Differential cohomology in an $(\infty,1)$-topos
We discuss examples for and applications of the theory presented at differential cohomology in an (∞,1)-topos -- survey .
Every (∞,1)-topos $\mathbf{H}$ comes with its intrinsic notion of cohomology. At differential cohomology in an (∞,1)-topos we define an intrinsic notion of differential cohomology whenever $\mathbf{H}$ has the special property that it is a (locally) ∞-connected (∞,1)-topos so that it has an intrinsic notion of path ∞-groupoids.
Here we discuss examples of such $\infty$-connected $(\infty,1)$-toposes.
The (∞,1)-category of (∞,1)-sheaves on the site CartSp of Cartesian spaces and smooth functions between them is an $\infty$-connected $(\infty,1)$-topos. Its objects we may think of as ∞-Lie groupoids.
Also $\infty$-connected is the infinitesimally thickened (∞,1)-topos of $(\infty,1)$-sheaves on the site ThCartSp of infinitesimally thickened Cartesian spaces. This is the $(\infty,1)$-version of what is called the Cahiers topos. It contains in the manner of synthetic differential geometry also $\infty$-Lie groupoids with infinitesimally extended morphisms, which we may think of as ∞-Lie algebroids.
For a list of examples related to this $(\infty,1)$-topos see
Most examples of differential cohomology in an $(\infty,1)$-topos discussed below take place in $\infty LieGrpd$.
The archetypical $(\infty,1)$-topos is ∞Grpd itself. This is $\infty$-connected, for trivial reasons.
For interpreting the following statements, it is useful to think of the objects in $\infty Grpd$ as precisely those objects in ∞LieGrpd with discrete smooth structure, under the embedding $LConst : \infty Grpd \to \infty Lie Grpd$.
In $\infty Grpd$ the intrinsic path ∞-groupoid-functor is the identity – $\mathbf{\Pi} = Id$ – and hence every cocycle is uniquely a flat differential cocycle in the sense of flat differential cohomology in an (∞,1)-topos.
To get an impression for what this statement amounts to, notice that for instance the analog of a (not necessarily flat) $U(1)$-principal bundle in $\inftyLieGrpd$ over the 0-truncated group object $U(1) \in \infty LieGrpd$ (the circle group) is, in this terminology, a flat $\mathbf{B}\mathbb{Z}$-principal 2-bundle over the 1-truncated group object $\mathbf{B}\mathbb{Z}$ in $\infty Grpd$ (the strict 2-group coming from the crossed module $[\mathbb{Z} \to 0]$). While often these concepts are conflated, their conceptual difference is important. For instance, while their classification happens to match over good enough base objects, already their cocycle groupoids differ: for $X$ a smooth manifold we have $\pi_1 \infty LieGrpd(X,\mathbf{B}U(1)) = C^\infty(X,U(1))$, while $\pi_1 \infty Grpd(Sing X,\mathbf{B}\mathbf{B}\mathbb{Z}) \simeq *$.
Accordingly, for groupal objects $A \in \infty Grpd$ the curvature characteristic class $curv_A : A \to \mathbf{\flat}_{dR} \mathbf{B}A \simeq *$ is trivial in $\infty Grpd$ and the notion of differential cohomology with groupal coefficients as the $curv_A$-twisted cohomology reduces to just the ordinary intrinsic cohomology in $\mathbf{H} = \infty Grpd$.
Nevertheless, the intrinsic Chern character in an (∞,1)-topos in $\mathfb{H} = \infty Grpd$ is nontrivial: it coincides precisely with the rationalization map
Accordingly, differential cohomology in $\infty Grpd$ in the sense of Chern-character twisted cohomology is essentially the (unstabilized) definition of traditional differential cohomology following Hopkins-Singer, Freed, Bunke-Schick and others, if for $X$ a smooth manifold in forming the homotopy fibers of
we do so explicitly over over singular cohomology-cocycles with values in $\mathbb{R}$ that come from smooth differential forms on $X$.
A potentially noteworthy point here is that from the intrinsic perspective of the $(\infty,1)$-topos $\infty Grpd$, neither the notion of smooth manifold nor that of differential forms exists, so that the pairing of homotopy theory and differential geometry in the standard definition of differential cohomology is somewhat ad hoc .^{1}
However, using the richer context of the $(\infty,1)$-topos $\mathbf{H} =$ ∞LieGrpd we may lift bare $\infty$-groupoids $A \in \infty Grpd$ to ∞-Lie groupoids $\tilde A \in \infty Lie Grpd$, in that under geometric realization $\Pi : \infty Lie Grpd \to \infty Grpd$ we have $\Pi(\tilde A) \simeq A$. For instance $\mathbf{B}\mathbb{Z} \in \infty Grpd$ may be lifted to $U(1) \in \infty Lie Grpd$. But quite generally we also always have the lift $\tilde A := LConst A$ to be the $\infty$-Lie groupoid structure on $A$ with discrete smooth structure.
