode differential cocycles with groupal coefficients $A$ in terms of explicit Cech cocycles with values in “pseudo-connection”-forms that are constrained to take values in genuine connection forms.
The following makes statements about explicit representatives of differential cocycles in $\mathbf{H}$ in terms of the model by simplicial presheaves $[C^{op}, sSet_{Quillen}]_{proj,cov}$. Moreover, we invoke the model for the arrow category of $\mathbf{H}$: write $I = \Delta[1]$ for the interval, regarded as a category with two objects and a single nontrivial morphism. Then on the functor category $[I,[C^{op}, sSet]]$ we have the global projective model structure on functors $[I,[C^op, sSet]_{proj,cov}]_{cov}$.
An object in this model is cofibrant precisely if it is given by a morphism in $[C^{op}, sSet]_{proj,cov}$ which is a cofibration between cofibrant objects. We use in particular that for $U \in C$ a representable simplicial presheaf, the morphism $U \to \mathbf{\Pi}_R(U)$ is such a cofibration between cofibrant objects, as discussed at path ∞-groupoid.
An object in this model is fibrant if it is a morphism between fibrant objects. We assume we have fixed a fibrant model for $A$ in $[C^{op}, sSet]_{proj,cov}$ which we denote by the same letter. We write $\mathbf{B}A$ for its fibrant delooping model and $\mathbf{E}A := (\mathbf{B}A)^I \times_{\mathbf{B}A} *$ for the standard model for the point, that makes $\mathbf{E}A \to \mathbf{B}A$ a fibration. We use the standard properties of this resolution as described at category of fibrant objects.
We have
The object $A \in [C^{op}, sSet]$ is isomorphic to the simplicial presheaf
The object $\mathbf{\flat}A$ is represented by the simplicial presheaf
and similarly $\mathbf{\flat}\mathbf{B}A$ by
The object $\mathbf{\flat}_{dR}\mathbf{B}A$ is represented by the simplicial presheaf
The simplicial presheaf $A_{diff}$ defined by
is weakly equivalent to $A$ in $[C^{op}, sSet]_{proj}$, where the weak equivalence $A_{diff} \stackrel{\simeq}{\to} A$ is induced by postcomposition with the morphism
in $[I,[C^{op}, sSet]]_{proj}$.
In terms of these models the curvature characteristic class $curv_A : A \to \mathbf{\flat}_{dR}\mathbf{B}A$ is modeled by the span
where the morphism on the right is given by postcomposition with the morphism
in the arrow category.
We observe that the defining pasting diagram of (∞,1)-pullbacks
in $\mathbf{H}$ is modeled by the pasting diagram of ordinary pullbacks
Here we use that the diagram of arrows on the right is a pullback diagram of fibrant objects in $[I,[C^{op}, sSet]_{proj, cov}]_{proj}$ with the two right vertical morphisms being fibrations, and that homming a cofibrant arrow $(U \to \mathbf{\Pi}_R(U))$ into this makes this (by the defining property of the sSet-enriched model category $[I,[C^{op}, sSet]_{proj,cov}]_{proj}$) a diagram with the same properties in $sSet_{Quillen}$. Therefore the entrie diagram above is in turn a pullback diagram of fibrant objects with the two right morphisms being fibrations in $[C^{op}, sSet]_{proj,cov}$.
Now the right bottom corner of this diagram in the model clearly models the right bottom corner of the above diagram in $\mathbf{H}$. Its pullback is therefore the correct homotopy pullback modelling the bottom (∞,1)-pullback. Analogously then for the top pullback diagram.
The next lemma asserts that the curvature characteristic morphism is also globally modeled by just precomposition with the fibration $\array{ A &\to& * \\ \downarrow && \downarrow \\ \mathbf{E}A \to \mathbf{B}A}$ in the arrow category.
For any $Y \in [C^{op}, sSet]_{proj}$ cofibrant, the morphism
in sSet induced by the curvature characteristic class $A \leftarrow A_{diff} \to \mathbf{\flat}_{dR}\mathbf{B}A$ as modeled above is given by the morphism
in that domain and codomain objects are isomorphic and the two morphisms are equal under this isomorphism.
Using end-calculus for describing the sSet-hom-objects in the enriched functor categories we have
where we set $U_0 := U$, $U_1 := \mathbf{\Pi}_R(U)$, $A_0 := A$ and $A_1 := \mathbf{E}A$ and where we have used that the enriched hom preserves ends in the second argument and that by the Fubini theorem for ends the two ends commute.
Now we compute for $i = 0$ in the integrand
where in the first step we use the tensoring of $sPSh(C)$ over $sSet$, in the second step that the enriched hom takes coends in the first argument to ends, and finally the co-Yoneda lemma that expresses a presheaf as a coend over representables.
Analogously we compute for $i = 1$
where now in the last step we used the definition of the path ∞-groupoid functor $\mathbf{\Pi}_R$ by left Kan extension.
This shows the isomorphism of the simplicial sets in question. Since all of this is entirely natural in all variables involved, we have the equality of morphism as claimed.
