differential cohomology

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###Context###
#### Differential cohomology
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[[!include differential cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Differential cohomology is a refinement of plain [[cohomology]] such that a differential cocycle is to its underlying ordinary [[cocycle]] as a [[connection on an infinity-bundle|bundle with connection]] is to its underlying [[principal infinity-bundle|bundle]].

The best known version of differential cohomology is a differential refinement of [[generalized (Eilenberg-Steenrod) cohomology]], hence of [[cohomology]] in [[stable homotopy theory]] (as opposed to more general [[nonabelian cohomology]], cf. *[[nonabelian differential cohomology]]*). For [[ordinary cohomology]] the refinement to [[ordinary differential cohomology]] is modeled for instance by [[complex line bundles]] with [[connection on a bundle]], [[bundle gerbes with connection]], etc, and generally by [[Deligne cohomology]]. Similarly, differential refinements of [[topological K-theory]] to [[differential K-theory]] may be presented by [[vector bundles]] [[connection on a bundle|with connection]] (e.g [Simons-Sullivan 08](#SimonsSullivan08)).

A systematic characterization and construction of differential [[generalized (Eilenberg-Steenrod) cohomology]] in terms of  suitable [[homotopy fiber products]] of the [[mapping spectra]] [[Brown representability theorem|representing]] the underlying [[cohomology theory]] with [[differential form]] data was then given in ([Hopkins-Singer 02](#HopkinsSinger02)) (motivated by discussion of the [[quantization]] of the [[M5-brane]] via holographically dual [[7d Chern-Simons theory]] by [[Edward Witten]]).

In this stable case one characteristic property of differential cohomology that was first observed empirically in [[ordinary differential cohomology]] ([Simons-Sullivan 07](#SimonsSullivan07)) and [[differential K-theory]] ([Simons-Sullivan 08](#SimonsSullivan08)) is that it is a kind of cohomology theory which fits into a _[[differential cohomology diagram]]_ which is an interlocking system of two [[fiber sequences]] that expresses how the data of a geometric bundle with connection may be decomposed into the underlying bundle and the [[curvature]] of the connection and how connections on trivial bundles as well as bundles with [[flat infinity-connection|flat connections]] are examples of the general concept. (The [[homotopy fiber product]] used in ([Hopkins-Singer 02](#HopkinsSinger02)) provides one of the two commuting squares in this diagram.)

Schematically the characteristic [[differential cohomology hexagon]] (see there for details) looks as follows

$$
  \array{
    &&  connection\;on\;trivial\;bundle && \stackrel{de\;Rham\;differential}{\longrightarrow} && differential\;forms
    \\
    & \nearrow & & \searrow & & \nearrow_{curvature} && \searrow
    \\
    flat\;differential\;forms  && && geometric\;bundle\;with \;connection && && rational\;shape\;of\;bundle
    \\
    & \searrow &  & \nearrow & & \searrow^{\mathrlap{}} && \nearrow_{\mathrlap{Chern\;character}}
    \\
    && geometric\;bundle\;with\;flat\;connection && \underset{forget\;connection}{\longrightarrow} && shape\;of\;bundle
  }
$$

It turns out  that such [[differential cohomology diagrams]] are exactly what all [[stable homotopy types]] in [[cohesive (∞,1)-toposes]] $\mathbf{H}$ naturally sit in ([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)). All the traditional differential cohomology theories and their joint generalization due to ([Hopkins-Singer 02](#HopkinsSinger02)) are special cases of this for the case that $\mathbf{H} = T Smooth \infty Grpd$ is the [[tangent cohesive (∞,1)-topos]] of that of [[smooth ∞-groupoids]] ([[∞-stacks]] over the [[site]] of [[smooth manifolds]]).

It is therefore natural to _define_ differential cohomology to be the intrinsic [[cohomology]] of [[cohesive (∞,1)-toposes]]. (Notice that while the [[differential cohomology diagram]] itself only involves the [[shape modality]] and the [[flat modality]] of [[cohesion]], the [[sharp modality]] is needed to produce, via [[differential concretification]], the correct [[moduli stacks]] of differential cocycles over a given base object from the [[mapping stack]] of that into the representing [[stable homotopy type]]).

Viewed in this generality, differential cohomology makes sense for instance also in [[supergeometry]] where it subsumes structures such as [[super line 2-bundles]] and more generally [[Super Gerbes|higher super gerbes]] with connection, which naturally appear as differential refinements of the geometric [[twisted cohomology|twists]] of (differential) [[K-theory]] and [[tmf]].

## Examples

* The [[ordinary differential cohomology]] $\bar H^\bullet(-,\mathbb{Z})$ is modeled by

  * [[Deligne cohomology]];

  * [[Cheeger-Simons differential character]]s;

  * higher circle [[bundle gerbe]]s with connection;

  * [[circle n-bundles with connection]].

* apart from that people studied mainly [[differential K-theory]].

In [[physics]] differential cocycles model [[gauge theory|gauge fields]].

* Cocycles in [[ordinary differential cohomology]] (e.g. [[Deligne cohomology]]) model

  * in degree 2: the [[electromagnetic field]]

  * in degree 3: the [[Kalb-Ramond field]]

  * in degree 4: the [[supergravity C-field]]

* Cocycles in [[differential K-theory]] model the

  * [[RR-field]] .

For $c \in \bar \Gamma^\bullet(X)$ a differential cocycle representing a [[gauge field]], one says that

* its image $F(c)$ in differential forms is the corresponding [[field strength]];

* its image $cl(c)$ in non-differential cohomology is the "topological twist" of the [[gauge theory|gauge field]]. In special cases this can be identified with [[magnetic charge]].

Other examples:

* [[differential algebraic K-theory]]

## Differential stable cohomology

> the following are ancient notes that need to be brushed up and polished.

### Characterization following Hopkins-Singer
 {#HopSin}

A standard definition of differential cohomology is in terms of a [[homotopy pullback]] of a [[generalized (Eilenberg-Steenrod) cohomology theory]] with the complex of [[differential forms]] over [[real cohomology]]:

Let $\Gamma^\bullet$ be a generalized cohomology theory in the sense of the generalized Eilenberg--Steenrod axioms, and let
$\Gamma^\bullet \to H^\bullet(-,\mathbb{R}) \otimes \Gamma^\bullet(*)$ be a morphism to real singular cohomology with coefficients in the ring of $\Gamma$-cohomology of the point. Then the differential refinement of $\Gamma^q$, the degree $q$**differential $\Gamma$-cohomology** is the [[homotopy pullback]] $\bar \Gamma^\bullet$ in

$$
  \array{
  \bar \Gamma^\bullet(-)
  &\stackrel{F}{\to}&
  \Omega^{\geq q}(-)\otimes \Gamma^\bullet(*)
  \\
  \downarrow^{cl} && \downarrow
  \\
  \Gamma^\bullet(-)
  &\stackrel{ch}{\to}&
  Z^\bullet(-, \mathbb{R}) \otimes \Gamma^\bullet(*)
  }
  \,,
$$

where

* $Z^\bullet(-,\mathbb{R})$ is the complex underlying real singular cohomology $H^\bullet(-,\mathbb{R})$

* $\Omega^\bullet(-,\mathbb{R})$ is the complex underlying deRham cohomology

* $\Omega^\bullet(-) \to H^\bullet(-,\mathbb{R})$ is [[deRham theorem]] morphism

* $ch: \Gamma^\bullet(-)
  \to
  Z^\bullet(-, \mathbb{R}) \otimes \Gamma^\bullet(*)$ is the [[Chern character]] map.

For more details on this see at _[[differential function complex]]_.

### Characterization following Bunke-Schick

> The following are old notes once taken in a talk by [[Thomas Schick]] at _[[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]_. Needs to be brushed-up, polished, improved, rewritten...

The following theory of differential cohomology
(also called **smooth cohomology**) is developed and used
in the work of [[Ulrich Bunke]] and [[Thomas Schick]]. Contrary to the above, it does not take the notion of homotopy limit as fundamental, but instead characterizes the universality of the above commuting diagram by other means. On the other hand, this means that their axiomatization at the moment only capture cohomology classes, not the representing cocycles. It is sort of known that for various applications specific cocycle representatives do play an important role, and one may imagining refining to discussion below eventually to accommodate for that.

* idea:

  * combine cohomology + differential forms

main diagram

$$
  \array{
    \hat H(M) &\stackrel{I}{\to}& H^\bullet(M)
    \\
    \downarrow^{R} && \downarrow
    \\
    \Omega^\bullet_{d=0}(M)
    &\stackrel{}{\to}&
    H^\bullet_{dR}(M)
    \simeq H^\bullet(M,\mathbb{R})
  }
$$

so differential cohomology $\hat H^\bullet(M)$ combines the ordinary cohomology $H^\bullet(M)$
with a differential form representative of its image in [[real cohomology]].

* $I$ projects a differential cohomology to its underlying ordinary cohomology class;

* $R$ send the differential cohomology class to its _curvature_ differential form data

we want an exact sequence

$$
 \array{
   H^{\bullet-1}(M)
   &\stackrel{ch}{\to}&
   \Omega^{\bullet-1}(M)/{im(d)}
   &\stackrel{d}{\to}&
   \hat H(M)
   &\stackrel{I}{\to}&
   H^\bullet(M) \to 0
   \\
   &&&
   {}_{d}\searrow
   &
   \downarrow^R
   \\
   &&&&
   \Omega^\bullet_{d=0}(M)
 }
$$

+-- {: .num_defn}
###### Definition

Given cohomology theory $E^\bullet$, a smooth refinement
$\hat E^\bullet$ is a functor $\hat E : Diff \to Grps$
with transformations $I, R$ such that

$$
  \array{
    \hat E(M) &\stackrel{I}{\to}& E^\bullet(M)
    \\
    \downarrow^{R} && \downarrow
    \\
    \Omega^\bullet_{d=0}(M, V)
    &\stackrel{}{\to}&
    E^\bullet_{dR}(M)
    \simeq E^\bullet(M,\mathbb{R})
  }
$$

where

$V = E^\bullet(pt)\otimes \mathbb{R}$

is the graded non-torsion cohomology of $E$ on the point (so now all the
gradings above denote total grading)

and such that there is a transformation

$$
  a : \Omega^{\bullet -1}(M)/{im(d)}
  \to
  \hat E^*(M)
$$

that gives the above kind of exact sequence.
=--

+-- {: .num_defn}
###### Definition

If $E^*$ is multiplicative, we say $\hat E^*$ is multiplicative with
product $\vee$
if $\hat E$ takes values in graded rings and the transformations are
compatible with multiplicative structure, where

$$
  a(\omega) \vee x = a(\omega \wedge R(x))
$$
=--

+-- {: .num_defn}
###### Definition

$\hat E$ has $S^1$-integration if there is a natural (in $M$) transformation

$$
  \int : \hat E^*(M \times S^1) \to \hat E^{\bullet -1}(M)
$$

compatible with $\int$ of forms  and for $E$ it is given by the suspension isomorphism

$$
  \int \circ p^* = 0
$$

for $p : M \times S^1 \to M$ and

$$
  \int \circ (  id \times (z \mapsto \bar z) )^* = - \int
$$
=--

+-- {: .num_remark}
###### Remark

Ordinary cohomology theories are supposed to be homotopy
invariant, but differential forms are not, so in general the differential
cohomology is not.
=--

+-- {: .num_lemma}
###### Lemma
Given $\hat E$ a smooth cohomology theory.
The **homotopy formula**:

given $h : M \times [0,1] \stackrel{smooth}{\to} N$
a [[smooth homotopy]] we have

$$
  h^*_1(X) - h^*_0(X)
  =
  a( \int_{M \times [0,1]/M}  h^*(R(x)))
$$
=--

+-- {: .num_cor}
###### Corollary
$ker(R)$ (i.e. flat cohomology) is a homotopy invariant
functor.
=--

+-- {: .num_defn}
###### Definition
$\hat H{flat} := ker(R)$
=--

+-- {: .proof}
###### Proof of lemma

It suffices to show

$$
  \iota_1^*(x) - \iota_0^*(x) = a(\int_{M\times [0,1]/M} R(x))
$$

for all $x \in \hat E(M \times [0,1])$.

Observe if $x = p^* y$ the left hand side vanishes, $\int R(p^* y) = 0$.

For general $x$ $\exists y \in \jhat E(M)$; $x - p^*(y) = a (\omega)$
$\omega \in \Omega(M \times [0,1])$.

Stokes' theorem gives $i^*_1 \omega - i^*_0 \omega = \int_{[0,1]} d \omega$
$= \int R(a(\omega)) = \int R(x-p^* \omega) = \int R(x)$.

On the other hand

$$
  i^*_1(x) - i^*_0(x) = i^*_1(a(\omega))
    - i^*_0(a(\omega))
    = a(\int R(x))
$$

A calculation: $\hat H^1_{flat}(pt) = \hat H^1(pt) = \mathbb{R}/\mathbb{Z} = \hat K^1(pt)$.
=--

+-- {: .num_theorem}
###### Theorem
(Hopkins--Singer)

For each generalized cohomology theory $E^*$
a differential version  $\hat E^*$ as in the above definition does exist.