Then the intrinsic definition of differential cohomology with groupal coefficients applies and may give the expected answers (which is the case for the lift to $U(1)$ as we show below in the section abelian n-gerbes with connection) or at least the homotopy fibers of the intrinsic Chern-character map are naturally computed over cocycles represented by smooth differential forms, since that concept does exists intrinsically in $\infty Lie Grpd$ (as we discuss below in Ordinary de Rham cohomology).
In summary, this suggests that we think of the tradiional definition of differential cohomology as being the intrisic $(\infty,1)$-topos theoretic differential cohomology in $\mathbf{H} = \infty LieGrpd$ restricted to coefficient objects that happen to be in the image of $LConst : \infty Grpd \to \infty LieGrpd$.
Here we discuss examples for Differential cohomology with groupal coefficients.
This applies when the coeficcient object $A \in \mathbf{H}$ is an ∞-group-object in that it has a delooping $\mathbf{B}A$. In that case there is a canonical morphism
representing a class in the intrinsic de Rham cohomology of $A$ – the ∞-Maurer-Cartan form on $A$ – and we define differential cocycles to be the objects in the homotopy fiber of the induced morphism
of cocycle ∞-groupoids.
This section is at
The abstract topos-theoretic definition of holonomy of differential cocycles in an $\infty$-connected $(\infty,1)$-topos $\mathbf{H}$ on an object $X$ with coefficients in an ∞-group $A$ along a morphism $\phi : \Sigma \to X$ such that $\mathbf{H}_{dR}(\Sigma,\mathbf{B}A) \simeq *$ identifies this notion with the canonical morphism
We demonstrate now how for $\mathbf{H} =$ ∞LieGrpd, for $X$ a closed smooth manifold and $A = \mathbf{B} \mathbf{B}^{n-1}U(1)$ the delooping of the circle Lie n-group this abstract definition reproduces the ordinary definition of holonomy of $U(1)$-principal bundles and their higher analogs.
The first proposition asserts that in this case the set $H_{flat}(X,A)$ is indeed canonically identified with the circle group $U(1)$, which we expect the ordinary holonomy of circle $n$-bundles to take values in. The proof that we present uses an insight pointed out in joint discussion by Domenico Fiorenza.
Let $k \leq n \in \mathbb{N}$ and let $X_{n-k} \in \mathbf{H} =$ ∞LieGrpd be a closed paracompact smooth manifold of dimension $n-k$. Write $\tau_k$ for $k$-truncation.
Then we have an equivalence
of ∞-group objects in ∞Grpd. In particular for $X_n$ $n$-dimensional, we have
Here, as usual, we write $\mathbf{B}^n U(1)$ for the $n$th delooping of the circle group $U(1)$ regarded naturally as a group object in ∞LieGrpd, and $\mathcal{B}^n U(1)$ for the delooping in ∞Grpd, whose homotopy type is that of the Eilenberg-MacLane space $K(n, U(1))$.
By definition of flat differential cohomology and using the path ∞-groupoid adjunction we have
Since $X$ is assumed to be a paracompact manifold, we have by the discussion at geometric realization that $\Pi(X) \simeq Sing X$ is equivalent to the singular simplicial complex of $X$. Accordingly
is the cocycle $\infty$-groupoid of ordinary singular cohomology $H^\bullet(X,U(1))$ of $X$. So far this is just a special case of the general statement about cohomology with constant coefficients in ∞LieGrpd.
Observe that from general considerations (recalled for instance at cohomology and at fiber sequence) we have for the homotopy groups of the cocycle $\infty$-groupoid that
We can evaluate this using that $X$ is an $(n-k)$-dimensional manifold in the universal coefficient theorem, which asserts that we have an exact sequence
Since the circle group $U(1)$ is an injective $\mathbb{Z}$-module the left term vanishes, $Ext^1(-, U(1)) = 0$.
Therefore we have an isomorphism
between the cohomology group in question and the abelian group of homomorphisms from the homology group $H_{n-i}(X,\mathbb{Z})$ to $U(1)$.
