twist

The term **twist** or **twisted** is one of the hugely overloaded terms in math. Among the various meaning it may have is

* [[twisted cohomology]]

  * [[twisted bundle]]

  * [[twisted K-theory]]

* [[twisting cochain]]

* [[twisted complex]]

* [[twisted tensor product]]

* [[twisted module of homomorphisms]]

* the twist in a [[balanced monoidal category]].

[[!redirects twists]]


twisted ad-equivariant K-theory

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#### Representation theory
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## Idea

The [[twisted equivariant K-theory]] of suitable [[compact Lie groups]] [[equivariant K-theory|equivariant]] with respect to the [[adjoint action]] of the group on itself.

By [FHT](#FHT) this is equivalent to the [[Verlinde ring]] of [[positive energy representations]] of the corresponding [[loop group]].

(...)

## Related concepts

* [[twisted ad-equivariant Tate K-theory]]

## References

On [[twisted ad-equivariant K-theory]] of [[compact Lie groups]] and the identification with the [[Verlinde ring]] of [[positive energy representations]] of their [[loop group]]:

* [[Daniel S. Freed]], _The Verlinde algebra is twisted equivariant K-theory_, Turk. J. Math 25 (2001) 159-167 ([arXiv:math/0101038](https://arxiv.org/abs/math/0101038))

{#FHT}
* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],

  1. _[[Loop Groups and Twisted K-Theory]] I_,

     J. Topology, 4 (2011), 737-789

     ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))

  1. _[[Loop Groups and Twisted K-Theory]] II_,

     J. Amer. Math. Soc. 26 (2013), 595-644

     ([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))

  1. _[[Loop Groups and Twisted K-Theory]] III_,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     ([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))

* [[Daniel S. Freed]], [[Constantin Teleman]],

  _Dirac families for loop groups as matrix factorizations_,

  Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

  ([arxiv/1409.6051](http://arxiv.org/abs/1409.6051), [doi:10.1016/j.crma.2015.02.011](https://doi.org/10.1016/j.crma.2015.02.011))

Discussion in terms of [[differentiable stacks]]:

* [[Ulrich Bunke]], Ingo Schroeder, _Twisted K-Theory and TQFT_ ([arXiv:math/0504458](https://arxiv.org/abs/math/0504458), [spire:681555](https://inspirehep.net/literature/681555))

Review:

* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin  Schottenloher]], Section 22 of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))

* [[Valentin Zakharevich]], _K-Theoretic Computation of the Verlinde Ring_, thesis 2018 ([hdl:2152/67663](http://hdl.handle.net/2152/67663), [pdf](http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf), [[ZakharevichKTheoryAndVerlindeRing.pdf:file]])

Generalization to [[KR-theory]]:

* [[Chi-Kwong Fok]], *The Real K-theory of compact Lie groups*, SIGMA 10 (2014), 022, ([arXiv:1308.3871](https://arxiv.org/abs/1308.3871), [doi:10.3842/SIGMA.2014.022](https://doi.org/10.3842/SIGMA.2014.022))

* [[Chi-Kwong Fok]], *Equivariant twisted Real K-theory of compact Lie groups*,   Journal of Geometry and Physics 124 (2018) 325-349 ([arXiv:1503.00957](https://arxiv.org/abs/1503.00957), [doi:10.1016/j.geomphys.2017.11.013](https://doi.org/10.1016/j.geomphys.2017.11.013))


twisted arrow (∞,1)-category

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## Idea

This is the generalization of [[twisted arrow category]] to (∞,1)-categories.

## Definition

In the case of [[quasi-category|quasicategories]], we can give an explicit definition. For a quasi-category $C$ we define the twisted arrow quasi-category $Tw(C)$ to be
$$
  Tw(C)_n = sSet((\Delta^n)^{op} \star (\Delta^n), C)
$$
The insertions into the components of the join determine natural transformations
$src : Tw(C) \to C^{op}$ and $tgt : Tw(C) \to C$.

By [Kerodon, tag 03K9](#Kerodon), this definition respects equivalences between quasi-categories, and thus descends to a functor $Tw : (\infty,1)Cat \to (\infty,1)Cat$.

Since $(\Delta^n)^{op} \star (\Delta^n)$ is preserved by taking opposites, this construction satisfies $Tw(C) = Tw(C^{op})$, and swaps the source and target maps.

The opposite convention often appears in the literature,
$$
  \overline{Tw}(C)_n = sSet((\Delta^n) \star (\Delta^n)^{op}, C)
$$
and comes with natural transformations
$\overline{src} : \overline{Tw}(C) \to C$ and $\overline{tgt} : \overline{Tw}(C) \to C^{op}$.

This satisfies $\overline{Tw}(C) \simeq Tw(C)^{op}$.

## Properties

If $C$ is a 1-category, then $Tw(C)$ agrees with the 1-categorical version. In the quasi-category model we have an even stronger statement

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

If $C$ is a 1-category, there is an isomorphism $T : N(Tw(C)) \cong Tw(N(C))$ of simplicial sets, uniquely determined by the properties that $T(f) = f$ and that the source and target maps are consistent with the obvious map $N(C^{op} \times C) \cong N(C)^{op} \times N(C)$.

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

This is [Kerodon, tag 03JN](#Kerodon).

=--

For a quasi-category $C$, both $Tw(C)$ and $\overline{Tw}(C)$ are quasi-categories. This follows from the following characterization:

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

* $Tw(C) \to C^{op} \times C$ is the left fibration classfied by the hom-space functor $C^{op} \times C \to \infty Gpd$.

* $\overline{Tw}(C) \to C \times C^{op}$ is the right fibration classified the hom-space functor $C^{op} \times C \to \infty Gpd$.

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

This description of $\overline{Tw}(C)$ is [Lurie 4.2.5](#Lurie). The left fibration classified by a map is the opposite of the right fibration classified
by the same map. Alternatively, this is [Kerodon, tag 03JQ](#Kerodon).

=--

This means that we have the formulas $Tw(C) \simeq el(\hom_C) \simeq ({*} \downarrow \hom)$, just as in the 1-category version.

## Related concepts

* [[(∞,1)-end]]

* [[lax (∞,1)-colimit]]

* [[twisted arrow category]]

## References

* {#Lurie} [[Jacob Lurie]], _[[Formal moduli problems]]_

* {#Kerodon} [[Kerodon]], *The Twisted Arrow Construction* $[$[kerodon:03JF](https://kerodon.net/tag/03JF)$]$

[[!redirects twisted arrow (∞,1)-categories]]

[[!redirects twisted arrow (infinity,1)-category]]
[[!redirects twisted arrow (infinity,1)-categories]]


twisted arrow category

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## Terminology

A twisted arrow category is an alternative name for a [[category of factorisations]]. That latter name is applied when discussing natural systems  and [[Baues-Wirsching cohomology]], whilst the name _twisted arrow category_ is more often used in discussing Kan extensions and within the categorical literature[^Law].