Moreover $\hat E_{flat}^* = E \mathbb{R}/\mathbb{Z}^{\bullet -1}$.
=--

+-- {: .num_remark}
###### Remark
It's not evident how to obtain more structure like multiplication.
=--

+-- {: .num_theorem}
###### Theorem
Using geometric models, multiplicative smooth extensions
with $S^1$-integration are constructed for

* K-theory (Bunke--Schick)

* MU-bordisms (unitary bordisms)
  (Bunke--Schröder--Schick--Wiethaupts; and from there Landweber exact cohomology theories)
=--

+-- {: .num_theorem}
###### Theorem
(Uniqueness theorem due to Bunke--Schick. Simons--Sullivan proved this for ordinary integral cohomology.)

Assume $E^*$ is _rationally even_, meaning that

$$
  E^k(pt)\otimes \mathbb{Q} = 0 \;\; for odd k
$$

plus one further technical assumption.

Then any two smooth extensions $\hat E^*$, $\tilde E^*$ are naturally
isomorphic.

If required to be compatible with integration the ismorphism is unique.

If $\hat E, \tilde E$ are multiplicative, then this isomorphism is,
as well.
=--

+-- {: .num_example}
###### Example

If we don't require compatibility with $S^1$-integration, then there
are "exotic" abelian group structures on $\hat K^1$.
=--

#### Examples

##### Differential cobordism cohomology theory

A model for a multiplicative differential refinement of [[complex cobordism cohomology theory]], the theory represented by the [[Thom spectrum]] is in

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schröder,  Moritz Wiethaup,  _Landweber exact formal group laws and smooth cohomology theories_ ([arXiv:0711.1134](http://arxiv.org/abs/0711.1134))

See [[differential cobordism cohomology theory]]

## Related concepts

* [[nonabelian differential cohomology]]

* [[tangent cohesion]], [[smooth spectrum]]

* [[differential orientation]]

* [[twisted differential cohomology]]

* [[Hodge-filtered differential cohomology]]

[[!include gauge field - table]]

## References
 {#References}

### General

Differential cohomology was first developed for the special cases of [[ordinary differential cohomology]]  and of [[differential K-theory]] (interest into which came from discussion of [[D-brane charge]] in [[type II superstring theory]]). See the references there. A survey is in

* [[Ulrich Bunke]], _Differential cohomology in geometry and analysis_ ([pdf slides](http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/Vortrag-Erlangen.pdf))

The suggestion that differential cohomology is or should be characterized by the [[differential cohomology diagram]] is due to

* {#SimonsSullivan07} [[James Simons]], [[Dennis Sullivan]], _Axiomatic Characterization of Ordinary Differential Cohomology_ ([arXiv:math/0701077](http://arxiv.org/abs/math/0701077))

  (discussed there for [[ordinary differential cohomology]])

and

* {#SimonsSullivan08} [[James Simons]], [[Dennis Sullivan]], _Structured vector bundles define differential K-theory_ ([arXiv:0810.4935](http://arxiv.org/abs/0810.4935))

  (discussed there for [[differential K-theory]]).

A first systematic account characterizing and constructing stable differential cohomology in terms of [[homotopy fiber products]] of bare [[spectra]] with [[differential form]] data (which is really the homotopy-theoretic refinement of one of right square in the [[differential cohomology diagram]]) was given in

* {#HopkinsSinger02} [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_, J. Differential Geom. Volume 70, Number 3 (2005), 329-452.   ([arXiv:math/0211216](http://arxiv.org/abs/math/0211216), [euclid:1143642908](https://projecteuclid.org/euclid.jdg/1143642908))

motivated by applications in [[higher gauge theory]] and [[string theory]], as explained further in

* [[Dan Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_ ([arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220))

with further reviews including

* [[Alessandro Valentino]], _Differential cohomology and quantum gauge fields_ ([pdf](http://www.uni-math.gwdg.de/sandro/seminarGoett.pdf))

* {#Szabo12} [[Richard Szabo]], _Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology_, contribution to: *7th International Conference on Mathematical Methods in Physics (ICMP 2012)* [[arXiv:1209.2530](http://arxiv.org/abs/1209.2530), [doi:10.22323/1.175.0009](https://doi.org/10.22323/1.175.0009), [inspire:1185286](https://inspirehep.net/literature/1185286)]

The homotopy-pullback definition of differential generalized cohomology from

* [Hopkins-Singer 02, Section 4.1](#HopkinsSinger02)

is picked up in

* [[Ulrich Bunke]], [[David Gepner]], Section 2.2 of: _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_ ([arXiv:1306.0247](https://arxiv.org/abs/1306.0247))

* {#Bunke12} [[Ulrich Bunke]], Section 4.3 of: _Differential cohomology_ ([arXiv:1208.3961](http://arxiv.org/abs/1208.3961))

* [Bunke-Nikolaus-Völkl 13, Section 4.4](#BunkeNikolausVoelkl13)

The suggestion that differential cohomology is naturally the intrinsic [[cohomology]] of those [[(∞,1)-toposes]] which are [[cohesive (∞,1)-topos|cohesive]] appeared in

* {#Schreiber13} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

and the observation, working with [[sheaves of spectra]], that indeed every [[stable homotopy type]] in a [[cohesive (∞,1)-topos]] canonically sits inside a [[differential cohomology diagram]] is due to

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_,  J. Homotopy Relat. Struct. **11** (2016) 1-66 [[arXiv:1311.3188](http://arxiv.org/abs/1311.3188), [doi:10.1007/s40062-014-0092-5](https://doi.org/10.1007/s40062-014-0092-5)]

Discussion in the generality of differential [[non-abelian cohomology]]:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalizaed]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

Another book-length overview:

* {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250))

On differential cohomology in [[mathematical physics]]:

* [[Arun Debray]], *Differential Cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.14338](https://arxiv.org/abs/2312.14338)]

### Lectures and talks

* Simons Center Workshop on Differential Cohomology January 10, 2011- January 14, 2011 ([web](http://www.scgp.stonybrook.edu/?q=node/21))

* [Bunke 2012](#Bunke12)

* {#DG18} [[Daniel Grady]], _Differential cohomology and Applications_, talk at _Geometry, Topology & Physics_, NYU Abu Dhabi, April 2018 ([[DiffCohomologyAndApplications18.pdf:file]])

[[!redirects differential cohomology theory]]
[[!redirects differential cohomology theories]]

[[!redirects differential generalized cohomology]]


differential cohomology - contents

**[[differential cohomology]]**

## Ingredients

* [[cohomology]]

* [[differential geometry]]

## Connections on bundles

* [[connection on a bundle]]

  * [[parallel transport]], [[holonomy]]

* [[curvature]]

  * [[curvature characteristic form]], [[Chern character]]

* [[Chern-Weil theory]]

  * [[Chern-Weil homomorphism]]

  * [[secondary characteristic class]]

  * [[differential characteristic class]]

## Higher abelian differential cohomology

* [[differential function complex]]

* [[differential orientation]]

* [[ordinary differential cohomology]]

  * [[differential Thom class]]

  * [[differential characters]],

  * [[Deligne cohomology]]

  * [[circle n-bundle with connection]],

  * [[bundle gerbe|bundle gerbe with connection]]

* [[differential K-theory]]

* [[differential elliptic cohomology]]

* [[differential cobordism cohomology]]

## Higher nonabelian differential cohomology

* [[principal 2-bundle]], [[principal ∞-bundle]]

* [[connection on a 2-bundle]], [[connection on an ∞-bundle]]

* [[Chern-Weil theory in Smooth∞Grpd]]

  * [[∞-Lie algebra cohomology]]

* [[∞-Chern-Simons theory]]

## Fiber integration

* [[higher holonomy]]

* [[fiber integration in differential cohomology]]

  * [[fiber integration in ordinary differential cohomology]]

  * [[fiber integration in differential K-theory]]

## Application to gauge theory

* [[gauge theory]]

* [[gauge field]]

  * [[electromagnetic field]]

  * [[Yang-Mills field]]

  * [[Kalb-Ramond field]]/[[B-field]]

  * [[RR-field]]

  * [[supergravity C-field]]

  * [[D'Auria-Fre formulation of supergravity|supergravity]]

* [[quantum anomaly]]

[Edit this sidebar](/nlab/edit/differential+cohomology+-+contents)

[[!redirects differential cohomology -- contents]]


differential cohomology diagram

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In abstract generality, a _differential cohomology diagram_ is the hexagonal [[diagram]] in a [[cohesive (∞,1)-topos]] formed by the two [[fracture squares]] of the [[unit of a monad|unit]] and [[counit of a comonad|counit]] of, respectively, the [[shape modality]] $\Pi$ and the [[flat modality]] $\flat$. The exactness properties of this diagram for any [[stable homotopy type]] $\hat E$ in the corresponding [[tangent cohesive (∞,1)-topos]] express $\hat E$ as being the [[coefficients]] for a [[differential cohomology]] refinement by differential form data $\flat_{dR} \hat E$ of the [[generalized (Eilenberg-Steenrod) cohomology theory]] which is [[Brown representability theorem|represented]] by the [[shape modality|shape]] [[spectrum]] $E \coloneqq \Pi(\hat E)$.

Historically, it had been shown in ([Simons-Sullivan 07](#SimonsSullivan07)) for [[ordinary differential cohomology]] and in ([Freed-Lott 10](#FreedLott10), [Simons-Sullivan 08](#SimonsSullivan08)) for [[differential K-theory]] that these cohomology theories are characterized as sitting in the middle of a hexagonal [[diagram]] of interlocking [[exact sequences]] of [[cohomology groups]], which expresses, on the one hand, how every differential cohomology class has underlying it a non-differential cohomology class ("of a [[principal ∞-bundle]]") as well as a [[curvature]] [[differential form]] datum, and, on the other hand, how the special cases of trivial underlying classes equipped with differential form datum and of [[flat connection|flat]] differential form data sit inside the differential cohomology classes.

At this schematic conceptual level a differential cohomology diagram looks as follows (where all unlabelled arrows are meant to be read as "evident inclusions");

$$
  \array{
    &&  {{connection\;forms}\atop{on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} && {{curvature}\atop{forms}}
    \\
    & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}}
    \\
    {{flat}\atop{differential\;forms}}  &&
    && {{geometric\;bundles}\atop{with\;connection}} && &&
    {{rationalized}\atop{bundle}}
    \\
    & \searrow &  & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}}
    \\
    && {{geometric\;bundles}\atop{with\;flat\;connection}} && \underset{comparison/regulator\;map}{\longrightarrow} && {{shape}\atop{of\;bundle}}
  }
$$

One characteristic property is that the two outer sequences are [[exact sequences]]. This expresses (at this rough schematic level) for instance (for the upper part) that the connections $A$ on trivial bundles whose curvature vanishes in that $\mathbf{d}A = 0$, are exactly the flat connections; as well as (for the lower part) that bundles with [[flat connections]] have [[torsion subgroup|torsion]] Chern-clases.

So a [[differential cohomology]] theory would be one whose [[cocycles]]/[[cohomology classes]] have the interpretation of (stable) [[principal ∞-bundles]] with [[connection on an ∞-bundle|connection]] such as in the middle of this diagram.

The characterization/construction of [[differential cohomology]] via [[homotopy fiber products]] (of [[mapping spectra]] with [[differential form]] data) due to ([Hopkins-Singer 02](#HopkinsSinger02)) provides an incarnation of this kind of diagram genuinely in [[stable homotopy theory]], so that the outer parts are indeed [[homotopy fiber sequences]] and the two squares are homotopy cartesian (are [[homotopy pullbacks]]/[[homotopy pushouts]]).

In ([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)) it was observed (see [Schreiber 13, section 4.1.2](#Schreiber13) for the generality in which we present this here) that every [[stable homotopy type]] $\hat E$ in a [[cohesive (∞,1)-topos]] $\mathbf{H}$ canonically sits inside a diagram of this form, being formed from the [[fracture squares]] of the units and counits of the [[shape modality]] $\Pi$ and the [[flat modality]] $\flat$, which in the right part are interpreted as the [[Maurer-Cartan form]] $\theta_E$ and the [[Chern character]] $ch_E$:

$$
  \array{
    &&  \Pi_{dR} {\hat E} && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}{\hat E}
    \\
    & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_{\hat E}}} && \searrow
    \\
    \Pi_{dR} \flat  {\hat E}  && && {\hat E} && && \Pi \flat_{dR}  \hat E
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_E}}
    \\
    && \flat {\hat E} && \longrightarrow && \Pi \hat E
  }
  \,.
$$

This is theorem \ref{TheDifferentialDiagram} below.

For the special case that $\mathbf{H} = T Smooth\infty Grpd$ is the [[tangent (∞,1)-topos]] of [[smooth ∞-groupoids]] ([[∞-stacks]] over the [[site]] of [[smooth manifolds]]) this subsumes the cases mentioned above. But there are many examples of cohomology theories not of the form of ([Hopkins-Singer 02](#HopkinsSinger02)) but represented by [[stable homotopy types]] in a [[cohesive (∞,1)-topos]] (see [Schreiber 13](#Schreiber13), [BunkeNikolausVölkl 13](#BunkeNikolausVoelkl13)) and hence fitting into such a diagram, where the interpretation of the pieces of the diagram is just as it should be.