By assumption on $X$ we have for $i \lt k$ that $H_{n-i}(X,\mathbb{Z}) = 0$ and for $i = k$ that $H_{n-i}(X,\mathbb{Z}) = \mathbb{Z}$. It follows that
Here we discuss realization of the theory of Differential cohomology in an (∞,1)-topos with general coefficients.
We had seen above that a connection on a $\mathbf{B}^n U(1)$-principal ∞-bundle is an intermediate step in a model for the curvature characteristic class morphism
namely a cocycle with values in the simplicial presheaf $\mathbf{B}^n U(1)_{diff}$ that appears as the correspondence space of a span
(an $\infty$-anafunctor) of simplicial presheaves that represents $curv_{\mathbf{B}^n U(1)}$ in the model $[CartSp^{op}, sSet]_{proj,cov}$. We had also seen above that an equivalent model for $\mathbf{B}^n U(1)_{diff,simp}$ is given by relative Lie integration of the morphism $b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R})$ of ∞-Lie algebras.
That latter model has an evident generalization $\mathbf{B}G_{diff}$ to arbitrary ∞-Lie algebras and the ∞-Lie groups $G$ integrating them. Cocycles with values in such $\mathbf{B}G_{diff,simp}$ we discuss below as
These generalize the ordinary notion of a connection on a bundle to a notion of connections of $G$-principal ∞-bundles, including connections on nonabelian gerbes.
As before in the abelian case, such connections may be understood as an interpolating step in a concrete model for an intrinsic (∞,1)-topos-theoretic notion of curvature characteristic classes, namely for the intrinsic Chern character?
in $\mathbf{H} =$ ∞LieGrpd. However, for general non-abelian $G$, the relation between the intrinsic notion of curvature and its model by connections is a bit more involved than in the abelian case: instead of using a single span, we need to model a collection of morphisms
into the stages of the Postnikov tower-decomposition of $(\mathbf{\Pi}\mathbf{B}G) \otimes \mathbb{R}$;
and out of the stages of a smooth refinement of the intrinsic Whitehead tower of $\mathbf{B}G$.
This we discuss below in
Each smoothly refined stage in this decomposition corresponds to a invariant polynomial on the ∞-Lie algebra $\mathfrak{g}$ of $G$ and yields a curvature characteristic form obtained by evaluting this polynomial on the curvature? of the $\infty$-Lie algebra valued connection. The resulting map from $G$-principal ∞-bundles to curvature characteristic forms subsumes and generalizes the ordinary Chern-Weil homomorphism. This we discuss in
But more is true: the combination of the underlying characteristic class in degree $n+1$ with its curvature differential form lifts to a differential character: a cocycle in the abelian differential cohomology $\mathbf{H}_{diff}(-, \mathbf{B}^n U(1))$. These refined differential characteristic classes
provide a differential refinement of the twisted cohomology induced by the bare underlying characteristic classes. As examples for this we discuss
The content of this section is at
We summarize here for reference and for convenience a handful of standard computational tools that we use to extract concrete models from the abstract $(\infty,1)$-topos-theoretic definitions.
The main tools for abelian differential cohomology are those of abelian sheaf cohomology.
Dold-Kan correspondence for presenting $\infty$-stacks with values in strictly abelian strict $\infty$-groupoids in terms of chain complexes of sheaves;
Cech cohomology for computing derived hom-spaces between $\infty$-stacks;
Partitions of unity for finding coboundaries between Cech hypercohomology cocyces.
We recall a basic tool for the discussion of stable differential cohomology: the Dold-Kan correspondence. This serves to model ∞-stacks with values in strictly abelian strict ∞-groupoids equivalently as sheaves with values in chain complexes of abelian groups. For more discussion of this standard procedure in cohomology-theory see the discussion at abelian sheaf cohomology.
We write
for the Dold-Kan map from non-negatively graded chain complexes of abelian groups to sSet. Notice that this map factors through crossed complexes and then through Kan complexes, hence in particular lands in fibrant objects in the standard model structure on simplicial sets. More generally, a fundamental property of the Dold-Kan correspondence that we make use of repeatedly is that $\Xi$ preserves projective fibrations and weak equivalences: a degreewise surjection of chain complexes is sent to a Kan fibration and a quasi-isomorphism of chain complexes is sent to a weak homotopy equivalence of Kan complexes.