[^Law]: [[Lawvere]] ('Equality in hyperdoctrines', 1970) uses the term 'twisted _morphism_ category'.

## Definition

The twisted arrow category $Tw(C)$ of $C$ a category is the [[category of elements]] of its [[hom-functor]]:
\[
Tw(C) = el(hom) = * / hom
\]

### Explicit description

Unpacking the well-known explicit construction of [[comma object|comma objects]] in $\mathbf{Cat}$ as [[comma category|comma categories]], we get that $Tw(C)$ has

* objects: $f$ an arrow in $C$, and

* morphisms: between $f$ and $g$ are pairs of arrows $(p,q)$ such that the following diagram commutes:
\[
\begin{matrix}
  A             & \overset{p}{\leftarrow}   & C            \\
  f \downarrow  &                           & \downarrow g \\
  B             & \underset{q}{\to}         & D
\end{matrix}
\]
you could view then morphisms from $f$ to $g$ as *factorizations* of $g$ through $f$: $g = q f p$.

From the description above, $Tw(C)$ is the same as $Arr(C)$ the [[arrow category]] of $C$, but with the direction of $p$ above in the definition of morphism *reversed*, hence the twist.

### As a lax colimit

The (opposite of the) category $Tw(C)$ can be described as the [[2-limit#lax|lax colimit]] of the diagram $C \to Cat\colon c\mapsto C/c$.

## Example

If $C$ is a partially ordered set, then $Tw(C)$ is isomorphic to the set of nonempty [[interval|intervals]] $[a,b]=\{c\in C| a\leq c\leq \b\}$ with $a\leq b$ in C ordered by inclusion (cf. [Johnstone 1999](#JT99)).

## Properties

* From its definition as a comma category, there is a functor (a [[discrete opfibration]], in fact)
\[
\pi_C \colon Tw(C) \to C^{op} \times C
\]
which at the level of objects forgets the arrows:
\[
\pi_C(f \colon A \to B) = (A,B)
\]
and keeps everything at the morphisms level.

### $Tw(C)$ and wedges

One could say that $Tw(C)$ *classifies* [[wedge|wedges]], in the sense that for any functor $F \colon C^{op} \times C \to B$,

* [[extranatural transformation|extranatural transformations]] $ e \stackrel{\bullet}{\to} F $

are the same as

* [[natural transformation|natural transformations]] $ e \to F \circ \pi_C $

This can be used to give a proof of the reduction of [[end|ends]] to [[conical limit|conical limits]] in the $\mathbf{Set}$-enriched case, and is used in the construction of [[ends in a derivator]].

## Generalizations

* The twisted arrow category is a special case of a **category of judgments** in the sense of ([Melliès-Zeilberger 15](#MZ15)).

* The construction generalizes to [[(infinity,1)-category-theory|$(\infty,1)$-category theory]], see at *[[twisted arrow (∞,1)-category]]*.

## Related entries

* [[category of factorizations]]

* [[twisted arrow (∞,1)-category]]

## References

* {#BN00}[[Marta Bunge|M. Bunge]], [[Susan Niefield|S. Niefield]], _Exponentiability and single universes_ , JPAA **148** (2000) pp.217-250.

* L. Errington, _Twisted Systems_ , PhD thesis Imperial College London 1999.  ([doi](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.6369))

* [[Peter Johnstone]], _A Note on Discrete Conduché Fibrations_ , TAC **5** no.1 (1999) pp.1-11. ([pdf](http://www.tac.mta.ca/tac/volumes/1999/n1/n1.pdf))

* [[Fred Linton]], _Autonomous categories and duality of functors_ , J. Algebra **2** no.3 (1965) pp.315-349.

* {#DAGX} [[Jacob Lurie]], _Derived Algebraic Geometry X: Formal Moduli Problems_ , ms. (2011).

* [[Saunders Mac Lane]], _Categories for the Working Mathematician_ , Springer Heidelberg 1998². (cf. exercise IX.6.3, p.227)

* [[Paul-André Melliès]], [[Noam Zeilberger]], _Isbell Duality for Refinement Types_ , [arXiv:1501.05115](http://arxiv.org/abs/1501.05115) (2015). {#MZ15}

[[!redirects twisted arrow category]]
[[!redirects twisted arrow categories]]


twisted Bredon cohomology

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## Idea

Twisted Bredon cohomology, or _equivariant ordinary cohomology with local coefficients_ is the [[twisted cohomology|twisted]] generalization of [[Bredon cohomology]], and the [[equivariant cohomology|equivariant]]-generalization of [[twisted ordinary cohomology]].

## Related concepts

* [[twisted ordinary cohomology]]

* [[twisted K-theory]]

* [[twisted Cohomotopy]]

## References

* [[Ieke Moerdijk]], J.-A. Svensson, _The Equivariant Serre Spectral Sequence_, Proceedings of the American Mathematical Society Vol. 118, No. 1 (May, 1993), pp. 263-278 ([jstor:2160037](https://www.jstor.org/stable/2160037))

* Amiya Mukherjee, [[Goutam Mukherjee]], _Bredon-Illman cohomology with local coefficients_, The Quarterly Journal of Mathematics, Volume 47, Issue 2, June 1996, Pages 199–219 ([doi:10.1093/qmath/47.2.199](https://doi.org/10.1093/qmath/47.2.199))

* [[Marek Golasiński]], _Equivariant cohomology with local coefficients_, Mathematica Slovaca (1997) Volume: 47, Issue: 5, page 575-586 ([dml:34467](https://eudml.org/doc/34467))

* [[Goutam Mukherjee]], Neeta Pandey, _Equivariant cohomology with local coefficients_, Proc. Amer. Math. Soc. 130 (2002), 227-232 ([doi:10.1090/S0002-9939-01-06377-8](https://doi.org/10.1090/S0002-9939-01-06377-8))

* [[Goutam Mukherjee]], Debasis Sen, _Equivariant simplicial cohomology with local coefficients and its classification_, Topology and its Applications, Volume 157, Issue 6, 15 April 2010, Pages 1015-1032 ([doi:10.1016/j.topol.2010.01.004](https://doi.org/10.1016/j.topol.2010.01.004))

[[!redirects twisted ordinary equivariant cohomology]]


twisted bundle

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## Idea

A _twisted principal-bundle_ is the object classified by a cocycle in [[twisted cohomology]] the way an ordinary [[principal bundle]] is the object classified by a cocycle in plain [[cohomology]] (generally in [[nonabelian cohomology]]).

For $\hat G$ a [[group]], a $\hat G$-[[principal bundle]] is classified in degree 1 [[nonabelian cohomology]] with coefficients in the [[delooping|delooped]] [[groupoid]] $\mathbf{B} \hat G$.