Therefore it makes sense to define generally that a _differential cohomology diagram_ is the above combined [[fracture squares]] with its outer [[homotopy fiber sequences]] for [[shape modality]] and [[flat modality]] in any [[cohesive (∞,1)-topos]].

## Definition/Construction

### Differential coefficients and Maurer-Cartan forms

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]], write $T\mathbf{H}$ for its [[tangent cohesive (∞,1)-topos]] and write $T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ for the [[stable (∞,1)-category]] of [[spectrum objects]] inside it.

As usual, we use the following notation.

+-- {: .num_defn #DifferentialCoefficients}
###### Definition (Notation)

Write

* $\Pi$ for the  [[shape modality]]

* $\flat$ for the [[flat modality]]

restricted to $Stab(\mathbf{H}) \hookrightarrow  T \mathbf{H}$, respectively.

Given $E \in $ [[Spectra]] we say that $\hat E \in Stab(\mathbf{H})$ with an [[equivalence]]

$$
  \Pi(\hat E) \simeq E
$$

is a _cohesive refinement_ or _differential refinement_ of $E$.

Write

* $\flat_{dR}$ for the [[homotopy fiber]] of the [[suspension]] of the $\flat$-[[counit of a comonad|counit]]

* $\Pi_{dR}$ for the [[homotopy cofiber]] of the looping of the [[unit of a monad|unit]] of $\Pi$.

* $\theta \;\colon\; id \longrightarrow \flat_{dR}$  for the canonical morphism (itself the [[homotopy fiber]] of $\flat_{dR} \to \flat$) which has the interpretation of being the [[Maurer-Cartan form]].

Notice that on [[stable homotopy types]] we have

$$
  \flat_{dR}A \simeq cofib(\flat A \to A) =  A/(\flat A)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

Warning: Elsewhere we often write $\flat_{dR} \Sigma$ for $\flat_{dR}$ above and $\Pi_{dR}\Omega$ for what is $\Pi_{dR}$ here. That other convention has its advanages in the context of unstable cohesion. Here with stable cohesion the present convention is more natural.

=--

+-- {: .num_remark}
###### Remark

As discussed at _[structures in a cohesive ∞-topos -- de Rham cohomology](cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology)_ there (and as is discussed below at [de Rham coefficients](#DeRhamCoefficients)) for $A \in \mathbf{T}_\ast H$ a [[cohesive homotopy type]] then we may think of $\Pi_{dR} A$ and of  $\flat_{dR}  A$  as being the [[de Rham complex]] with [[coefficients]] in $\Pi(\flat_{dR} \Sigma A)$, truncated to negative degree and to non-negative degree, respectively; the canonical map

$$
  \array{
    \Pi_{dR}A && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}A
    \\
    & \searrow && \nearrow
    \\
    && A
  }
$$

interpreting as the [[de Rham differential]].

Beware that this is a very general conceptualization of de Rham coefficients. In standard examples $\flat_{dR}$ does come from traditional [[differential form]] data (see at _[Deligne coefficients](#DeligneCoefficients)_ and at _[Hopkins-Singer coefficients](#HopkinsSingerCoefficients)_ below), but generally it may have quite different looking models. But in any case $\flat_{dR} \Sigma A$ always has the interpretation of the home of the _[[curvature forms]]_ of cohomology with coefficients in $A$, which makes thinking of $\flat_{dR}$ as producing generalized form data useful.

=--

### Chern character and the differential fracture squares
 {#ChernCharacterAndFractureSquares}

+-- {: .num_defn #GeneralChernCharacter}
###### Definition

The [[shape modality|shape]] of the [[Maurer-Cartan form]]  $\theta$, def. \ref{DifferentialCoefficients}, we call the _[[Chern character]]_

$$
  ch \coloneqq \Pi \theta
  \,.
$$

Its component on $\hat E\in T_\ast \mathbf{H}$ we write $ch_{\hat E}$ or $ch_E$, for short.

=--

+-- {: .num_prop #TheDifferentialFractureSquare}
###### Proposition
**(differential fracture square)**

For every $\hat E \in T_\ast \mathbf{H}$ the [[natural transformation|naturality square]]

$$
  \array{
    \hat E &\stackrel{}{\longrightarrow}& \flat_{dR} \hat E
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    \Pi(\hat E) &\stackrel{ch_E}{\longrightarrow}& \Pi \flat_{dR} \hat E
  }
$$

(of the [[shape modality]] applied to the [[homotopy cofiber]] of the [[counit of a comonad|counit]] of the [[flat modality]]) is an [[(∞,1)-pullback]] square (hence also an [[(∞,1)-pushout]]).

Dually, also the square

$$
  \array{
    \Pi_{dR} \flat \hat E &\longrightarrow& \Pi_{dR} \hat E
    \\
    \downarrow && \downarrow
    \\
    \flat \hat E & \longrightarrow & \hat E
  }
$$

is homotopy cartesian.

=--

This fact was observed in ([Bunke-Nikolaus-Völkl 13, prop. 3.5](#BunkeNikolausVoelkl13)). It may be thought of as an incarnation of the concept of a _[[fracture theorem]]_.

+-- {: .proof}
###### Proof

By [[cohesion]] and [[stable (infinity,1)-category|stability]] we have the diagram

$$
  \array{
    \flat \hat E &\longrightarrow & \hat E &\stackrel{}{\longrightarrow}& \flat_{dR} \hat E
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow
    \\
    \Pi(\flat \hat E) &\longrightarrow& \Pi(\hat E) &\stackrel{}{\longrightarrow}& \Pi(\flat_{dR} \hat E)
  }
$$

where both rows are [[homotopy fiber sequences]]. By [[cohesion]] the left vertical map is an [[equivalence in an (infinity,1)-category|equivalence]]. The claim now follows with the [homotopy fiber characterization](homotopy+pullback#HomotopyFiberCharacterization) of [[homotopy pullbacks]].

The second statement follows dually:

$$
  \array{
    \Pi_{dR} \flat \hat E &\longrightarrow& \Pi_{dR} \hat E &\longrightarrow& \Pi_{dR} \flat_{dR} \hat E
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    \flat \hat E & \longrightarrow & \hat E &\longrightarrow&\flat_{dR} \hat E
  }
  \,.
$$

=--

Notice that generally $\flat_{dR} \hat E$ is a [[flat modality]] "[[anti-modal type]]" in that it is annihilated by $\flat$:

$$
  \flat (\flat_{dR} \hat E) \simeq 0
  \,.
$$

($\flat$-anti-modal types are called "pure" in ([Bunke-Nikolaus-Völkl 13, prop. 3.5](#BunkeNikolausVoelkl13))).

The following says that not only does, by prop.\ref{TheDifferentialFractureSquare}, the $(\Pi \dashv \flat)$-[[fracture square]] exhibit any stable cohesive homotopy type as the pullback of a $\flat$-[[anti-modal type]] along a map of $\Pi$-modal types into its [[shape modality|shape]], but that conversely all homotopy pullbacks of this form are $(\Pi \dashv \flat)$-fracture squares.

+-- {: .num_lemma #ReconstructionOfFractureSquare}
###### Lemma

For

1. $E$ a [[shape modality|shape]]-[[modal type]]

1. $\Omega^{\bullet \geq 0}$ a [[flat modality|flat]]-[[anti-modal type]];

1. $E \stackrel{f}{\longrightarrow} \Pi \Omega^{\bullet \gt 0}$ a morphism;

then the [[homotopy pullback]] square for the [[homotopy fiber product]]

$$
  \hat E \coloneqq E \underset{\Pi\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0}
$$

is equivalently the $(\Pi\dashv \flat)$-[[fracture square]] of $\hat E$
according to prop. \ref{TheDifferentialFractureSquare}:

$$
  \array{
    & \hat E &\longrightarrow& \flat_{dR} \hat E & \simeq \Omega^{\bullet \geq 0}
    \\
    & \downarrow && \downarrow
    \\
    E \simeq & \Pi \hat E &\stackrel{f \simeq ch_{\hat E}}{\longrightarrow}&
    \Pi \flat_{dR} \hat E & \simeq \Pi \Omega^{\bullet \geq 0}
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

First we observe that indeed $\Omega^{\bullet\geq 0}\simeq \flat_{dR} \hat E$. For that consider the following morphisms of homotopy pullback diagrams

$$
  \array{
    0 \simeq & \flat \Omega^{\bullet \geq 0} &\longrightarrow& \Omega^{\bullet \geq 0} &\longrightarrow& \Omega^{\bullet \geq 0}
    \\
    & \downarrow && \downarrow && \downarrow
    \\
    & \flat \Pi \Omega^{\bullet \geq 0} &\stackrel{\simeq}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} &\longrightarrow& 0
    \\
    & \uparrow && \uparrow^{\mathrlap{f}} && \uparrow
    \\
    & \flat E &\stackrel{\simeq}{\longrightarrow}& E &\longrightarrow& 0
    \\
    \\
    \underset{\leftarrow}{\lim}(-) \simeq& \flat \hat E &\longrightarrow& \hat E &\longrightarrow& \flat_{dR}\hat E
  }
$$

Here in the middle column we are showing the homotopy fiber product defining $\hat E$. On the left we have its image under $\flat$, with the $\flat$-counits running horizontally and using that $\flat$ preserves [[(∞,1)-limits]] and that $E$ and $\Pi \Omega^{\bullet \geq 0}$ are $\flat$-[[modal types]] by assumption and by [[cohesion]], while $\Omega^{\bullet \geq 0}$ is $\flat$-[[anti-modal type|anti modal]] by assumption. This and using that by [[stability]] all [[finite (∞,1)-limits]] and [[finite (∞,1)-colimits]] commute produces the [[homotopy cofiber]] morphisms going to the right.

This shows that $\Omega^{\bullet \geq 0} \simeq \flat_{dR} \hat E$ and that the [[projection]] $E \underset{\Pi_{dR}\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0} \to \Omega^{\bullet \geq 0}$ is equivalently the $\flat_{dR}$-[[unit of a monad|unit]] on $\hat E$.

Applying the [[shape modality]] to the right half of this diagram, using that by stability it preserves the homotopy pullbacks gives

$$
  \array{
    & \Pi \Omega^{\bullet \geq 0}
      &\stackrel{\simeq}{\longrightarrow}&
    \Pi \Omega^{\bullet \geq 0}
    \\
    & \downarrow^{\mathrlap{\simeq}} && \downarrow
    \\
    & \Pi \Omega^{\bullet \geq 0} &\longrightarrow& 0
    \\
    & \uparrow^{\mathrlap{f}} && \uparrow
    \\
    & E &\longrightarrow& 0
    \\
    \\
    \underset{\leftarrow}{\lim}(-) \simeq
    &
    E
   &\stackrel{ch_{\hat E}}{\longrightarrow}&
   \Pi \flat_{dR}\hat E
  }
  \,.
$$

This shows that $ch_{\hat E} \simeq f$.

Finally, the dual argument

$$
  \array{
    & \Omega^{\bullet \geq 0}
    &\stackrel{}{\longrightarrow}&
    \Pi \Omega^{\bullet \geq 0}
    \\
    & \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    & \Pi \Omega^{\bullet \geq 0}
    &\stackrel{}{\longrightarrow}&
    \Pi \Omega^{\bullet \geq 0}
    \\
    & \uparrow && \uparrow
    \\
    & E
    &\stackrel{}{\longrightarrow}&
    \Pi E
    & \simeq E
    \\
    \\
    \underset{\leftarrow}{\lim}(-)
    \simeq
    &
    \hat E &\longrightarrow& \Pi(\hat E) & \simeq E
  }
$$

shows that the other [[projection]] $E \underset{\Pi_{dR}\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0} \to E$ is equivalently the [[shape modality|shape]]-[[unit of a monad|unit]] of $\hat E$.

=--

The following proposition says that the construction in lemma \ref{ReconstructionOfFractureSquare} extends to a decomposition of the [[(∞,1)-category]] of cohesive  stable types as a [[homotopy fiber product]] of the [[(∞,1)-category]] of moprhisms of stable [[shape modality|shape]]-[[modal types]] with that of [[flat modality|flat]]-[[anti-modal types]]:

+-- {: .num_prop}
###### Proposition

There is an [[(∞,1)-pullback]] of [[(∞,1)-categories]] of the form

$$

  \array{
    Stab(\mathbf{H}) &\stackrel{\flat_{dR} \Sigma}{\longrightarrow}&
    ker(\flat)
    \\
    \downarrow^{\mathrlap{ch}} && \downarrow^{\mathrlap{\Pi}}
    \\
    Spectra^I &\stackrel{cod}{\longrightarrow}& Spectra
  }
  \,,
$$

where the bottom map is the [[codomain fibration]] of the [[(∞,1)-category of spectra]], $\Pi$ denotes the [[shape modality]] restricted to stable $\flat$-[[anti-modal types]],  $ch$ assigns the [[Chern character map]] of def. \ref{GeneralChernCharacter} and $\flat_{dR} \Sigma$ assigns de Rham coefficients as in def. \ref{DifferentialCoefficients}.