(characterization of differential cocycles)
Let $A \in [C^{op},sSet]_{proj,cov}$ be a fibrant representative of a group object in $\mathbf{H}$ and let $Y \in [C^{op},sSet]$ be a cofibrant representative of an object $X \in \mathbf{H}$.
Let $[P] \in H_{dR}(X,\mathbf{B}A)$ be a fixed curvature class, represented by some fixed $P : Y \to \mathbf{\flat}_{dR}\mathbf{B}A$. Then the cocycles and coboundaries in the differential cohomology of $X$ with coefficients in $A$ for curvature characteristic $P$, i.e. the (∞,1)-pullback
is given as follows:
a cocycle is a commuting diagram
in $[C^{op}, sSet]$ such that the bottom composite in
equals $P$.
The underlying cocycle is $\eta(g,\nabla) = g : Y \to A$.
We call $\mathbf{\Pi}_R(Y) \to \mathbf{E}A$ the connection encoded by the differential cocycle and $P_j$ the curvature of this connection.
A coboundary between two such cocycles is a strictly commuting diagram of 2-cells in $sPSh(C)$ of the form
such that this induces the identity on $P$, i.e. such that
commutes strictly.
Using the above lemmas we have that the (∞,1)-pullback in question is modeled in $sSet_{Quillen}$ by the ordinary pullback
All homs involved here go from cofibrant to fibrant objects. Since in a simplicial model category the enriched hom out of a cofibrant object preserves fibrations, this is a fibration in sSet, i.e. a Kan fibration. Therefore the pullback diagram of the point along this morphism is a fibrant diagram and the homotopy pullback in question is computed by the ordinary pullback in $sSet$. The cells of this ordinary pullback in degrees 0 and 1 is what the above proposition describes.
(connections)
Conceptually the crucial aspect here is the way $A_{diff}$ serves to produce the span $A \stackrel{\simeq}{\leftarrow} A_{diff} \to \mathbf{\flat}_{dR}\mathbf{B}A$ and induced from that the span
that exhibits a map from $A$-cocycles to de Rham-cocycles obtained by first lifting any given $A$-cocycle $g : Y \to A$ to an $A_{diff}$-cocycle $(g,\nabla)$ and then computing from the lift $\nabla$ the corresponding characteristic class $[P(\nabla)] \in H(X,\mathbf{\flat}_{dR}\mathbf{B}A)$.
This lift is a choice of connection on the underlying $A$-cocycle $g$, respectively on the principal ∞-bundle that it classifies. With this interpretation we can label the components of the diagram that characterizes a differential cocycles according to the above corollary as follows
The fact that $sPSh(Y,A_{diff}) \to sPSh(Y,A)$ is a weak equivalence says that every equivalence class of $A$-cocycles admits a connection. But in fact this morphism is even an acyclic fibration, since $A_{diff} \to A$ is, and therefore surjective on objects. This implies in particular that every single $A$-cocycle admits a lift to $A_{diff}$, which in light of the above we may state as:
Every $A$-principal ∞-bundle admits a connection.
Notice from the above, following the logic of twisted cohomology, that a choice of connection may be understood as an attempt to extend the underlying cocycle $g : Y \to A$ to a flat differential cocycle $\mathbf{\Pi}_R(Y) \to A$. Because the attempt may fail, $\nabla$ takes values not in $A$, but in the contractible $\mathbf{E}A$. The composite curvature $F(\nabla) : \mathbf{\Pi}_R(Y) \stackrel{\nabla}{\to} \mathbf{E}A \stackrel{p_A}{\to} \mathbf{B}A$ projects out, precisely up to equivalence, the failure of this attempt:
by the general logic of twisted cohomology, an $A$-cocycle $g \in \mathbf{H}(X,A)$ admits a lift through $\mathbf{H}(\mathbf{\Pi}_R(X), A) \to \mathbf{H}(X,A)$ to a flat differential cocycle – i.e. the principal ∞-bundle it classifies admits a flat connection – precisely if its curvature characteristic class, its image $[P(A)]$ under the morphism $\mathbf{H}(X,A) \to H(X,\mathbf{\flat}_{dR}\mathbf{B}A)$, is the trivial class.
For $\mathb{B}^n A$ an Eilenberg-MacLane object it typically happens that $\mathbf{H}(\mathbf{B}^n A, \mathbf{\flat}_{dR}) \mathbf{B}^{n+1}A)$ is 0-truncated. Equivalently, that all the intrinsic de Rham cohomologies in degree $k \lt n+1$ are trivial, $H(\mathbf{B}^n A, \mathbf{\flat}_{dR}\mathbf{B}^k A) = *$.
Example. This is the case in $\mathbf{H} =$ ∞LieGrpd for $A = U(1)$ the circle group. The discussion is at Intrinsic de Rham cohomology of Bu U(1).
In this case we have a 2-cell
By the $(\infty,1)$-pullback definition of differental cohomology, this induces a morphism
This image of $Id_{\mathbf{B}^n A}$ under this map we call the universal $\mathbf{B}^{n-1}A$-connection on $\mathbf{B}^n A$:
For $c : \mathbf{B}G \to \mathbf{B}^n A$ any morphism, we get an induced diagram
in $\mathbf{H}$. This morphism of diagrams induces a morphism of the (∞,1)-pullbacks
and the composite
produces a canonical $\mathbf{B}^{n-1}A$-connection on $\mathbf{B}G$.