Given a realization of $\hat G$ as an abelian extension

$$
  A \to \hat G \to G
$$

of groups, i.e. given a [[fibration sequence]]

$$
  \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G
$$

of [[groupoid]]s such that $\mathbf{B}A$ is once [[delooping|deloopable]] so that the [[fibration sequence]] continues to the right at least one step as

$$
  \mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A
$$

the general mechanism of [[twisted cohomology]] induces a notion of _twisted_ $\hat G$-cohomology. The fibrations classified by this are the twisted $\hat G$-bundles.

## Definition

We give a discussion of twisted bundles as a realization of [[twisted cohomology]] in any [[cohesive (∞,1)-topos]] $\mathbf{H}$ as described in the section cohesive (∞,1)-topos -- twisted cohomology. For the cases that $\mathbf{H} = $ [[ETop∞Grpd]] or $\mathbf{H} = $ [[Smooth∞Grpd]] this reproduces the traditional notion of [[topology|topological]] and [[smooth structure|smooth]] twisted bundles, respectively, whose twists are correspondingly topological or smooth [[bundle gerbe]]s/[[circle n-bundle]]s.

### Setup
 {#Setup}

Let $\mathbf{B}^{n-1}U(1) \in \mathbf{H}$ be the [[circle n-group]]. We shall concentrate here for definiteness on twists in $\mathbf{B}^2 U(1)$-[[cohomology]], since that reproduces the usual notions of twisted bundles found in the literature. But every other choice would work, too, and yield a corresponding notion of twisted bundles.

Fix once and for all an [[∞-group]] $G \in \mathbf{H}$ and a [[cocycle]]

$$
  \mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 U(1)
$$

representing a [[characteristic class]]

$$
  [\mathbf{c}] \in H_{Smooth}^2(\mathbf{B}G,U(1))
$$

Notice that if $G$ is a [[compact space|compact]] [[Lie group]], as usual for the discussion of twisted bundles where $G = P U(n)$ is the [[projective unitary group]] in some dimension $n$, then by this theorem we have that

$$
  H_{Smooth}^2(\mathbf{B}G, U(1)) \simeq H^3(B G, \mathbb{Z})
  \,,
$$

where on the right we have the ordinary [[integral cohomology]] of the [[classifying space]] $B G \in$ [[Top]] of $G$.

### The abstract definition

Let $G$ and $\mathbf{c}$ be as [above](#Setup).

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

Write

$$
  \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 U(1)
$$

for the [[homotopy fiber]] of $\mathbf{c}$.

=--

This identifies $\hat G$ as the [[group extension]] of $G$ by the 2-[[cocycle]] $\mathbf{c}$.

+-- {: .num_note #TheGroupExtensionAsACircleBundle}
###### Note

Equivalently this means that

$$
  \mathbf{B}U(1) \to \mathbf{B}\hat G \to \mathbf{B}G
$$

is the smooth [[circle n-bundle|circle 2-bundle]]/[[bundle gerbe]] classified by $\mathbf{c}$; and its [[loop space object]]

$$
  U(1) \to \hat G \to G
$$

the corresponding [[circle group]] [[principal bundle]] on $G$.

=--

Let $X \in \mathbf{H}$ be any object. From _[[twisted cohomology]]_ we have the following notion.

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

The degree-1 **total twisted cohomology** $H_{tw}^1(X, \hat G)$
of $X$ with coefficients in $\hat G$, def. \ref{TheGroupExtension}, relative to the characteristic class $[\mathbf{c}]$ is the set

$$
  H^1_{tw}(X, \hat G) := \pi_0 \mathbf{H}_{tw}(X, \mathbf{G}\hat H)
$$

of connected components of the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{H}_{tw}(X, \mathbf{B}\hat G) &\stackrel{tw}{\to}&
      H_{Smooth}^2(X,U(1))
    \\
     \downarrow && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}G)
     &\stackrel{\mathbf{c}_*}{\to}&
    \mathbf{H}(X, \mathbf{B}^2 U(1))
  }
  \,,
$$

where the right verticsl morphism is any [[section]] of the truncation projection from cocycles to cohomology classes.

Given a twisting class $[\alpha] \in H^2_{Smooth}(U(1))$ we say that

$$
  H_{[\alpha]}^1(X,\hat G)
  :=
  H^1_{tw}(X, \hat G) \times_{[\alpha]} *
$$

is the $[\alpha]$-**twisted cohomology** of $X$ with coefficients in $\hat G$ relative to $\mathbf{c}$.

=--

+-- {: .num_note #VanishingTwistGivesOrdinaryBundles}
###### Note

For $[\alpha] = 0$ the trivial twist, $[\alpha]$-twisted cohomology coincides with ordinary cohomology:

$$
  H^1_{[\alpha] = 0}(X, \hat G)
  \simeq
  H^1_{Smooth}(X, \hat G)
  \,.
$$

=--

By the discussion at _[[principal ∞-bundle]]_ we may identify the elements of $H^1_{Smooth}(X, \hat G)$ with $\hat G$-[[principal ∞-bundle]]s $P \to X$. In particular if $\hat G$ is an ordinary [[Lie group]] and $X$ is an ordinary [[smooth manifold]], then these are ordinary $\hat G$-[[principal bundle]]s over $X$. This justifies equivalently calling the elements of $H^1_{tw}(X,\hat G)$ **twisted principal $\infty$-bundles**; and we shall write

$$
  \hat G TwBund(X) := H^1_{tw}(X, \hat G)
  \,,
$$

where throughout we leave the characteristic class $[\mathbf{c}]$ with respect to which the twisting is defined implcitly understood.

### Explicit cocycles

We unwind the abstract definition, def. \ref{AbstractDefinition}, to obtain the explicit definition of twisted bundles by [[Cech cohomology|Cech cocycles]] the way they appear in the traditional literature (see the [General References](#GeneralReferences) below).

+-- {: .num_prop #CechCocyclesForTwisted1Bundles}
###### Proposition

Let $U(1) \to \hat G \to G$ be a [[group extension]] of [[topological group]]s.

Let $X \in $ [[Mfd]] $\hookrightarrow$ [[ETop∞Grpd]] $=: \mathbf{H}$ be a [[paracompact topological space|paracompact]] [[topological manifold]] with [[good open cover]] $\{U_i \to X\}$.