=--

([Bunke-Nikolaus-Völkl 13, prop. 3.5](#BunkeNikolausVoelkl13))

Dually:

+-- {: .num_prop}
###### Proposition

For every $A \in T_\ast \mathbf{H}$ the [[natural transformation|naturality square]]

$$
  \array{
    \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Sigma^{-1} A)
    \\
    \downarrow && \downarrow
    \\
    \flat A &\stackrel{}{\longrightarrow}& A
  }
$$

(of the [[flat modality]] applied to the [[homotopy fiber]] of the [[unit of a monad|unit]] of the [[shape modality]]) is an [[(∞,1)-pullback]] square.

=--

+-- {: .proof}
###### Proof

As before but dually, the diagram extends to a morphism of
[[homotopy cofiber]] diagrams of the form

$$
  \array{
    \flat(\Pi_{dR}  A) &\longrightarrow & \Pi_{dR}(A)
    \\
    \downarrow && \downarrow
    \\
    \flat A &\stackrel{}{\longrightarrow}& A
    \\
    \downarrow && \downarrow
    \\
    \flat \Pi(A) &\stackrel{\simeq}{\longrightarrow}& \Pi(A)
  }
  \,,
$$

and by [[cohesion]] the bottom horizontal morphism is an [[equivalence in an (infinity,1)-category|equivalence]].

=--

### The hexagon diagram
 {#TheHexagonDiagram}

Combining these two statements yields the following ([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)).

+-- {: .num_theorem #TheDifferentialDiagram}
###### Theorem

For $\mathbf{H}$ a [[cohesive (∞,1)-topos]] with [[shape modality]] $\Pi$ and [[flat modality]] $\flat$, then for every [[stable homotopy type]] $A \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}$ the canonical hexagon diagram

$$
  \array{
    &&  \Pi_{dR} A && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR} A
    \\
    & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow
    \\
    \Pi_{dR} \flat  A  && \Downarrow^{\mathrlap{a}} && A && \Downarrow^{\mathrlap{b}} && \Pi \flat_{dR} A
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_A}}
    \\
    && \flat A && \longrightarrow && \Pi A
  }
  \,,
$$

formed from the $\Pi$-[[unit of a monad|unit]] and $\flat$-counit  -- the "differential cohomology hexagon" -- is homotopy exact in that

1. the two squares are [[homotopy pullback]] squares ("[[fracture squares]]");

1. the two diagonals are the [[homotopy fiber sequences]] of the [[Maurer-Cartan form]] $\theta_A$ and its dual;

1. the bottom morphism is the canonical [[points-to-pieces transform]];

1. the top and bottom outer sequences are long [[homotopy fiber sequences]].

=--

+-- {: .proof}
###### Proof

Only the last statement remains to be shown, for that use the [[pasting law]]: this gives the following diagram in which every square and every [[pasting diagram|pasting]] rectangle is a [[homotopy pullback]]

$$
  \array{
    \Pi_{dR} \flat A  &\to & \flat A  &\to& 0
    \\
    \downarrow^{\mathrlap{f_1}} &\swArrow_{\mathrlap{a^{-1}}}& \downarrow && \downarrow
    \\
    \Pi_{dR} A  &\stackrel{f_{2}}{\to}& A &\stackrel{\theta_A}{\to}& \flat_{dR} A
    \\
    \downarrow && \downarrow & \swArrow_{\mathrlap{b}} & \downarrow^{\mathrlap{f_3}}
    \\
    0 &\to& \Pi A &\to& \Pi \flat_{dR} A
  }
  \,.
$$

This exhibits the sequence  $\stackrel{f_1}{\to}\stackrel{\theta_A \circ f_{2}}{\longrightarrow} \stackrel{f_3}{\to}$, which is the top part of the hexagon, as a [[homotopy fiber sequence]].

The dual argument shows that the bottom part of the hexagon is a [[homotopy cofiber sequence]].

=--

+-- {: .num_remark}
###### Remark

Theorem \ref{TheDifferentialDiagram} in particular implies that for [[stable homotopy types|stable]] [[cohesive homotopy types]] $A$ there are natural equivalences

* $\Pi_{dR} A \simeq \Pi_{dR} \flat_{dR} A$

* $\flat \Pi_{dR} A\simeq \Pi \flat_{dR} A$

Given that conceptually, as made explicit in the [Idea-section](#Idea) above, we may think of cocycles in $\Pi \flat_{dR} A$ as the rationalized characteristic classes and of coycles in $\flat \Pi_{dR} A$ as flat differential forms, this expresses the conceptual content of the [[de Rham theorem]].

=--

## Examples

### Ordinary differential cohomology
 {#OrdinaryDifferentialCohomology}

We discuss differential refinements in $\mathbf{H} =$ [[Smooth∞Grpd]] of [[ordinary cohomology]], hence of cohomology with [[coefficients]] in [[chain complexes]] or equivalently, via the [[stable Dold-Kan correspondence]], cohomology [[Brown representability theorem|represented]] by [[spectra]] underlying [[Eilenberg-MacLane spectrum]]-[[module spectra]].

#### De Rham coefficients
 {#DeRhamCoefficients}

We briefly recall the [[smooth spectra]] given by the [[de Rham complex]] (tensored with any [[chain complex]]) from _[smooth spectrum -- Examples -- De Rham spectra](smooth+spectrum#ExamplesDeRhamSpectra)_.

Write $Ch_\bullet$ for the [[(∞,1)-category of chain complexes]] (of [[abelian groups]], hence over the [[ring]] $\mathbb{Z}$ of [[integers]]). It is convenient to choose for $A_\bullet \in Ch_\bullet$ the grading convention

$$
  \array{
    \vdots
    \\
    \downarrow
    \\
    A_{-1}
    \\
    \downarrow
    \\
    A_0
    \\
    \downarrow
    \\
    A_1
    \\
    \downarrow
    \\
    \vdots
   }
$$

such that under the [[stable Dold-Kan correspondence]]

$$
  DK \;\colon\; Ch_\bullet \stackrel{}{\longrightarrow} Spectra
$$

the [[homotopy groups]] of spectra relate to the [[homology groups]] by

$$
  \pi_n(DK(A_\bullet)) \simeq  H_{-n}(A_\bullet)
  \,.
$$

In particular for $A \in $ [[Ab]] an [[abelian group]] then $A[n]$ denotes the chain complex concentrated on $A$ in degree $-n$ in this counting.

The grading is such as to harmonize well with the central example of a sheaf of chain complexes over the site of [[smooth manifolds]], which is the [[de Rham complex]], regarded as a [[smooth spectrum]] via the discussion at _[smooth spectrum -- from chain complexes of smooth modules](smooth+spectrum#FromChainComplexesOfSmoothModules)_

$$
  \Omega^\bullet \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}
$$

$$
  \Omega^{\bullet} \;\colon\;  X\mapsto (\cdots \to 0 \to 0 \to \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X)\stackrel{\mathbf{d}}{\to} \cdots)
$$

with $\Omega^0(X) = C^\infty(X, \mathbb{R})$ in degree 0.

Throughout we keep the inclusion

$$
  C \in Ch_\bullet \stackrel{DK}{\longrightarrow}
  Spectra \stackrel{Disc}{\hookrightarrow}
  Stab(\mathbf{H})
$$

notationally implicit, regarding bare chain complexes as geometrically discrete spectrum objects in smooth differential geometry.

We consider for $n \in \mathbb{N}$ the truncated sheaves of de Rham complexes with [[coefficients]] in a given [[chain complex]]:

+-- {: .num_defn}
###### Definition

Write

$$
  \Omega^{\bullet \geq n} \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}
$$

for the [[smooth spectrum]] given under the [[stable Dold-Kan correspondence]] by the sheaf of truncated [[de Rham complexes]]

$$
  \Omega^{\bullet \geq n} \;\colon\;  X\mapsto (\cdots \to 0 \to 0 \to \Omega^n(X) \stackrel{\mathbf{d}}{\to} \Omega^{n+1}(X)\stackrel{\mathbf{d}}{\to} \cdots)
$$

with $\Omega^n(X)$ in degree $n$ (the $p$th stage in the [[Hodge filtration]]).

=--

More genereally:

+-- {: .num_defn #DeRhamComplexesWithCoefficients}
###### Definition

For $C \in Ch_\bullet$ a [[chain complex]],

write

$(\Omega \otimes C)^{\bullet \geq n}$ for the [[smooth spectrum]] given over each manifold $X$ by the [[tensor product of chain complexes]] followed by truncation as indicated:

$$
  \begin{array}
  (\Omega \otimes C)^{\bullet \geq n}
  &=&
  (\cdots &\to& 0 &\to& 0 &\to& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k} &\stackrel{\mathbf{d} \pm d_{C}}{\to}& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k+1}&\stackrel{\mathbf{d}\pm d_{C}}{\to}& \cdots)
  \\
  \\
   deg && && && n-1 && n && n+1 &&
  \end{array}
 \,.
$$

where the first non-trivial term displayed is in taken to be in degree $n$;

and write $(\Omega \otimes C)^{\bullet \lt n}$ for

$$
  \begin{array}
  (\Omega \otimes C)^{\bullet \lt n}
  &=&
  (
    \cdots
    &\stackrel{\mathbf{d} \pm d_{C}}{\to}&
    \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-2-k}
    &\stackrel{\mathbf{d} \pm d_{C}}{\to}&
    \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-1-k}
    0 &\to& 0 &\to& \cdots )
    \\
    \\
    deg &&  && n-1 && n && n+1  &&
  \end{array}
$$

where again the first non-trivial term displayed is taken to be in degree $n$.

=--

+-- {: .num_remark}
###### Remark

Beware of the degree conventions in
def. \ref{DeRhamComplexesWithCoefficients}: in the first clause the tensor product of complexes is truncated without any shifting, while in the second case the truncation is shifted by one. This notation turns out to well reflect the way that the hexagon, theorem \ref{TheDifferentialDiagram}, decomposes these [[smooth spectra]]  by prop. \ref{DifferentialComponentsOfDeRhamTensorC} and remark \ref{OnDecompositionOfDeRhamWithCoefficients} below.

=--

+-- {: .num_prop #DifferentialComponentsOfDeRhamTensorC}
###### Proposition

For $C \in Ch_\bullet(\mathbb{R})$ a [[chain complex]] of [[real number|real]] [[vector spaces]] and
for all $n \in \mathbb{Z}$

1. $\Pi ((\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n})\simeq C^\bullet$

1. $\flat ((\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n})
   \simeq
   C^{\bullet \geq n} $

1. $\flat_{dR}  (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n}
   \simeq
   (\Omega \otimes_{\mathbb{R}} C^{\bullet \leq n-1})^{\bullet \geq n}$

1. $\Pi_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n}
   \simeq
   (\Omega \otimes_{\mathbb{R}} C )^{\bullet \leq n-1}[-1]$

1. $\Pi \flat_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} \simeq C^{\bullet \leq n-1}$

1. $ch \;\colon\; C \to C_{\bullet\leq n-1} $ is the canonical projection.

=--

([Bunke-Nikolaus-Völkl 13, lemma 4.4](#BunkeNikolausVoelkl13))

+-- {: .num_remark #OnDecompositionOfDeRhamWithCoefficients}
###### Remark

Prop. \ref{DifferentialComponentsOfDeRhamTensorC} means first of all that the [[de Rham complex]] $(\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n}$ is a differential refinement, def. \ref{DifferentialCoefficients}, of the chain complex $C$ (for all $n$)

Moreover it says that $\flat_{dR} \Sigma (-)$ and $\Pi_{dR} \Omega(-)$ are similarly the high degree and low degree truncation, respectively, of the [[de Rham complex]] with coefficients in $\Pi \flat_{dR} \Sigma (-)$.

Specifically for $n = 0$, only the connected part

$$
  D C_{\bullet \leq -1}
$$

appears in the de Rham coefficients, and we have

$$
  \array{
    \Pi_{dR}(\cdots) && \stackrel{\mathbf{d}}{\longrightarrow} &&
    \flat_{dR}(\cdots)
    \\
    \simeq && \simeq  && \simeq
    \\
    \Omega(-,D)^{\bullet \lt 0}
    &&
      \stackrel{\mathbf{d}_{dR} \pm d_{C}}{\longrightarrow}
    &&
    \Omega(-,D)^{\bullet \geq 0}
  }
  \,.
$$

=--

#### Deligne coefficients
 {#DeligneCoefficients}

We discuss the differential cohomology hexagon for smooth [[Deligne cohomology]] and hence smooth [[circle n-bundles with connection]] realizedin $\mathbf{H}= $ [[Smooth∞Grpd]].

+-- {: .num_defn}
###### Definition

For $C \in Ch_\bullet(\mathbb{Z})$ be a [[chain complex]] of [[abelian groups]]. For $n \in \mathbb{N}$ an [[integer]], write $C_{conn,-n}$ for the [[homotopy fiber product]] in

$$
  \array{
    C_{conn,-n} &\longrightarrow& (\Omega \otimes_{\mathbb{Z}} C)^{\bullet \geq n}
    \\
    \downarrow && \downarrow
    \\
    C
     &\longrightarrow&
    C \otimes \mathbb{R} & \simeq \Pi((\Omega \otimes_{\mathbb{Z}} C)^{\bullet \geq n})
  }
  \,,
$$

where the right map is the [[unit of a monad|unit]] of the [[shape modality]] according to prop. \ref{DifferentialComponentsOfDeRhamTensorC}.