Example See circle n-bundle with connection.
An ordinary connection on a bundle $(P,\nabla)$ over a space $X$ induces a holonomy-map from loops in $X$ to the structure group: the parallel transport of the connection along the loops. One may think of the holonomy of a loop $\gamma : [0,1] \to X$ as associated to the pullback bundle with connection $(\gamma^* P , \gamma^* \nabla)$ over $S^1$. From this perspective the fact that a notion of holonomy exists and is well defined is crucially related to the fact that the curvature of $\gamma^* \nabla$ necessarily vanishes. Similarly, a connection $\nabla$ on a principal 2-bundle $P$ induces a notion of holonomy over 2-dimensional surfaces $\phi : \Sigma_2 \to X$, and this too may be understood as induced from the flat pullback 2-connection $\phi^* \nabla$.
The following definition generalizes this to a notion of holonomy of differential cocycles with groupal coefficients in an $\infty$-connected $(\infty,1)$-topos $\mathbf{H}$. That in $\mathbf{H} =$ ∞LieGrpd this reproduces the ordinary notion of holonomy of circle bundles and bundle $n$-gerbes is discussed at differential cohomology in an (∞,1)-topos -- examples.
Consider $\Sigma, X, A \in \mathbf{A}$ such that the intrinsic de Rham cohomology of $\Sigma$ with coefficients in $\mathbf{B}A$ is trivial,
Then every morphism $phi : \Sigma \to X$ induces a morphism
from the differential cohomology of $X$ to the flat differential cohomology of $\Sigma$, from the diagram
For $\nabla \in \mathbf{H}_{diff}(X,A)$ a differential cocycle, we call $hol_\phi(\nabla)$ the holonomy of $\nabla$ along $\phi$.
Above we described differential cohomology in an with ∞-connected (∞,1)-topos $\mathbf{H}$ with groupal coefficients $A \in \mathbf{H}$ as the twisted cohomology for the canonical curvature characteristic class $A \to \mathbf{\flat}_{dR} \mathbf{B}A$. We showed that this is an exact obstruction theory, in that the curvature class $X \to A \stackrel{curv_A}{\to} \mathbf{\flat}_{dR}A$ of an $A$-cocycle $g : X \to A$ is trivial in cohomology if and only if $g$ extends to flat differential cohomology.
If $A$ fails to be deloopable, we can find various approximations to obstruction classes against flat refinements.
A characteristic class on $\mathbf{B}G$ is simply any morphism
in $\mathbf{H}$.
The composition operation with the canonical curvature characteristic form $curv_{\mathbf{B}^{n} K}$
induces a square
By the universal propery of the (∞,1)-pullback this yields a morphism
from $G$-principal ∞-bundles to abelian differential cohomology.
This is the curvature characteristic of the given characteristic class.
Above in Connections we had seen that cocycles in differential cohomology with grobal coefficients may be modeled in $[C^{op}, sSet]$ by connections
For $\mathbf{B}G \to A$ a given curvature characteristic class there typically is a groupal model for the universal G-principal ∞-bundle $\mathbf{E}G$ and a diagram
covering the model for the characteristic class. In this case the curvature characteristic class is modeled by a diagram
($\infty$-Chern-Weil homomorphism)
In $\mathbf{H} =$ ∞LieGrpd such constructions may be obtained by Lie integration. This is described at ∞-Chern-Weil homomorphism.
We say a line object $R$ in an ∞-connected (∞,1)-topos $\mathbf{H}$ is compatible with the $\infty$-connectivity if its geometric realization is contractible
Being in particular an abelian ∞-group-object, the line object induces a notion of $R$-cohomology
Say that a moprhism $f : X \to Y$ in $\mathbf{H}$ is trivial in $R$-cohomology if
is an isomorphism for all $n \in \mathbb{N}$. We have the universal localization of an (∞,1)-category at these morphisms, giving a reflective sub-(∞,1)-category
We say that this exhibits on $\mathbf{H}$ the structure of rational homotopy theory in an (∞,1)-topos if combined with the terminal geometric morphism
we have that the composite (∞,1)-adjunction
is rationalization $X \mapsto X \otimes R$ over the field $R$.
We have then the triple composite adjunction
We say that its unit
is the intrinsic Chern-character on $A$ in $\mathbf{H}$.
For $f : B \to C$ a morphism – a characteristic class – with homotopy fiber $A \to B$, we may think of the $\mathbf{H}(X,A) \times_{\mathbf{H}(X,B)} H(X,B)$ as the $f$-twisted cohomology with coefficients in $A$.
Above we conceived differential cohomology as curvature-twisted or Chern-character-twisted cohomology. This may be paired with the twisting $f$ simply by forming the product characteristic class
For suitable choices of $f$ this then discribes twisted bundles with connection, twisted 2-bundles with connection, etc.
This has been considered in
where connection data on twisted String-principal bundles and twisted Fivebrane-principal bundles is worked out.
(…)
Context
differential cohomology