1. Relative to this every twisting cocycle $[\alpha] \in H^2_{ETop}(X, U(1))$ is a [[Cech cohomology]] representative given by a collection of functions

   $$
     \{ \alpha_{i j k} : U_i \cap U_j \cap U_k \to U(1) \}
   $$

   satisfying on every quadruple intersection the equation

   $$
     \alpha_{i j k} \alpha_{i k l} = \alpha_{j k l} \alpha_{i j l}
     \,.
   $$

1. I terms of this cocycle data the twisted cohomology $H^1_{[\alpha]}(X, \hat G)$ is given by [[equivalence class]]es of [[cocycle]]s consisting of

   1. collections of functions

      $$
        \{g_{i j} : U_i \cap U_j \to \hat G \}
      $$

      subject to the condition that on each triple overlap the equation

      $$
        g_{i j} \dot g_{j k} = g_{i k} \cdot \alpha_{i j k}
      $$

      holds, where on the right we are injecting
      $\alpha_{i j k}$ via $U(1) \to \hat G$ into $\hat G$
      and then form the product there;

   1. subject to the [[equivalence relation]] that identifies two such collections of cocycle data $\{g_{i j}\}$ and $\{g'_{i j}\}$ if there exists functions

      $$
        \{h_i : U_i \to \hat G\}
      $$

      and

      $$
        \{\beta_{i j} : U_i \cap U_j \to \hat U(1)\}
      $$

      such that

      $$
        \beta_{i j} \beta_{j k} = \beta_{i k}
      $$

      and

      $$
        g'_{i j} = h_i^{-1} \cdot g_{i j} \cdot h_j \cdot \beta_{i j}
        \,.
      $$

=--

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

We pass to the standard [[presentable (∞,1)-category|presentation]] of [[ETop∞Grpd]] by the projective local [[model structure on simplicial presheaves]] over the [[site]] [[CartSp]]. We then compute the defining [[(∞,1)-pullback]] by a [[homotopy pullback]] there.

Write $\mathbf{B}\hat G_{c}, \mathbf{B}^2 U(1)_c \in [CartSp^{op}, sSet]$ etc. for the standard models of the abstract objects of these names by simplicial presheaves. Write accordingly $\mathbf{B}(U(1) \to \hat G)_c$ for the delooping of the [[crossed module]] associated to the central extension $\hat G \to G$.

In terms of this the [[characteristic class]] $\mathbf{c}$ is represented by the  [[∞-anafunctor]]

$$
  \array{
    \mathbf{B}(U(1) \to \hat G)_c &\stackrel{\mathbf{c}}{\to}&
     \mathbf{B}(U(1) \to 1)_c
     =
     \mathbf{B}^2 U(1)_c
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}G_c
  }
  \,,
$$

where the top horizontal morphism is the evident projection onto the $U(1)$-labels.
Moreover, the [[Cech nerve]] of the good open cover $\{U_i \to X\}$ forms a cofibrant [[resolution]]

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

and so $\alpha$ is presented by an [[∞-anafunctor]]

$$
  \array{
     C(\{U_i\}) &\stackrel{\alpha}{\to}& \mathbf{B}^2 U(1)_c
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     X
  }
  \,.
$$

Using that $[CartSp^{op}, sSet]_{proj}$ is a [[simplicial model category]] this means in conclusion that the [[homotopy pullback]] in question is given by the ordinary [[pullback]] of [[simplicial set]]s

$$
  \array{
     \mathbf{H}^1_{[\alpha]}(X,\hat G) &\to& *
     \\
     \downarrow && \downarrow^{\mathrlap{\alpha}}
     \\
     [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}(U(1) \to \hat G)_c)
     &\stackrel{\mathbf{c}_*}{\to}&
     [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}^2 U(1)_c)
  }
  \,.
$$

An object of the resulting simplicial set is then seen to be a simplicial map $g : C(\{U_i\}) \to \mathbf{B}(U(1) \to \hat G)_c$ that assigns

$$
  g
  \;\;
  :
  \;\;
  \array{
    && (x,j)
    \\
    & \nearrow &\Downarrow& \searrow
    \\
    (x,i)
    &&\to&&
    (x,k)
  }
  \;\;\;\;
  \mapsto
  \;\;\;\;
  \array{
    && \bullet
    \\
    & {}^{\mathllap{g_{i j}(x)}}\nearrow
     &\Downarrow^{\alpha_{i j k}(x)}&
     \searrow^{\mathrlap{g_{j k}(x)}}
    \\
    \bullet
    &&\underset{g_{i k}(x)}{\to}&&
    \bullet
  }
$$

such that projection out along $\mathbf{B}(U(1) \to \hat G)_c \to \mathbf{B}(U(1) \to 1)_c = \mathbf{B}^2 U(1)_c$ produces $\alpha$.

Similarily for the morphisms. Writing out what these diagrams in $\mathbf{B}(U(1) \to \hat G)_c$ mean in equations, one finds the formulas claimed above.

=--

## Properties

### General

(...)

Consider the extension $U(1) \to U(n) \to P U(n)$ of the [[projective unitary group]] to the [[unitary group]] for all $n$. Then [[direct sum]] of matrices gives a sum operation

$$
  H^1_{[\alpha]}(X, P U(n_1))
  \times
  H^1_{[\alpha]}(X, P U(n_2))
  \to
  H^1_{[\alpha]}(X, P U(n_1 + n_2))
$$

and a tensor product operation

$$
  H^1_{[\alpha_1]}(X, P U(n))
  \times
  H^1_{[\alpha_2]}(X, P U(n))
  \to
  H^1_{[\alpha_1]+ [\alpha_2]}(X, P U(n_1 \cdot n_2))
$$

(...)

### Twisted K-theory

[[equivalence class|Equivalence classes]] of twisted $U(n)$-bundles for fixed $\mathbf{B}U(1)$-twist $\alpha$ form a model for [[topological K-theory|topological]] $\alpha$-[[twisted K-theory]]. See there for details.

## Related concepts

* [[twisted K-theory]]

* [[Chan-Paton bundle]]

* [[twisted spin structure]]

* [[twisted ∞-bundle]]

* [[equivariant bundle]]

## References
 {#References}

### General
 {#GeneralReferences}

The concept of twisted vector bundles was introduced, as a model for [[twisted K-theory]] (in the generality of [[equivariant bundles]] and [[orbifold K-theory]]) in:

* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], Section 7.2 of: *Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489.  ([arXiv:math/0105039](http://arxiv.org/abs/math/0105039), [doi:10.1007/s00220-003-1035-x](https://doi.org/10.1007/s00220-003-1035-x))

The [[Cech cohomology|Cech cocycle]]-incarnation of twisted vector bundles (effectively due to [Lupercio & Uribe 2001, Def. 7.2.1](#LupercioUribe01)) was considered in:

* {#Mackaay03} [[Marco Mackaay]], _A note on the holonomy of connections in twisted bundles_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 44 (2003) no. 1, pp. 39-62.  ([arXiv:math/0106019](http://arxiv.org/abs/math/0106019), [numdam:CTGDC_2003__44_1_39_0](http://www.numdam.org/item/?id=CTGDC_2003__44_1_39_0))