=--

([Bunke-Nikolaus-Völkl 13, 4.3](#BunkeNikolausVoelkl13))

+-- {: .num_prop #DecompositionForGeneralSmoothDeligneCoefficients}
###### Proposition

For all $n \in \mathbb{Z}$

1. $\Pi(C_{conn,-n}) \simeq C$

1. $\flat_{dR} C_{conn,-n} \simeq (\Omega \otimes C_{\leq n-1})^{\bullet \geq n}$

=--

([Bunke-Nikolaus-Völkl 13, lemma 4.5](#BunkeNikolausVoelkl13))

+-- {: .num_example #ExampleOrdinaryDeligneCohomology}
###### Example
**(ordinary Deligne cohomology)**

For $C = \mathbb{Z}[n+1]$ we have

$$

  (\Omega \otimes (\mathbb{Z}[n+1]\otimes \mathbb{R}))^{\bullet \geq 0}
  \simeq
  (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq 0}
  \simeq
  \Omega^{n+1}_{cl}(-)
$$

(with the last term being in degree 0) and more generally for $n \in \mathbb{N}$

$$
  (\Omega \otimes (\mathbb{Z}[n+1]\otimes \mathbb{R}))^{\bullet \geq -n}
  \simeq
  (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -n}
  \simeq
  (\Omega^1(-)
     \stackrel{\mathbf{d}}{\to}
   \Omega^2(-)
      \stackrel{\mathbf{d}}{\to}
      \cdots
      \stackrel{\mathbf{d}}{\to}
   \Omega^{n+1}_{cl}(-)
  )
$$

(with the last term being in chain degrees $-n$ to 0 (hence homotopy group degrees $n$ to 0)).

{#PastingForDeligneTower} We have a pasting diagram of [[homotopy pullbacks]]
$$
  \array{
    B^n U(1)_{conn} &\to& \Omega^{n+1}_{cl}
    \\
    \downarrow && \downarrow
    \\
     &\stackrel{}{\longrightarrow}& (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -1}
    \\
    \downarrow && \downarrow
    \\
    \vdots && \vdots
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}^n U(1) &\stackrel{}{\longrightarrow}& (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -n}
    \\
    \downarrow && \downarrow
    \\
    B^{n+1}\mathbb{Z} &\to& B^{n+1} \mathbb{R}
  }
$$

where on the right runs the [[Hodge filtration]],
with the notation $\mathbf{B}^n U(1) = \mathbb{Z}[n+1]_{conn,n}$ as at [[circle n-group]] and  $\mathbf{B}^n U(1)_{conn} = \mathbb{Z}[n+1]_{conn,0}$ as at _[[circle n-bundle with connection]]_.

Hence for all $n \in \mathbb{N}$

$$
  \mathbb{Z}[n+1]_{conn,n} \simeq \mathbf{B}^n U(1)
  \simeq
  (C^\infty(-,U(1)) \to 0 \to \cdots\to 0)
$$

and

$$
  \mathbb{Z}[n+1]_{conn,0} \simeq \mathbf{B}^n U(1)_{conn}
  \simeq
  (C^\infty(-,U(1))\stackrel{\mathbf{d}log}{\to}\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to}\Omega^n)
  \,.
$$

is represented by the [[Deligne complex]].

The intermediate cases in between these two as in ([Schreiber 13, def. 3.9.46](#Schreiber13), [FRS 13, remark 2.3.15](#FRS13)) (discussed also for instance at _[Courant algebroid -- Relation to Atiyah Lie 2-algebroid](http://ncatlab.org/nlab/show/Courant+algebroid#RelationToAtiyahGroupoids)_).

Notice that by prop. \ref{DecompositionForGeneralSmoothDeligneCoefficients} the above homotopy pullback diagrams are all indeed the right part of the differential cohomology hexagon, theorem \ref{TheDifferentialDiagram}, e.g.

$$
  \array{
     \mathbf{B}^n U(1)_{conn}
     &\stackrel{\theta_{\mathbf{B}^n U(1)_{conn}}}{\longrightarrow}&
     \Omega^{n+1}_{cl}
     \\
     \downarrow && \downarrow
     \\
     \vdots && \vdots
     \\
     \downarrow && \downarrow
     \\
     \mathbf{B}^n U(1)
     &\stackrel{\theta_{\mathbf{B}^n U(1)}}{\longrightarrow}&
     (\Omega^1 \to \cdots\to \Omega^{n+1}_{cl})
     \\
     \downarrow && \downarrow
     \\
     \mathbf{B}^{n+1}\mathbb{Z} &\stackrel{\Pi(\theta_{\mathbf{B}^n U(1)})}{\longrightarrow}&\mathbf{B}^{n+1}\mathbb{R}
  }
  \,.
$$

The differential cohomology hexagon obtained from this is on cohomology classes for each [[smooth manifold]] $X$ and each $n \in \mathbb{N}$

$$
  \array{
    && \Omega^{n}(X)/im(\mathbf{d}) && \stackrel{\mathbf{d}}{\longrightarrow} &&  \Omega^{n+1}_{cl}(X)
    \\
    & \nearrow && \searrow^{\mathrlap{a}} && \nearrow && \searrow
    \\
    H^{n}(X, \mathbb{R})
    && &&
    H^0(X,\mathbf{B}^n U(1)_{conn})
    && &&
    H^{n+1}(X,\mathbb{R})
    \\
    & \searrow && \nearrow && \searrow && \nearrow
    \\
    && H^{n}(X,U(1)) && \underset{}{\longrightarrow} && H^{n+1}(X,\mathbb{Z})
  }
$$

It is common (e.g. [Simons-Sullivan 07](#SimonsSullivan07)) to display this after quotienting out the [[kernel]] of $a$ (which is the group $\Omega^n(X)_{\mathbb{Z}}$ of [[closed differential forms]] with [[integer|integral]] [[periods]]), which is such as to make the top-left to bottom-right diagonal sequence be not just [[exact sequence|exact]] at the middle term, but be a [[short exact sequence]]:

$$
  \label{OrdinarCohomologyHexagon}
  \array{
    0 & && && && & 0
    \\
    & \searrow && && &&  \nearrow
    \\
    && \Omega^{n}(X)/\Omega^n(X)_{\mathbb{Z}} && \stackrel{\mathbf{d}}{\longrightarrow} &&  \Omega^{n+1}_{cl}(X)
    \\
    & \nearrow && \searrow^{\mathrlap{a}} && \nearrow && \searrow
    \\
    H^{n}(X, \mathbb{R})
    && &&
    H^0(X,\mathbf{B}^n U(1)_{conn})
    && &&
    H^{n+1}(X,\mathbb{R})
    \\
    & \searrow && \nearrow && \searrow && \nearrow
    \\
    && H^{n}(X,U(1)) && \underset{\beta}{\longrightarrow} && H^{n+1}(X,\mathbb{Z})
    \\
    & \nearrow && && &&  \searrow
    \\
    0 & && && && & 0
  }
  \,.
$$

Here the diagonals are now the "curvature exact sequence" and the "characteristic class exact sequence" as discussed at _[ordinary differential cohomology -- Properties -- curvature and characteristic class](ordinary+differential+cohomology#CurvatureAndCharClass)_.

The bottom horizontal map is the [[Bockstein homomorphism]] of the [[exponential sequence]] ([this example](Bockstein+homomorphism#Mod2BocksteinAndExponentialExactSequence)).

=--

### Hopkins-Singer coefficients
 {#HopkinsSingerCoefficients}

We discuss how the [[differential function complexes]] of ([[Quadratic Functions in Geometry, Topology,and M-Theory|Hopkins-Singer 05]]), providing differential refinements of geometrically discrete [[spectra]], and how they fit into their differential cohomology hexagon (following [Bunke-Nikolaus-Völkl 13, section 4.4.](#BunkeNikolausVoelkl13)).

Let again $\mathbf{H} = $ [[Smooth∞Grpd]].

Throughout, we leave notationally implicit

1. the [[stable Dold-Kan correspondence]] $DK \;\colon\;Ch_\bullet \longrightarrow Spectra$;

1. the inclusion $Spectra \stackrel{Stab}{\hookrightarrow} Stab(\mathbf{H})$;

hence always regard a [[chain complex]] as the corresponding [[spectrum]] and regard any bare spectrum always as a [[geometrically discrete infinity-groupoid|geometrically discrete]] [[smooth spectrum]].

+-- {: .num_defn #HopkinsSingerPullback}
###### Definition

For

1. $E \in Spectra \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H})$
a [[spectrum]],

1. $C \in Ch_\bullet(\mathbb{R})$ a [[chain complex]] of [[real number|real]] [[vector spaces]];

1. $c \colon E \longrightarrow C$ a [[homomorphism]] of [[spectra]] from $E$ to the image of $C$ under the [[stable Dold-Kan correspondence]];

1. $n \in \mathbb{N}$

write $E_{conn_c,-n}$ for the [[homotopy fiber product]] in

$$
  \array{
    E_{conn_c,-n} &\longrightarrow& (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq -n}
    \\
    \downarrow && \downarrow
    \\
    E &\longrightarrow& C & \simeq \Pi((\Omega \otimes C)^{\bullet \geq -n})
  }
  \,,
$$

where the map on the right is the [[unit of a monad|unit]] of the [[shape modality]] according to prop. \ref{DifferentialComponentsOfDeRhamTensorC}.

We abbreviate

$$
  E_{conn_c} \coloneqq E_{conn_c,0}
  \,.
$$

=--

This appears in ([Bunke-Nikolaus-Völkl 13, 4.4](#BunkeNikolausVoelkl13)), with an earlier version in ([Bunke-Gepner 13, def. 2.1](#BunkeGepner13)).

+-- {: .num_remark}
###### Remark

Specifically when $C \simeq E \wedge DK(\mathbb{R})$ is a model for the rationalization (realification) of $E$ and $c \colon E \to C$ is the canonical map, then it is a model for the [[Chern character]] and def. \ref{HopkinsSingerPullback} is essentially the definition of ([[Quadratic Functions in Geometry, Topology,and M-Theory|Hopkins-Singer 05]]).

=--

+-- {: .num_prop #DifferentialComponentsOfHopkinsSingerCoefficients}
###### Proposition

For all $n \in \mathbb{Z}$

1. $\flat_{dR}  E_{conn_c,-n} \simeq (\Omega \otimes_{\mathbb{R}}  C_{\bullet \leq -n -1})^{\bullet\geq n}$

1. $\Pi(E_{conn_c,-n}) \simeq E$.

=--

([Bunke-Nikolaus-Völkl 13, lemma 4.7](#BunkeNikolausVoelkl13))

+-- {: .num_remark}
###### Remark

Beware that in prop. \ref{DifferentialComponentsOfHopkinsSingerCoefficients}
the intrinsic de Rham coefficients $\flat_{dR} \Sigma E_{conn_c,-n}$ in general differ by a truncation from the de Rham complex $(\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq -n}$ which is being pulled back in definition \ref{HopkinsSingerPullback}.

But by prop. \ref{DifferentialComponentsOfHopkinsSingerCoefficients} and theorem \ref{TheDifferentialDiagram} composition with the truncation map preserves the [[homotopy pullback]], hence in the differential cohomology hexagon the two differ only in the exactness property in the top right entry.

=--

+-- {: .num_cor}
###### Corollary

Given $E_{conn_c} \coloneqq E_{conn_c,0}$ as in def. \ref{HopkinsSingerPullback},
then for each [[smooth manifold]] $X$ (indeed for each [[smooth ∞-groupoid]] $X$) the differential cohomology diagram on cohomology classes is of the following form

$$
  \array{
    && (\Omega(X) \otimes_{\mathbb{R}} C)^{-1}/{im(\mathbf{d})}
    && \stackrel{\mathbf{d}}{\longrightarrow} &&
    (\Omega(X)\otimes_{\mathbb{R}} C)^0_{cl}
    \\
    & \nearrow && \searrow && \nearrow && \searrow
    \\
    H^{-1}(X,C)
    && &&
    E_{conn_c}^0(X)
    && &&
    H^0(X,C)
    \\
    & \searrow && \nearrow && \searrow && \nearrow
    \\
    && \flat E^0(X)
    && \longrightarrow &&
    E^0(X)
  }
  \,.
$$

=--

([Bunke-Nikolaus-Völkl 13, (25)-(26)](#BunkeNikolausVoelkl13))

### Differential K-theory
 {#DifferentialKTheory}

We discuss several differential refinements, def. \ref{DifferentialCoefficients}, to [[smooth spectra]], of the complex [[topological K-theory]] spectrum [[KU]] or of its [[connective cover]] $ku$, hence versions of [[differential K-theory]].

#### Hopkins-Singer differential $ku$
 {#HSDifferentialKU}

Let $\mathbb{C}[b]$ be the [[polynomial ring]] with [[complex numbers]] [[coefficients]] on a single generator $b$. With $b$ regarded as in degree 2, regard this as a [[chain complex]] with vanishing [[differentials]].