By convenient abuse of notation, we use the same symbol $\Xi$ also for the evident objectwise extension of this functor to presheaves on a small category $C$:
With the above remark we have the following
The functor $\Xi : [C^{op}, Ch_\bullet]_{proj} \to [C^{op}, sSet_{Quillen}]_{proj}$ is a right Quillen functor that preserves all weak equivalences:
it is a right adjoint and hence preserves all limits;
For $f : A_\bullet \to B_\bullet$ a morphims in $[C^{op}, Ch_\bullet]$ that is a surjection for each object $U \in C$ and in each degree $n \in \mathbb{N}$ we have that $\Xi(f)$ is a fibration in $[C^{op}, sSet]_{proj}$.
If $f$ is over each $U \in C$ a quasi-isomorphism then $\Xi(f)$ is a weak equivalence in $[C^{op}, sSet]_{proj}$.
The left adjoint functor
to $\Xi$ is the normalized chain complex functor. Of the various isomorphic definitions of this for our purposes the following is the most useful one:
for a simplicial abelian group $A$ with face and degeneracy maps $\{\delta^k_j\}, \{\sigma^k_j\}$, respectively, we have that the corresponding normalized chain complex $N_\bullet(A)$ is
in degree $k$ given by the group $A_k$ modulo the images of the degeneracy maps
with differential given by the alternating sum of the face maps
Again, by convenient abuse of notation we also write
for the evident objectwise prolongation of $N_\bullet$ to presheaves of simplicial abelian groups.
For computing the intrinsic cohomology of an (∞,1)-topos $\mathbf{H}$ – and its cocycle ∞-groupoids, we may present it by a model structure on simplicial presheaves on some sSet-site $C$ (which in many of our examples is an ordinary 1-categorical site) as
This is reviewed at models for ∞-stack (∞,1)-toposes. In terms of this, the cocycle $\infty$-groupoids are derived hom-spaces whose computation is effectively that of (possibly nonabelian) Cech? hypercohomology.
In the model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,cov}$ modelling the $(\infty,1)$-topos $\mathbf{H} =$ ∞LieGrpd a cofibrant resolution of a paracompact smooth manifold $X$ is given by the Cech nerve $C(\{U_i\}) \to X$ of any good open cover of $X$.
If $A = \Xi \mathcal{A}_\bullet \in \inftyLieGrpd$ is represented by a fibrant object in $[CartSp^{op}, sSet]_{proj}$ in the image of the Dold-Kan map as discussed above, then the intrinsic cocycle $\infty$-groupoid on a paracompact manifodl $X$ with coefficients in $A$ coinicides with the abelian sheaf cohomology cocycle $\infty$-groupoid for $\mathcal{A}_\bullet$ computed as the Cech hypercohomology with respect to a good open cover $\{U_i \to X\}$ of $X$.
An $n$-cocycle $c$ then is given by a sequence of elements
with $C_i \in \mathcal{A}_0(U_i)$, $B_{i j} \in \mathcal{A}_1(U_i \cap U_j)$, etc., which is closed under the double differential $D = d_{\mathcal{A}} \pm \delta$, where $\delta$ is the alternating sum of the pullbacks along the face maps of the Cech nerve.
(…) partition of unity (…)
The Cech nerve $C(\{U_i\}) \to X$ of a manifold has itself trivial cohomology in positive degree, with respect to the differential on degreewise functions
given by the alternating sum of the pullbaks along the face maps.
An explicit choice of coboundaries $\delta \lambda = \eta$ for a given $\delta \eta = 0$ is obtained from a choice of partition of unity
$(\rho_i \in C^\infty(U_i, \mathbb{R}))$ subordinate to the cover $\{U_i \to X\}$.
(…)
The familiar tools for embedding abelian sheaf cohomology into the intrinsic cohomolog of the (∞,1)-topos over the given site that we recalled above have well known – albeit maby not as widely known – generalizations into the regime of nonabelian cohomology. These are known as tools in nonabelian algebraic topology and nonabelian homological algebra. We briefly recall what we need below.
(…)
Let $\Xi : CrsdCplx \to sSet$ now denote the inclusion of crossed complexes into all ∞-groupoids.
This generalizes the functor $\Xi$ used in the abelian case above as we have a sequence of inclusions
See also the cosmic cube of higher category theory.
Again, we use the same notation for the objectwise extension of this function to presheaves
(…)
(…)
see Cech cohomology
(…)
Context
At the Fourth Annual Meeting of the Dutch Quantum Theory and Geometry Cluster conference in Amsterdam in June 2010, Sir Michael Atiyah raised this point in the discussion session after a talk about differential K-theory: he argued that the standard definition is “ugly” in that it mixes concepts from “different worlds” and suggested to look for a modified definition built coherently in a single context. ↩