An equivalent characterization of twisted vector bundles (identified in [Lupercio & Uribe 2001, (v2-) Prop. 7.2.2 ](#LupercioUribe01)) appeared under the name *[[bundle gerbe module]]* in:

* {#BCMMS02}  [[Peter Bouwknegt]], [[Alan Carey]], [[Varghese Mathai]], [[Michael Murray]], [[Danny Stevenson]], Section 4 of: _K-theory of bundle gerbes and twisted K-theory_ ,  Commun Math Phys, 228 (2002) 17-49 ([arXiv:hep-th/0106194](http://arxiv.org/abs/hep-th/0106194), [doi:10.1007/s002200200646](https://doi.org/10.1007/s002200200646))

Discussion of a [[splitting principle]] for twisted vector bundles (phrased in terms of [[gerbe modules]]) is in

* {#Tomoda07} [[Atsushi Tomoda]], _On the splitting principle of bundle gerbe modules_, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. ([Euclid](https://projecteuclid.org/euclid.ojm/1174324334), talk slides [pdf](http://ton.prosou.nu/official/ryousi2005.pdf))

### In twisted K-theory

Just as [[vector bundles]] model cocycles in [[K-theory]], twisted vector bundles model cocycles in [[twisted K-theory]].

* [Lupercio & Uribe 2001](#LupercioUribe01)

* [BCMMS 2002](#BCMMS02)

The generalization of this construction to non-torsion twists requires using [[vectorial bundles]] instead of plain [[vector bundles]]. Full twisted K-theory in terms of twisted vectorial bundles was realized in

* [[Kiyonori Gomi]], p. 7 of _Twisted K-theory and finite-dimensional approximation_ ([arXiv:0803.2327](http://arxiv.org/abs/0803.2327))

Then there is

* {#Karoubi} [[Max Karoubi]], _Twisted bundles and twisted K-theory_, Clay Mathematics Proceedings, Volume 19 (2011) ([pdf](http://www.math.jussieu.fr/~karoubi/Publications/89.pdf))

* {#Pennig} [[Ulrich Pennig]], _Twisted K-theory with coefficients in $C^\ast$-algebras_, ([arXiv:1103.4096](http://arxiv.org/abs/1103.4096))

### As 2-sections of 2-bundles
 {#ReferencesAsSectionsOf2Bundles}

The observation that twisted vector bundles may be understood as higher-order [[sections]] of [[2-vector bundle]]s associated with [[circle n-bundle with connection|circle 2-bundles]]/[[bundle gerbes]] appears in

* [[Urs Schreiber]], _Quantum 2-States: Sections of 2-vector bundles_ Talk at _Higher categories and their applications_, Fields institute (2007) ([[Schreiber2States.pdf:file]]).

A discussion of this with [[connection on a bundle|2-connections]] taken into account is in section 4.4.3 of

* [[Urs Schreiber]], [[Konrad Waldorf]], _Connections on non-abelian Gerbes and their Holonomy_ ([arXiv:0808.1923](http://arxiv.org/abs/0808.1923))

A discussion in the context of [[principal infinity-bundles]] (as opposed to higher vector bundles), is in section "2.3.5 Twisted cohomology and sections" and then in section  "3.3.7.2 Twisted 1-bundles -- twisted K-theory"

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

The observation then re-appears independently in

* [[Chris Rogers]], _Higher geometric quantization_, talk at _Higher Structures_ in Göttingen (2011) ([pdf slides](http://www.crcg.de/wiki/Higher_geometric_quantization))

[[!redirects twisted bundles]]

[[!redirects twisted vector bundle]]
[[!redirects twisted vector bundles]]

[[!redirects twisted unitary bundle]]
[[!redirects twisted unitary bundles]]

[[!redirects connection on a twisted vector bundle]]
[[!redirects connections on a twisted vector bundle]]
[[!redirects connections on twisted vector bundles]]


twisted Chern character

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The ordinary [[Chern character]] for [[K-theory]] sends K-classes to [[ordinary cohomology]] with real coefficients. Over a [[smooth manifold]] the [[de Rham theorem]] makes this equivalently take values in [[de Rham cohomology]].

The **twisted Chern character** analogously goes from [[twisted K-theory]] to [[twisted de Rham cohomology]].

## Details

### In terms of twisted curvature characteristic forms
 {#InTermsOfTwistedCurvatureCharacteristicForms}

For a degree-3 [[ordinary cohomology]]-class $\tau_3 \in \,H^3(X;, \mathbb{Z})\,$ (e.g. modeled as a [[bundle gerbe]]) on a [[compact topological space|compact]] [[smooth manifold]] $X$, a class in the $\tau_3$-[[twisted K-theory]] $KU^{\tau_3}(X)$ may be represented as a suitable [[Grothendieck group]]-[[equivalence class]] $[V_1,V_2]$ of a [[pair]] $V_1, V_2 \in TwVectBund^{\tau_3}(X)$ of [[complex vector bundles|complex]] [[twisted vector bundles]] (e.g. modeled as [[bundle gerbe modules]], possibly of infinite [[rank of a vector bundle|rank]]).

Then for any lift of the twist $\tau_3$ [to](ordinary+differential+cohomology#CurvatureAndCharClass) a [[Deligne cohomology|Deligne cocycle]] $[h_0, A_1, B_2, H_3]_U$ (a [[connection on a bundle gerbe]]) with respect to some [[surjective submersion]] $p \colon U \twoheadrightarrow X$, hence in particular with a [[differential 2-form]] (the local [[B-field]])

\[
  \label{BFieldForTheTwist}
  B_2
    \,\in\,
  \Omega^2(U)
  \,,
\]

one may make a choice of [[connections on twisted vector bundles]] $(\nabla_1, \nabla_2)$ on $(V_1, V_2)$, and this induces [[endomorphism ring]]-valued [[curvature 2-forms]]

\[
  \label{TwistedCurvatureForms}
  F_i
    \,\in\,
  \Omega^2
  \big(
    U,
    \,
    End(V_i)
  \big)
\]

on the cover.