Then the complex ordinary [[Chern character]] on K-theory is a map

$$
  ch\;\colon\; ku \longrightarrow \mathbb{C}[b]
  \,.
$$

+-- {: .num_defn #HSDifferentialKu}
###### Definition

Write

$$
  ku_{conn_ch}
  \in
  Stab(Smooth\infty Grpd)
$$

for the [[smooth spectrum]] induced by def. \ref{HopkinsSingerPullback}) from the standard [[Chern character]] $c \coloneqq ch$ on connective [[KU|ku]].

=--

([Hopkins-Singer 02, section 4.4](#HopkinsSinger02), [Bunke-Nikolaus-Völkl 13, section 6.1](#BunkeNikolausVoelkl13))

#### Algebraic K-theory of smooth manifolds and the $e$-invariant
 {#SoothVectorBundlesWithConnectionAndEInvariant}

Write

1. $\mathbf{Vect}^{\oplus} \in Sh(SmthMfd) \hookrightarrow \mathbf{H} = Sh_\infty(SmthMfd)$ for the [[stack]] of (complex) [[vector bundles]] equipped with the [[direct sum]];

1. $\mathbf{Vect}_{conn}^{\oplus} \in Sh(SmthMfd)$ for the [[stack]] of (complex) [[vector bundles]] with [[connection on a vector bundle|connection]], equipped with direct sum;

Write

$$
  \mathcal{K} \;\colon\;  CMon_\infty(\infty Cat) \longrightarrow Spectra
$$

for the [[(∞,1)-functor]] from [[symmetric monoidal (∞,1)-categories]]
into [[spectra]] which produces the [[algebraic K-theory of symmetric monoidal (∞,1)-categories]].

+-- {: .num_defn #KTheoryOfStackOfVectorBundles}
###### Definition

Forming objectwise the [[algebraic K-theory of symmetric monoidal (∞,1)-categories]] and then [[∞-stackification|∞-stackifying]] (which we leave notationally implicit) produces [[smooth spectra]]

$$
  \mathcal{K}(\mathbf{Vect}_{conn}^\oplus),\;\mathcal{K}(\mathbf{Vect}^\oplus)
  \in
  Stab(Smooth\infty Grpd)
  \,.
$$

This may be called the _[[algebraic K-theory of smooth manifolds]]_.

=--

+-- {: .num_prop #FlatAndPiForDifferentialKTheory}
###### Proposition

The [[smooth spectra]] $\mathcal{K}(\mathbf{Vect})$ and $\mathcal{K}(\mathbf{Vect}_{conn})$ are both differential refinements, def. \ref{DifferentialCoefficients}, of connective [[KU|ku]], both whose underlying [[geometrically discrete infinity-groupoid|geometrically discrete spectrum]] is the [[algebraic K-theory]] $K \mathcal{C}$ of the [[complex numbers]]:

1. $\Pi(\mathcal{K}(Vect^\oplus)) \simeq \Pi(\mathcal{K}(Vect_{conn}^\oplus))\simeq ku$;

1. $\flat(\mathcal{K}(Vect^\oplus)) \simeq \flat(\mathcal{K}(Vect_{conn}^\oplus))\simeq K \mathbb{C}$.

1. in both cases the [[points-to-pieces transform]] $\flat \to \Pi$ is the [[comparison map between algebraic and topological K-theory]] $K \mathbb{C}\to ku$.

=--

([Bunke-Nikolaus-Völkl 13, lemma 6.3, corollary 6.5](#BunkeNikolausVoelkl13))

+-- {: .num_prop #SmoothRegulator}
###### Proposition

The traditonal [[Chern character]] induces, via the [[universal property]] of the [[homotopy pullback]] in def. \ref{HopkinsSingerPullback}, a morphism

$$
  \mathcal{K}(\mathbf{Vect}_{conn}^\oplus)
  \longrightarrow
  ku_{conn_{ch}}
$$

from the differential refinement of $ku$ given by def. \ref{KTheoryOfStackOfVectorBundles}, to that given by def. \ref{HSDifferentialKu}.

Its [[secondary characteristic class]], hence its image under the [[flat modality]], is in [[homotopy groups]] the [[regulator]]

$$
  \array{
    K\mathbb{C}_{tor} &\stackrel{\simeq}{\longrightarrow}& \mathbb{Q}/\mathbb{Z}
    \\
    \downarrow && \downarrow
    \\
    K\mathbb{C} &\stackrel{\pi_{2k-1}}{\longrightarrow}& \mathbb{C}/\mathbb{Z}
  }
$$

related to the [[Adams e-invariant]] via ([Bunke 11, section 5.3](#Bunke11)).

=--

([Bunke-Nikolaus-Völkl 13, example 6.9](#BunkeNikolausVoelkl13))

#### Via smooth Snaith's theorem
 {#ViaSmoothSnaithTheorem}

The ordinary [[Snaith theorem]] realizes [[KU]] as the localization of the [[∞-group ∞-ring]] of the [[circle 2-group]] $B U(1)$ at the [[Bott element]] $b\in \pi_2(KU)$.

$$
  KU \simeq (\Sigma_+^\infty B U(1))[b^{-1}]
  \,.
$$

With due care to define smooth geometric looping and taking the [[Hopf fibration]] as an actual smooth bundle over the smooth 2-sphere, this lifts to the smooth circle 2-group $\mathbf{B}U(1)$ and with more care to $\mathbf{B}U(1)_{conn}$ to produce [[smooth spectra]]

$$
  (\Sigma_+ \mathbf{B}U(1))[b^{-1}]
$$

and

$$
  (\Sigma_+ \mathbf{B}U(1)_{conn})[b^{-1}]
$$

which both are differential refinements, def. \ref{DifferentialCoefficients}, of [[KU]].

([Bunke-Nikolaus-Völkl 13, section 6.3](#BunkeNikolausVoelkl13))

### Complex-analytic differential generalized cohomology

The above examples all take place in the $T$[[Smooth∞Grpd]], modelling [[higher differential geometry]]. Another fundamental example of a [[cohesive (∞,1)-topos]] is [[ComplexAnalytic∞Grpd]], modelling [[complex analytic higher geometry]].

Differential cohomology in the complex-analytic context is discussed in ([Hopkins-Quick 12](#HopkinsQuick12)), see at
_[complex analytic ∞-groupoid -- Properties -- Differential cohomology](complex+analytic+∞-groupoid#DifferentialCohomology)_ for more.

### $E_\infty$-Arithmetic geometry

[[!include arithmetic cohesion -- table]]

## Related concepts

* [[intermediate Jacobian]]

## References
 {#References}

The differential cohomology hexagon was maybe first highlighted in the context of [[ordinary differential cohomology]]

* {#SimonsSullivan07} [[James Simons]], [[Dennis Sullivan]], _Axiomatic Characterization of Ordinary Differential Cohomology_,  Journal of Topology 1.1 (2008): 45-56. ([arXiv:math/0701077](http://arxiv.org/abs/math/0701077))

and in the context of [[differential K-theory]] in

* {#SimonsSullivan08} [[James Simons]], [[Dennis Sullivan]], _Structured vector bundles define differential K-theory_ ([arXiv:0810.4935](http://arxiv.org/abs/0810.4935))

discussed also in

* {#FreedLott10} [[Daniel Freed]], [[John Lott]], _An index theorem in differential K-theory_, Geometry and Topology 14 (2010) ([pdf](http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf))

* Jiahao Hu, *Characterization of differential K-theory by hexagon diagram* [[arXiv:2209.04925](https://arxiv.org/abs/2209.04925)]

An artistic impression of the differential cohomology hexagon as a dance choreography inspired by a talk by [[James Simons]] is at

* Kyla Barkin, Aaron Selissen, _Differential Cohomology_, 2011 ([full video (37 min)](http://vimeo.com/32903887), [sequence of "The March Back" (4 min)](http://www.dancemedia.com/v/7044), [press release](http://www.businessinsider.com/jim-simons-mathematical-genius-inspired-a-ballet-2011-1))

That the differential cohomology hexagon exists not just at the level of [[exact sequences]] of [[cohomology classes]] but already at the level of [[homotopy fiber sequences]] of [[cocycle]] spaces was observed for [[ordinary differential cohomology]] and for [[differential K-theory]] in

* Man-Ho Ho, _Refined hexagons for differential cohomology_ ([arXiv:1310.0582](http://arxiv.org/abs/1310.0582))

That generally the [[homotopy fiber product]]-construction of differential refinements of [[generalized (Eilenberg-Steenrod) cohomology]] theories due to

* {#HopkinsSinger02} [[Mike Hopkins]] and [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_ ([arXiv:math/0211216](http://arxiv.org/abs/math/0211216))

constitutes the right one of the two squares in the homotopy-theoretic version of the diagram is discussed explicitly for instance in prop 4.57 of

* {#Bunke12} [[Ulrich Bunke]], _Differential cohomology_ ([arXiv:1208.3961](http://arxiv.org/abs/1208.3961))

That every [[stable homotopy type]] in a [[cohesive (∞,1)-topos]] naturally sits in a differential cohomology diagram was observed (focusing on [[Smooth∞Grpd]]) in

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], *Differential cohomology theories as sheaves of spectra*, Journal of Homotopy and Related Structures **11**  (2016) 1–66 $[$[arXiv:1311.3188](http://arxiv.org/abs/1311.3188), [doi:10.1007/s40062-014-0092-5](https://doi.org/10.1007/s40062-014-0092-5)$]$

following

* {#Schreiber13} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

where in section 4.1.2 a fully general abstract account is given. This in turn follows

* {#Schreiber09} [[Urs Schreiber]], _[[schreiber:Background fields in twisted differential nonabelian cohomology]]_, talk at Oberwolfach Workshop Strings, Fields, Topology (2009), Oberwolfach Report No. 28/2009 ([pdf](https://www.mfo.de/document/0924/OWR_2009_28.pdf))

Surveys/expositions of this:

* {#Schreiber14a} [[Urs Schreiber]], _[[schreiber:Differential generalized cohomology in Cohesive homotopy type theory]]_, talk at IHP trimester on _Semantics of proofs and certified mathematics_, _Workshop 1: Formalization of Mathematics_, Institut Henri Poincaré, Paris, 5-9 May 2014

* {#Schreiber14b} [[Urs Schreiber]], _[[schreiber:Differential cohomology is Cohesive homotopy theory]]_, talk at _[Higher Geometric Structures along the Lower Rhine June 2014](http://www.ru.nl/math/research/vmconferences/2014-higher/)_, 19-20 June  2014

* {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), Section 2.3 in: _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250))

and from the point of view of [[modal homotopy type theory|modal homotopy theory]]:

* [[Urs Schreiber]], _[[nLab:differential cohesion and idelic structure|Fractures, Ideles and the Differential Hexagon]]_, talk at _[Workshop on differential cohomologies](http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html)_, CUNY Graduate Center, August 12-14 2014 ([video recording](http://videostreaming.gc.cuny.edu/videos/video/1806/in/channel/55/))

* [[Urs Schreiber]], _[[schreiber:Differential cohesion and Arithmetic geometry]]_, talk at [Karslruher Weihnachts-Workshop 2015](http://www.math.kit.edu/iag3/~herrlich/seite/weihnachts-workshops)

{#ReferencesFormalizationInCohesiveHoTT} Formalization in [[cohesive homotopy type theory|cohesive]] [[modal homotopy type theory|modal homotopy type theory]] [[homotopy type theory]]:

* {#Myers21} [[David Jaz Myers]], _Modal Fracture of Higher Groups_,  talk at _[CMU-HoTT Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html), 2021 ([pdf](http://davidjaz.com/Talks/CMU_March_2021.pdf), [[MyersModalFracture2021.pdf:file]])

* [[David Jaz Myers]], _Modal Fracture of Higher Groups_ ([arXiv:2106.15390](https://arxiv.org/abs/2106.15390))

For discussion of the hexagon in an [[equivariant cohomology|equivariant]] setting, see

* [[Andreas Kübel]], [[Andreas Thom]], _Equivariant Differential Cohomology_, ([arXiv:1510.06392](https://arxiv.org/abs/1510.06392))

See also

* {#Bunke11} [[Ulrich Bunke]], _On the topological contents of eta invariants_ ([arXiv:1103.4217](http://arxiv.org/abs/1103.4217))

* {#HopkinsQuick12} [[Michael Hopkins]], [[Gereon Quick]], _Hodge filtered complex bordism_ ([arXiv:1212.2173](http://arxiv.org/abs/1212.2173))

* {#FRS13} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Chris Rogers]], _Higher geometric prequantum theory_ ([arxiv:1304.0236](http://arxiv.org/abs/1304.0236))

* {#BunkeGepner13} [[Ulrich Bunke]], [[David Gepner]], _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_ ([arXiv:1306.0247](http://arxiv.org/abs/1306.0247))

[[!redirects differential cohomology diagrams]]

[[!redirects differential cohomology hexagon]]

[[!redirects differential hexagon]]


differential cohomotopy

> under construction

#Contents#
* table of contents
{:toc}

## Idea

A _differential Cohomotopy cohomology theory_ is refinement of [[Cohomotopy cohomology theory]] to a [[differential cohomology theory]].

(...)

## Related concepts

[[!include flavours of cohomotopy -- table]]

## References

Discussion of differential [[stable Cohomotopy]]:

* {#Bunke12} [[Ulrich Bunke]], Ex. 4.120 of: _Differential cohomology_ [[arXiv:1208.3961](http://arxiv.org/abs/1208.3961)]

Discussion of differential non-abelian (unstable) [[Cohomotopy]]:

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §4 in: *[[schreiber:The WZW term of the M5-brane and differential cohomotopy]]*, J. Math. Phys. **56** 102301 (2015) [[arXiv:1506.07557](https://arxiv.org/abs/1506.07557), [doi:10.1063/1.4932618](https://doi.org/10.1063/1.4932618)]

* [[Daniel Grady]], [[Hisham Sati]], Def. 3.2 in: *Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers*, Journal of Geometry and Physics **165** (2021) 104203 [[arXiv:2001.07640](https://arxiv.org/abs/2001.07640), [doi:10.1016/j.geomphys.2021.104203](https://doi.org/10.1016/j.geomphys.2021.104203)]

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Ex. 9.3 in: _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalized]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

[[!redirects differential Cohomotopy]]

[[!redirects differential cohomotopy theory]]
[[!redirects differential cohomotopy theories]]

[[!redirects differential Cohomotopy theory]]
[[!redirects differential Cohomotopy theories]]

[[!redirects differential cohomotopy cohomology theory]]
[[!redirects differential cohomotopy cohomology theories]]

[[!redirects differential Cohomotopy cohomology theory]]
[[!redirects differential Cohomotopy cohomology theories]]


differential crossed module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _differential crossed module_ (or crossed module of/in [[Lie algebras]]) is a way to encode the structure of a [[strict Lie 2-algebra]] in terms of two ordinary [[Lie algebras]].

This is the [[infinitesimal object|infinitesimal]] version of how a smooth [[crossed module]] encodes a smooth [[strict 2-group]].

## Definition

### As crossed modules of Lie algebras

A **differential crossed module** $\mathfrak{g}$ is

* a pair of [[Lie algebra]]s $\mathfrak{g}_0$ and $\mathfrak{g}_1$

* equipped with two Lie algebra homomorphisms

  * $\partial : \mathfrak{g}_1 \to \mathfrak{g}_0$

  * $ \rho : \mathfrak{g}_0 \to Der(\mathfrak{g}_1)$

    (to the [[Lie algebra]] of Lie [[derivation]]s)

* such that for all $x \in \mathfrak{g}_0, b,b' \in \mathfrak{g}_1$ we have

  * $\partial  ( \rho(x)(b) ) = [x, \partial(b)]$

  * $\rho(\partial b)(b') = [b, b']$.

Notice that the Lie algebra structure on $\mathfrak{g}_1$ is already fixed by the rest of the data. So a differential crossed module may equivalently be thought of as extra structure on a Lie module $\mathfrak{g}_1$ of $\mathfrak{g}_0$. This leads over to the following perspective.

### As dg-Lie algebras

Equivalently, a  differential crossed module is a [[dg-Lie algebra]] structure on a [[chain complex]] $(\mathfrak{g}_1 \stackrel{\partial}{\to} \mathfrak{g}_0)$ concentrated in degrees 0 and 1.

The components of the dg-Lie bracket are

* the given bracket $[-,-] : \mathfrak{g}_0 \otimes \mathfrak{g}_0 \to \mathfrak{g}_0$;

* the mixed bracket

  $$
    [-,-] : \mathfrak{g}_0 \otimes \mathfrak{g}_1 \to \mathfrak{g}_1
  $$

  identifies with the action:

  $$
    [x,b] := \rho(x)(b)
    \,.
  $$

This way the respect of the dg-bracket for the differential

$$
  \partial [x,b] = [\partial x, b ] + [x, \partial b] = [x,\partial b]
$$

is equivalently the above condition

$$
  \partial \rho(x)(b) = \rho(x)(\partial b)
  \,.
$$

### As strict Lie 2-algebras

By the discussion there, [[dg-Lie algebra]]s are _strict_ [[L-∞ algebra]]s (those for which all the brackets of higher arity vanish). Therefore the above identification of differential crossed modules with 2-term [[dg-Lie algebra]]s identifies these also with [[strict Lie 2-algebra]]s.

## Related concepts

* [[Lie algebra]], [[Lie group]]

* [[Lie 2-algebra]], [[Lie 2-group]]

* [[L-infinity algebra]], [[smooth ∞-group]]

* [[Lie algebroid]], [[Lie groupoid]]

* [[L-∞ algebroid]], [[smooth ∞-groupoid]]

## References

See also the references at *[[Lie 2-algebra]]*.

On the history of this and related concepts:

* {#Huebschmann22} [[Johannes Huebschmann]], *On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras, J. Geom. Mech. 13 (2021), 385-402 * [[arXiv:2208.02539](https://arxiv.org/abs/2208.02539)]

The [[crossed modules]] (in groups or Lie groups) and the differential crossed modules are examples of the [[internal crossed module]]s. A good theory of them is developed in [[semiabelian categories]].

* [[George Janelidze]], _Internal crossed modules_, Georgian Math. J. 10 (2003) 99114 ([journal](http://www.heldermann.de/GMJ/GMJ10/GMJ101/gmj1008.htm)).

[[!redirects differential crossed modules]]
[[!redirects crossed module in Lie algebras]]
[[!redirects Lie algebra crossed module]]

[[!redirects Lie crossed module]]
[[!redirects Lie crossed modules]]


differential elliptic cohomology

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###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

_Differential elliptic cohomology_ is the refinement of [[elliptic cohomology]] to [[differential cohomology]].

This may be obtained from [[differential cobordism cohomology]] by tensoring the latter with a [[Landweber exactness|Landweber exact ring]].

## References

A model for a multiplicative differential refinement of [[complex cobordism cohomology theory]], the theory represented by the [[Thom spectrum]] is in

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schröder,  Moritz Wiethaup,  _Landweber exact formal group laws and smooth cohomology theories_ ([arXiv:0711.1134](http://arxiv.org/abs/0711.1134))

As discussed there, this model is compatible with tensoring over Landweber exact rings and hence induces in particular a notion of differential elliptic cohomology.

[[!redirects differential elliptic cohomology theory]]


differential equation

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### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _differential equation_ is an [[equation]] involving [[terms]] that are [[derivatives]] (or [[differentials]]).  One sometimes distinguishes _partial_ differential equations (which involve [[partial derivatives]]) from _[[ordinary differential equation|ordinary]]_ differential equations (which don\'t).

## As sub-$\infty$-groupoids of tangent and jet bundle

Analogous to how ordinary equations determine and are determined by their spaces of solutions -- the corresponding [[schemes]] -- accordingly differential equations $F(\partial^i x)$ on sections of a bundle $E \to X$ determine and are determined by their solution spaces, which are sub-[[D-schemes]] of the [[jet bundle]] D-scheme:

$$
  \array{
    Solutions(F) &&\hookrightarrow&& Jet(E)
    \\
    & \searrow && \swarrow
    \\
    && \Im(X)
  }
$$

In fact there is an [[equivalence of categories]] between the [[Eilenberg-Moore category]] of the [[jet comonad]] over $X$ and the category of partial differential equations with variables in $X$ ([Marvan 86](#Marvan86)).

At least in nice cases for differential equations on functions on $X$ this is equivalently modeled by sub-[[Lie algebroids]] of [[tangent Lie algebroids]].

$$
  \array{
    Solutions(F) &&\hookrightarrow&& T X
  }
$$

see [[exterior differential system]] for details

## In terms of synthetic differential geometry
  {#InSynthDiff}

A perspective on differential equations from the [[nPOV]] of [[synthetic differential geometry]] is given in

* [[William Lawvere]], _Toposes of laws of motion_ , transcript of a talk in Montreal, Sept. 1997 ([pdf](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf))

  (on the description of [[differential equation|differential equations]] in terms of synthetic differential geometry)

See also the appendix of

* _Outline of synthetic differential geometry_ , seminar notes (1998) ([pdf](http://www.acsu.buffalo.edu/~wlawvere/SDG_Outline.pdf))

### First order homogeneous ODEs as extension problems in a smooth topos

In a [[smooth topos]] with $X$ and $A$ any two objects and $D = \{x \in R | x^2 = 0\}$ the [[infinitesimally thickened point|abstract tangent vector]], a [[diagram]], of the form

$$
  \array{
    {*}&\to& D &\stackrel{v}{\to}& X
    \\
    &{}_{\mathllap{v(f)(x)}}\searrow& {}^{\mathllap{v(f)}}\downarrow & \swarrow_{\mathrlap{f}}
    \\
    && A
  }
$$

may be read as

* $f$ is a smooth function on $X$ with values in $A$;

* whose derivative along the [[tangent vector]] $v \in T_x X \subset T X = X^D$...

* ...is the tangent vector $v(f) \in T_{f(x)} A \subset T A = A^D$.

Accordingly, the diagram

$$
  \array{
    D &\stackrel{v}{\to}& X
    \\
    {}_{\mathllap{\alpha}}\downarrow
    \\
    A
  }
$$

may be read as encoding the differential equation $v(f)_x = \alpha$ (at one point) whose solutions $f \in A^X$ are the extensions that complete this diagram.

To get differential equations in the more common sense that they impose a condition on the derivative of $f$ at each single point of $X$ and varying smoothly with $X$, we think of  $f : X \to A$ in terms of the [[exponential object|exponential map]]

$$
  f^X : X^X \to A^X
$$

and consider

$$
  \array{
    &&{*}
    \\
    && \downarrow & \searrow^{\mathrlap{Id_X}}
    \\
    {*}&\to& D &\stackrel{v}{\to}& X^X
    \\
    &{}_{\mathllap{f}}\searrow&{}^{\mathllap{\alpha}}\downarrow & \swarrow_{\mathrlap{f^X}}
    \\
    &&A^X
  }
  \,.
$$

In this diagram now

* $v : D \to X^X$ is the [[adjunct]] of a [[vector field]] $X \to T X = X^D$ on $X$;

  (the commutativity of the top  right triangle ensures that indeed this $X \to T X$ is a [[section]] of the [[tangent bundle]] projection $T X \to X$);

* $\alpha : D \to A^X$ is the [[adjunct]] of a map $X \to T A = A^D$ that sends each point $x \in X$ to a tangent vector in $T_{f(x)} A$

  (enforced by the commutativity of the left bottom triangle)

* and the commutativity of the right lower triangle is the **differential equation**

  $$
    v(f) = \alpha
    \,.
  $$

## Examples

* [[heat equation]]

* [[wave equation]], [[Klein-Gordon equation]], [[sine-Gordon equation]]

* [[Navier-Stokes equation]]

* [[Schrödinger equation]]

* [[Dirac equation]]

* [[Green hyperbolic partial differential equation]]

## Related concepts

* [[differentiation]], [[differential calculus]]

* [[diffiety]]

* [[ordinary differential equation]], [[partial differential equation]]

* [[linear differential equation]]

* [[integrable differential equation]],  [[formally integrable differential equation]]

* [[generalized solution of a PDE]]

* [[geometric analysis]]

* [[Green's function]]

* [[Sturm–Liouville theory]]

* [[initial value problem]]

* [[differential Galois theory]]

* [[D-module]], [[D-geometry]]

* [[homotopy analysis method]]

## References

### General

General accounts:

* [[Jean-François Pommaret]], *Partial Differential Equations and Group Theory*, Springer (1994) [[doi:10.1007/978-94-017-2539-2](https://doi.org/10.1007/978-94-017-2539-2)]

* [[Sergiu Klainerman]], _PDE as a unified subject_ 2000 [[pdf](http://www.math.princeton.edu/~seri/homepage/papers/telaviv.pdf)]

* Boris Kruglikov, Valentin Lychagin, _Geometry of differential equations_ [[pdf](http://www.math.uit.no/seminar/Preprints/07-01-BKVL.pdf)]

* Arthemy Kiselev, _The twelve lectures in the (non)commutative geometry of differential equations_, preprint IHES M/12/13 [pdf](http://preprints.ihes.fr/2012/M/M-12-13.pdf)

* Yves Andre, _Solution algebras of differential equations and quasi-homogeneous varieties_, [arXiv:1107.1179](http://arxiv.org/abs/1107.1179)

* [[Mikhail Gromov]], _Partial differential relations_, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986. x+363 pp.

* [[Peter Olver]], _Applications of Lie groups to differential equations_, Springer; _Equivalence, invariants, and symmetry_, Cambridge Univ. Press 1995.