Now the twisted [[characteristic form]] (eq:TwistedCharacteristicFormOnCover) obtained as the [[trace]] (in the [[square matrix]]-[[coefficients]]) of the difference of the [[wedge product]]-[[exponential series]] of these two twisted [[curvature forms]] (eq:TwistedCurvatureForms), [[wedge product|wedged]] with the wedge-[[exponential series]] of the [[B-field]] (eq:BFieldForTheTwist)

\[
  \label{TwistedCharacteristicFormOnCover}
  \exp(B_2)
  \wedge
  tr
  \big(
    \exp(F_1) - \exp(F_2)
  \big)
  \;\;
  =
  p^\ast ch^{B_2}(\nabla_1, \nabla_2)
  \;\;
  \;\;\;
  \in
  \;
  \Omega^\bullet(U)
\]

turns out to

1. be well-defined, in that the [[traces]] all exist;

1. be the [[pullback of differential forms|pullback]] of an even-degree [[differential form]] on $X$

   $$
     ch^{B_2}(\nabla_1,\nabla_2)
     \;\;\;
     \in
     \;
     \Omega^{2\bullet}(X)
     \,,
   $$

1. which is closed in the $H_3$-[[twisted de Rham cohomology|twisted de Rham complex]] on $X$, in that

   $$
     \big( d - H_3 \wedge \big) ch^{B_2}(\nabla_1, \nabla_2)
     \;=\;
     0
     \,,
   $$

1. and whose [[twisted de Rham cohomology|twisted de Rham]] class

   \[
     \label{TwistedChernCharacterAsTwistedDeRhamClassOfTwistedCharClass}
     ch^{\tau_3}[V_1, V_2]
     \;\coloneqq\;
     \big[
       ch^{B_2}(\nabla_1, \nabla_2)
     \big]
     \;\;\;
     \in
     \;
     H^{3 + H_3}_{dR}(X)
   \]

   is independent of the choices ($B_2$, $\nabla_1$, $\nabla_2$) made.

([BCMMS 2002, Prop. 9.1](#BCMMS02))

This class (eq:TwistedChernCharacterAsTwistedDeRhamClassOfTwistedCharClass) is the *twisted Chern character* of the twisted K-theory class $[V_1, V_2]$ ([BCMMS 2002, p. 26](#BCMMS02)).

## Related concepts

* [[Chern character]]

* [[equivariant Chern character]]

* [[twisted equivariant Chern character]]

## References

* {#BCMMS02}  [[Peter Bouwknegt]], [[Alan Carey]], [[Varghese Mathai]], [[Michael Murray]], [[Danny Stevenson]], Section 6.3 in: *Twisted K-theory and K-theory of bundle gerbes*,  Commun Math Phys **228** (2002) 17-49 [[arXiv:hep-th/0106194](http://arxiv.org/abs/hep-th/0106194), [doi:10.1007/s002200200646](https://doi.org/10.1007/s002200200646)]

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], Section 2 of: _Twisted equivariant K-theory with complex coefficients_, Journal of Topology, Volume 1, Issue 1, 2007 ([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

* [[Varghese Mathai]], [[Danny Stevenson]], _Chern character in twisted K-theory: equivariant and holomorphic cases_, Commun. Math. Phys. **236** (2003) 161-186 [[arXiv:hep-th/0201010](https://arxiv.org/abs/hep-th/0201010), [doi:10.1007/s00220-003-0807-7](https://doi.org/10.1007/s00220-003-0807-7)]

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], Section 7 of: _Twisted K-theory_, Ukrainian Math. Bull. 1 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

* [[Varghese Mathai]], [[Danny Stevenson]], Section 6 of: _On a generalized Connes-Hochschild-Kostant-Rosenberg theorem_, Advances in Mathematics, vol. 200 no. 2 (2006) 303-335 ([arXiv:math/0404329](https://arxiv.org/abs/math/0404329))

* [[Paul Bressler]], A. Gorokhovsky, R. Nest, [[Boris Tsygan]], _Chern Character for Twisted Complexes_. In: Kapranov M., Manin Y.I., Moree P., Kolyada S., Potyagailo L. (eds.) Geometry and Dynamics of Groups and Spaces. Progress in Mathematics, vol 265. Birkhäuser Basel (2007) ([doi:10.1007/978-3-7643-8608-5_5](https://doi.org/10.1007/978-3-7643-8608-5_5))

* [[Kiyonori Gomi]], [[Yuji Terashima]], Section 4 of: _Chern-Weil Construction for Twisted K-Theory_,  Commun. Math. Phys. 299, 225–254 (2010) ([doi:10.1007/s00220-010-1080-1](https://doi.org/10.1007/s00220-010-1080-1))

  (in terms of [[vectorial bundles]])

* [[Max Karoubi]], Section 8.3 of: _Twisted bundles and twisted K-theory_, in:  Guillermo Cortiñas (ed.) _Topics in Noncommutative Geometry_ ([arXiv:1012.2512](https://arxiv.org/abs/1012.2512), [ISBN:978-0-8218-6864-5](https://bookstore.ams.org/cmip-16))

In twisted [[orbifold K-theory]]:

* [[Jean-Louis Tu]], [[Ping Xu]], _Chern character for twisted K-theory of orbifolds_, Advances in Mathematics Volume 207, Issue 2, 20 December 2006, Pages 455-483 ([arXiv:math/0505267](https://arxiv.org/abs/math/0505267), [doi:10.1016/j.aim.2005.12.001](https://doi.org/10.1016/j.aim.2005.12.001))

On [[bundle gerbe modules]] with [[higher structure|higher]] [[G-structures]] (including [[String structure]], etc.) and the corresponding [[twisted Chern character]]:

* [[Fei Han]], [[Ruizhi Huang]], [[Varghese Mathai]], *Fractional structures on bundle gerbe modules and fractional classifying spaces* [[arXiv:2203.14439](https://arxiv.org/abs/2203.14439)]


twisted cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

By the discussion at _[[cohomology]]_, plain cohomology is the study of

* [[(∞,1)-categorical hom-spaces]] in [[(∞,1)-toposes]] (for "[[nonabelian cohomology]]")

* or their [[stabilizations]]  to [[stable (∞,1)-categories]] of [[spectrum objects]] (for "[[generalized (Eilenberg-Steenrod) cohomology]]")

* or generally in  [[symmetric monoidal (∞,1)-categories]]

and maybe fully generally in any [[(∞,1)-category]] $\mathcal{C}$ whatsoever.

So for $A \in \mathcal{C}$ any [[object]] the [[cohomology]] of any other object $X$ with [[coefficients]] in $A$ is the [[mapping space]] $\mathcal{C}(X,A)$. Notice that this is equivalently the [[homotopy type]] of [[sections]] $\mathcal{C}_{/X}(X, X \times A)$ of the trivial $A$-[[fiber ∞-bundle]] over $X$. The idea of _twisted cohomology_ then is to consider general $A$-[[fiber ∞-bundles]] $\chi$ over $X$ and take the $\chi$-twisted cohomology of $X$ to the type of [[sections]] of this.

| [[cohomology]] | twisted cohomology |
|--|--|
| [[homotopy types]] of [[mapping spaces]] | [[homotopy types]] of spaces of [[sections]] |

Given an $\infty$-topos $\mathbf{H}$, then also its arrow $\infty$-category $\mathbf{H}^I$ is an $\infty$-topos, over $\infty Grpd^I$ and it also sits over $\mathbf{H}$ by the [[codomain fibration]], constituting an "[[extension]]" of $\mathbf{H}$ by itself:

$$
  \array{
    \mathbf{H}
    \\
    \downarrow^{\mathrlap{incl}}
    \\
    \mathbf{H}^I
    \\
   \downarrow^{\mathrlap{cod}}
     \\
    \mathbf{H}
  }
  \,.
$$

The intrinsic cohomology of $\mathbf{H}^I$ under this fibration is [[nonabelian cohomology|nonabelian]] twisted cohomology as discussed in some detail in _[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]_.