On the partial differential equations appearing in [[mathematical physics|mathematical]] [[physics]]:

* [[Cliff P. Burgess]], *Primer on Partial Differential Equations for Physicists*, lecture notes (1990) [[pdf](https://physics.mcmaster.ca/~cburgess/Notes/mathphys.pdf), [[Burgess-PDEs.pdf:file]]]

* [[Robert Geroch]], _Partial Differential Equations of Physics_, in: _[General Relativity: Proceedings](https://inspirehep.net/literature/328567)_ Edited by G.S. Hall and J.R. Pulham. Edinburgh, IOP Publishing, 1996. p. 19 ([arXiv:gr-qc/9602055](http://arxiv.org/abs/gr-qc/9602055))

*  Bijan Kumar Bagchi, *Partial Differential Equations for Mathematical Physicists*, Routledge (2020) [[ISBN:9780367227029](https://www.routledge.com/Partial-Differential-Equations-for-Mathematical-Physicists/Bagchi/p/book/9780367227029)]

### Via jet bundles and D-modules

Discussion of the spaces of solutions is in

* Batu Güneysu, [[Markus Pflaum]], _The profinite dimensional manifold structure of formal solution spaces of formally integrable PDE's_ ([arXiv:1308.1005](https://arxiv.org/abs/1308.1005))

Characterization of the category of partial differential equations as the [[Eilenberg-Moore category]] of [[coalgebras]] over the [[jet comonad]]:

* {#Marvan86} [[Michal Marvan]], _A note on the category of partial differential equations_, in _Differential geometry and its applications_, Proceedings of the Conference August 24-30, 1986, Brno ([[MarvanJetComonad.pdf:file]])

Discussion in [[differential cohesion]]:

* [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

Discussion in differentially cohesive [[modal homotopy type theory|modal]] [[homotopy type theory]]:

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

In the language of [[D-modules]] and hence for the special case of _linear_ differential equations, this appears as prop. 3.4.1.1 in

* {#GaitsgoryRozenblyum11} [[Dennis Gaitsgory]], [[Nick Rozenblyum]], _Crystals and D-modules_ ([arXiv:1111.2087](http://arxiv.org/abs/1111.2087))

### Other

A [[domain specific programming language]] for differential equations is presented in

* {#ALORW12} Martin S. Alnaes, Anders Logg, Kristian B. Oelgaard, Marie E. Rognes, Garth N. Wells, _Unified Form Language: A domain-specific language for weak formulations of partial differential equations_, [arXiv:1211.4047](http://arxiv.org/abs/1211.4047)

See also

* _[Dispersive Wiki](http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page)_

[[!redirects differential equation]]
[[!redirects differential equations]]

[[!redirects partial differential equation]]
[[!redirects partial differential equations]]
[[!redirects PDE]]
[[!redirects PDEs]]
[[!redirects PDE-s]]
[[!redirects PDE's]]
[[!redirects PDE\'s]]
[[!redirects PDE/'s]]
[[!redirects PDE's]]

[[!redirects PDE theory]]


differential fivebrane structure

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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Where a [[fivebrane structure]] is a trivialization of a class in [[integral cohomology]], a _differential fivebrane structure_ is the trivialization of this class refined to [[ordinary differential cohomology]]:

the second fractional [[Pontryagin class]]

$$
  \frac{1}{6} p_2 : B String \to B^8 \mathbb{Z}
$$

in the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]] has a refinement to $\mathbf{H} = $ [[Smooth∞Grpd]] of the form

$$
  \frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)
$$

-- the smooth second fractional Pontryagin class.

The induced morphism on [[cocycle]] [[∞-groupoid]]s

$$
  \frac{1}{6}\mathbf{p}_2
  :
  \mathbf{H}(X,\mathbf{B} String)
  \stackrel{}{\to}
  \mathbf{H}(X,\mathbf{B}^7 U(1))
$$

sends a [[string 2-group]]-[[principal 2-bundle]] $P$ to its corresponding [[Chern-Simons circle 7-bundle]] $\frac{1}{6}\mathbf{p}_2(P)$.

A choice of trivialization of $\frac{1}{6}p_2(P)$ is a [[fivebrane structure]]. The [[n-groupoid|6-groupoid]] of smooth fivebrane structures is the [[homotopy fiber]] of $\frac{1}{6}\mathbf{p}_2$ over the trivial [[circle n-bundle with connection|circle 7-bundle]].

By [[Chern-Weil theory in Smooth∞Grpd]] this morphism may be further refined to a [[differential characteristic class]] $\frac{1}{6}\hat \mathbf{p}_2$ that lands in the [[ordinary differential cohomology]]
$\mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))$,
classifying [[circle n-bundle with connection|circle 7-bundles with connection]]

$$
  \frac{1}{6}\hat \mathbf{p}_2
  :
  \mathbf{H}_{conn}(X,\mathbf{B} String)
  \stackrel{}{\to}
  \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))
$$

The [[n-groupoid|7-groupoid]] of **differential fivebrane structures** is the [[homotopy fiber]] of this refinement $\frac{1}{6}\hat \mathbf{p}_2$ over the [[circle n-bundle with connection|trivial circle 7-bundle with trivial connection]] or more generally over the trivial circle 7-bundles with possibly non-trivial connection.

Such a differential fivebrane structure over a [[smooth manifold]] $X$ is characterized by a tuple consisting of

1. a [[connection on a 2-bundle|2-connection]] $\nabla$ on a [[string 2-group|String]]-[[principal 2-bundle]] on $X$;

1. a choice of trivial [[circle n-bundle with connection|circle 7-bundle]] with connection $(0, H_7)$, hence a differential 7-form $H_7 \in \Omega^7(X)$;

1. a choice of [[equivalence in an (∞,1)-category|equivalence]] $\lambda$ of the [[Chern-Simons circle 7-bundle]] with connection $\frac{1}{6}\hat\mathbf{p}_2(\nabla)$ of $\nabla$ with this chosen 7-bundle

  $$
    \lambda : \frac{1}{6}\hat \mathbf{p}_2(\nabla) \stackrel{\simeq}{\to}
    (0,H_7)
    \,.
  $$

More generally, one can consider the [[homotopy fiber]]s of $\frac{1}{6}\hat \mathbf{p}_2$ over arbitrary circle 7-bundles with connection $\hat \mathcal{G}_8 \in \mathbf{H}_{diff}^8(X, \mathbf{B}^3 U(1))$ and hence replace $(0,H_7)$ in the above with $\hat \mathcal{G}_8$. According to the general notion of [[twisted cohomology]], these may be thought of as **twisted differential string structures**, where the class $[\mathcal{G}_8] \in H^8_{diff}(X)$ is the twist.

## Definition {#Definition}

We will assume that the reader is familiar with basics of the discussion at [[Smooth∞Grpd]]. We often write $\mathbf{H} := Smooth \infty Grpd$ for short.

Let $String(n) \in $ [[Smooth∞Grpd]] be the smooth [[String 2-group]], for some $n \gt 6$,[^low_rank] regarded as a [[Lie 2-group]] and thus canonically as an [[∞-group]] object in [[Smooth∞Grpd]]. We shall notationally suppress the $n$ in the following. Write $\mathbf{B}String$ for its [[delooping]] of $Spin$ in [[Smooth∞Grpd]]. (See the discussion here).
Let moreover $\mathbf{B}^6 U(1) \in Smooth \infty Grpd$ be the circle Lie 7-group and $\mathbf{B}^7 U(1)$ its [[delooping]].

[^low_rank]: for $n=3,4,5,6$ $String(n)$ is not 6-connected, so one must take extra care with the intervening stages in the [[Whitehead tower]].

At [[Chern-Weil theory in Smooth∞Grpd]] the following statement is proven ([FSS](#FSS)).

+-- {: .un_prop }
###### Proposition

The image under [[Lie integration]] of the canonical [[Lie algebra cohomology|Lie algebra 7-cocycle]]

$$
  \mu = \langle -,[-,-], [-,-], [-,-]\rangle : \mathfrak{so} \to b^6 \mathbb{R}
$$

on the [[semisimple Lie algebra]] $\mathfrak{so}$ of the [[Spin group]] -- the [[special orthogonal Lie algebra]] -- is a morphism in [[Smooth∞Grpd]] of the form

$$
  \frac{1}{6} \mathbf{p}_2 :
  \mathbf{B}String
    \to
  \mathbf{B}^7 U(1)
$$

whose image under the the fundamental ∞-groupoid (∞,1)-functor/ geometric realization $\Pi : Smooth \infty Grpd \to $ [[∞Grpd]] is the ordinary second fractional [[Pontryagin class]]

$$
  \frac{1}{6}p_2 : B String \to B^8 \mathbb{Z}
$$

in [[Top]]. Moreover, the corresponding refined differential characteristic class

$$
  \frac{1}{6}\hat \mathbf{p}_2 :
  \mathbf{H}_{conn}(-,\mathbf{B}String)
  \to
  \mathbf{H}_{diff}(-, \mathbf{B}^7 U(1))
$$

is in [[cohomology]] the corresponding refined [[Chern-Weil homomorphism]]

$$
  [\frac{1}{6}\hat \mathbf{p}_String] :
  H^1_{Smooth}(X,String) \to H_{diff}^8(X)
$$

with values in [[ordinary differential cohomology]] that corresponds to the second [[Killing form]] [[invariant polynomial]] $\langle - , - ,-,-\rangle $ on $\mathfrak{so}$.

=--

+-- {: .un_def #DifferentialStringStructure}
###### Definition

For any $X \in $ [[Smooth∞Grpd]], the [[n-groupoid|6-groupoid]]  of **differential fivebrane-structures** on $X$ -- $Fivebrane_{diff}(X)$ -- is the [[homotopy fiber]] of  $\frac{1}{6}\hat \mathbf{p}_2(X)$ over the trivial differential cocycle.

More generally (see [[twisted cohomology]]) the 6-groupoid of **twisted differential fivebrane structures** is the [[(∞,1)-pullback]] $Fivebrane_{diff,tw}(X)$ in

$$
  \array{
    Fivebrane_{diff,tw}(X) &\to& H_{diff}^8(X)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}String)
      &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}&
    \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))
  }
  \,,
$$

where the right vertical morphism is a choice of (any) one point in each [[connected component]] (differential cohomology class) of the [[cocycle]] [[∞-groupoid]] $\mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$ (the [[homotopy type]] of the [[(∞,1)-pullback]] is independent of this choice).

=--

+-- {: .un_remark }
###### Remark

In terms of local [[∞-Lie algebra valued differential forms]] data this has been considered in ([SSSIII](#SSSIII)), as we shall discuss [below](#ChernWeilTheory).

=--

## Construction in terms of $L_\infty$-Cech cocycles
  {#ChernWeilTheory}

We use the [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-topos]] [[Smooth∞Grpd]] (as described there) by the local [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ to give an explicit construction of twisted differential fivebrane structures in terms of [[Cech cohomology|Cech]]-cocycles with coefficients in [[∞-Lie algebra valued differential forms]].

Recall the following fact from [[Chern-Weil theory in Smooth∞Grpd]] ([FSS](#FSS)).

+-- {: .un_prop }
###### Proposition

The differential second fractional Pontryagin class $\frac{1}{6} \hat \mathbf{p}_2$ is presented in $[CartSp_{smooth}^{op}, sSet]_{proj}$ by the top morphism of simplicial presheaves in

$$
  \array{
    \mathbf{cosk}_7\exp(\mathfrak{so})_{ChW,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^7 \mathbb{R}/\mathbb{Z}_{ChW,smp}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_7\exp(\mathfrak{so})_{diff,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^7 \mathbb{R}/\mathbb{Z}_{smp}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}String_{c}
  }
  \,.
$$

=--

Here the middle morphism is the direct [[Lie integration]] of the [[infinity-Lie algebra cohomology|L-∞ algebra cocycle]] while the top morphisms is its restriction to coefficients for [[connection on an ∞-bundle|∞-connections]].

In order to compute the [[homotopy fiber]]s of
$\frac{1}{6}\hat \mathbf{p}_2$ we now find a [[resolution]] of this morphism $\exp(\mu,cs)$ by a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$. By the fact that this is a [[simplicial model category]] then also the hom of any cofibrant object into this morphism, computing the cocycle $\infty$-groupoids, is a fibration, and therefore, by the general discussion at [[homotopy pullback]], we obtain the [[homotopy fiber]]s as the ordinary [[fiber]]s of this fibration.

### Presentation of the differential class by a fibration

(...)

The discussion is analogous to that at [[differential string structure]]. To be filled in

(...)

## Related concepts

[[Whitehead tower]] of the [[orthogonal group]]

* [[orientation]]

* [[spin structure]]

* [[string structure]]

* [[fivebrane structure]]

* [[twisted differential c-structure]]

  * [[differential string structure]]

  * [[supergravity C-field]]

  * **differential fivebrane structure**

  * [[differential T-duality]]

## References

The local data for the [[∞-Lie algebra valued differential forms]] for the description of twisted differential fivebrane structures as above was given in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _Twisted differential string and fivebrane structures_ (web)
{#SSSIII}

The full Cech-Deligne cocycles induced by this and their homotopy fibers were discussed in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]]_

A general discussion is at

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

in section 4.2.

[[!redirects differential fivebrane structures]]

[[!redirects twisted differential fivebrane structure]]
[[!redirects twisted differential fivebrane structures]]

[[!redirects twisted differential Fivebrane structure]]
[[!redirects twisted differential Fivebrane structures]]

[[!redirects smooth second fractional Pontryagin class]]