Notice that "stable cohomology", which is traditionally called [[generalized (Eilenberg-Steenrod) cohomology]] may be thought of as the lowest order [[Goodwillie calculus|Goodwillie]] approximation to [[nonabelian cohomology]]: where a cocycle in nonabelian cohomology is a map to any [[homotopy type]], a cocycle in [[generalized (Eilenberg-Steenrod) cohomology]] is a map into a [[stable homotopy type]].

In this sense the [[tangent (infinity,1)-topos]] $T \mathbf{H}$ is the lowest order linear approximation to the [[codomain fibration]]

$$
  \array{
    \Stab(\mathbf{H})
    \\
    \downarrow^{\mathrlap{incl}}
    \\
    T\mathbf{H}
    \\
   \downarrow^{\mathrlap{cod}}
     \\
    \mathbf{H}
  }
  \,.
$$

Higher-order approximations should involve a notion of higher-order forms of the tangent (∞,1)-topos, in parallel with the relationship between the [[jet bundles]] and [[tangent bundle]] of a manifold. It is clear that whatever we may say in detail about the $k$th-[[jet (∞,1)-category|jet (∞,1)-topos]] $J^k \mathbf{H}$, its intrinsic cohomology is a version of twisted cohomology which is in between [[nonabelian cohomology]] and stable i.e. [[generalized (Eilenberg-Steenrod) cohomology]].

It seems that a layered analysis of nonabelian cohomology this way in higher homotopy theory should eventually be rather important, even if it hasn't received any attention at all yet. It seems plausible that a generalization of [[Chern-Weil theory]] which approximates classes of [[principal infinity-bundles]] not just by [[universal characteristic classes]] in [[ordinary cohomology]] and hence in stable cohomology, but that one wants to consider the whole Goodwillie Taylor tower of approximations to it.

## Definition
 {#Definition}

We discuss concrete realizations of the above [general idea](#Idea) in some cases of interest:

* _[In an ∞-topos](#InAnInfinityTopos)_

* _[In a stabilized ∞-topos ](#InAStabilizedInfinityTopos)_

* _[In a general symmetric monoidal ∞-category](#InASymmetricMonoidalInfinityCategory)_

### In an $\infty$-topos -- twisted nonabelian (sheaf) cohomology
 {#InAnInfinityTopos}

Let $\mathcal{C} = \mathbf{H}$ be an [[(∞,1)-topos]]. Let $A \in \mathbf{H}$ be any [[object]], to be called the [[coefficient]] object.

Write $\mathbf{Aut}(A) \in Grp(\mathbf{H})$ for the [[automorphism ∞-group]] of $A$ and $\mathbf{B}\mathbf{Aut}(A) \in \mathbf{H}$ for its [[delooping]]. There is a canonical  [[∞-action]] of $\mathbf{Aut}(A)$ on $A$ exhibited by the corresponding universal [[associated ∞-bundle]]

$$
  \array{
    A &\to& A//\mathbf{Aut}(A)
    \\
    && \downarrow^{\mathrlap{\rho_A}}
    \\
    && \mathbf{B}\mathbf{Aut}(A)
  }
  \,.
$$

Let $X \in \mathbf{H}$ be any object.

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

A **twist** for $A$-[[cohomology]] on $X$ is a morphism $\chi \colon X \to \mathbf{B}\mathbf{Aut}(A)$ in $\mathbf{H}$. The corresponding [[associated ∞-bundle|associated]] $A$-[[fiber ∞-bundle]] over $X$ which is the [[homotopy pullback]]

$$
  \array{
    \chi^\ast \rho_A &\to& A//\mathbf{Aut}(A)
    \\
    \downarrow && \downarrow^{\mathrlap{\rho_A}}
    \\
    X &\stackrel{\chi}{\to}& \mathbf{B}\mathbf{Aut}(A)
  }
$$

we call the [[local coefficient ∞-bundle]] for twisted $A$-cohomology classified by $\chi$.

The *[[cocycle]] [[∞-groupoid]] of $\chi$-twisted $A$-cohomology* is

$$
  \Gamma_X(\chi^\ast \rho_A) \in \infty Grpd
  \,.
$$

The **$\chi$-twisted cohomology set** of $X$ is

$$
  \pi_0 \Gamma_X(\chi^\ast \rho_A) \in Set
$$

=--

Special cases of this definition are implicit in traditional literature. The above statement appears in this form in ([Nikolaus-Schreiber-Stevenson 12](#NikolausSchreiberStevenson12)).

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

The $\chi$-twisted cohomology is equivalently the ordinary cohomology of $\chi$ with [[coefficients]] in $\rho_A$ in the [[slice (∞,1)-topos]] of $\mathbf{H}$ over $\mathbf{B}\mathbf{Aut}(A)$:

$$
  \Gamma_X(\chi^\ast \rho_A)
  \simeq
  \mathbf{H}_{/\mathbf{B}\mathbf{Aut}(A)}(\chi, \rho_A)
  \,.
$$

=--

### In a stabilized $\infty$-topos -- twisted ES-type (sheaf) cohomology
 {#InAStabilizedInfinityTopos}

Let now $\mathcal{C} = Stab(\mathbf{H})$ be an [[stable (∞,1)-category]] of [[spectrum objects]] in an ambient [[(∞,1)-topos]] $\mathbf{H}$. Let $E \in CRing_\infty(\mathbf{H})$ be a corresponding [[E-∞ ring]] object. Write

$$
  GL_1(E) \hookrightarrow \mathbf{Aut}(E)
  \in
  Grp(\mathbf{H})
$$

for the [[∞-group of units]] of $E$.

Now a twist $\chi \;\colon\; X \to \mathbf{B}GL_1(E)$ classifies an [[(∞,1)-module bundle]] of $E$-lines. The $\chi$-twisted $E$-cohomology is again the (stable) homotopy type of sections of this.

For the case of [[twisted K-theory]] (see the references there) this description goes back to [[Jonathan Rosenberg]]. The above general abstract description is developed in ([Ando-Blumberg-Gepner 10](#ABG10)).

For more details see

* at _[[(∞,1)-module bundle]]_ the section _[Sections and twisted cohomology](%28infinity%2C1%29-module+bundle#SectionsAndTwistedCohomology)_

* at _[[twisted bivariant cohomology]]_ the section _[Axiomatization in homotopy theory](bivariant+cohomology+theory#AxiomatizazionInHomotopyTheory)_ .

+-- {: .num_remark #PicardGeneralizingGL1}
###### Remark

There are canonical maps

$$
  \mathbf{B}GL_1(E) \simeq E Line \hookrightarrow Pic(E Mod)
  \hookrightarrow E Mod
  \,,
$$

where $Pic(E Mod)$ denotes the [[Picard ∞-groupoid]]. This suggest to speak not just of twists of the form $\chi \colon X \to \mathbf{B}GL_1(E) \simeq E Line \hookrightarrow E Mod$ but more generally of twists of the form $\chi \colon Pic(E Mod) \hookrightarrow E Mod$. While these in general no longer define $E$-[[fiber ∞-bundles]] (so that sections of them are strictly speaking in general no longer locally $E$-cohomology cocycles), this more general notion has the advantage that it makes sense also in [[symmetric monoidal (∞,1)-categories]] different from those of the form $Stab(\mathbf{H})$.

This we turn to [below](#InASymmetricMonoidalInfinityCategory).

=--

### In a general symmetric monoidal $\infty$-category
 {#InASymmetricMonoidalInfinityCategory}

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

If in the above situation we write $[X, E Mod]$ for the [[symmetric monoidal (∞,1)-category]] of $E$-[[(∞,1)-module bundles]] on $X$, then given an object $\chi \in [X,E Mod]$ its homotopy type of sections, hence the $\chi$-twisted cohomology of $X$ is equivalently

$$
  Hom_{[X, E Mod]}(\mathbb{I}_X, \chi)
  \in
  \infty Grpd
  \,,
$$

where $\mathbb{I}_X$ is the [[tensor unit]] object, the trivial $E$-[[(∞,1)-module bundle]] over $X$.

=--

In view of this and remark \ref{PicardGeneralizingGL1} one considers the following.

Let $(\mathcal{C}, \otimes)$ be a [[symmetric monoidal (∞,1)-category]].

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

An object $\chi \in Pic(\mathcal{C})$ of the [[Picard ∞-groupoid]] of $\mathcal{C}$ we call a _twist_ for [[cohomology]] in $\mathcal{C}$. For $X, A \in \mathcal{C}$ any two [[objects]], we say that the *$\chi-twisted$ cohomology of $X$ with [[coefficients]] in $A$* is

$$
  \mathcal{C}(X, \chi \otimes A) \in \infty Grpd
  \,.
$$

=--

## Properties

### Twisted cohomology with trivial twisting cocycle

> old material, to be harmonized...

Let  ${*} \to B$ be a [[pointed object]]. Then

* we say that the [[cocycle]]

  $(X \to * \to B) \in \mathbf{H}(X,B)$

  is the **trivial $B$-cocycle** on $X$.

*  the morphism $f:\hat{B}\to B$ induces a [[fibration sequence]]

   $A \to \hat B \stackrel{f}{\to} B$

   in $\mathbf{H}$.

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

The $([*],f)$-twisted cohomology with trivial twisting cocycle is equivalent to the ordinary [[cohomology]] with coefficients in the [[homotopy fiber]] $A$ of $f$:

$$
  \mathbf{H}_{[*]}(X,f) \simeq \mathbf{H}(X,A)
  \,.
$$

=--

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

By definition, the [[homotopy fiber]] of $A$ is the [[homotopy pullback]]

$$
  \array{
    A &\to& *
    \\
    \downarrow && \downarrow
    \\
    \hat B &\stackrel{f}{\to}& B
  }
$$

in $\mathbf{H}$. Since the $\infty$-groupoid valued [[derived hom-space|hom]] in an [[(∞,1)-category]] is [[exact functor|exact]] with respect ot [[homotopy limits]] (by definition of homotopy limits), it follows that for every object $X$, there is  fibration sequence of [[cocycle]] [[∞-groupoids]]

$$
  \array{
    \mathbf{H}(X,A) &\to& {*}
    \\
    \downarrow && \downarrow^{\mathrlap{const_{*}}}
    \\
    \mathbf{H}(X,\hat B) &\to& \mathbf{H}(X,B)
  }
  \,.
$$

By definition of twisted cohomology, this identifies

$$
  \mathbf{H}(X,A) \simeq \mathbf{H}_{[*]}(X,f)
  \,.
$$

=--

For this reason, when $B$ is pointed,
it is customary to call the set of equivalence classes $\pi_0\mathbf{H}_{[c]}(X;f)$ the **$c$-twisted $A$-cohomology of $X$**, and to denote it by the symbol

$$
  H_{[c]}(X,A)
$$

+-- {: .num_remark}
###### Remark
The cohomology fibration sequence $\mathbf{H}(X,A) \to \mathbf{H}(X,\hat B) {\to} \mathbf{H}(X,B)$ can be seen as an **obstruction problem** in cohomology:

* the **obstruction** to lifting a $\hat B$-cocycle to an $A$-cocycle is its image in $B$-cohomology (all with respect to the given [[fibration sequence]])

But it also says:

* $A$-cocycles are, up to equivalence, precisely those $\hat B$-cocycles whose class in $B$-cohomology is the class of the trivial $B$-cocycle.

=--

## Examples

### Sections as twisted functions... {#SectionsAsTiwstedFunctions}

For $V$ a [[vector space]] and $X$ a manifold, both regarded a 0-[[truncated]] objects in the $(\infty,1)$-topos on the site [[CartSp]] (that of [[Lie infinity-groupoid]]s), a [[cocycle]] $X \to V$ is simply smooth $V$-valued function on $X$.

Now let $G$ be a Lie group with smooth [[delooping]] groupoid $\mathbf{B}G$ and let $\rho : \mathbf{B}G \to Vect$ be a [[representation]] of $G$ on $V$, i.e. $\rho(\bullet) = V$. Then the corresponding [[action groupoid]] $V//G$ sits in the [[fibration sequence]]

$$
  V \to V//G \stackrel{p}{\to} \mathbf{B}G
  \,.
$$

Hence we can ask for the $p$-twisted cohomology of $X$ with values in $V$. Now, a cocycle $g : X \to \mathbf{B}G$ is one classifying a $G$-[[principal bundle]] on $X$. By looking at this in [[Cech cohomology]] it is immediate to convince onself that cocycles $X \to V//G$ such that the composite $X \to V//G \stackrel{p}{\to} \mathbf{B}G$ is equivalent to the given $g$ are precisely the [[section]]s of the $\rho$-[[associated bundle|associated vector bundle]]:

on a patch $U_i$ of a good cover over wich $P$ has been trivialized, the cocycle $X \to V//G$ is simply a $V$-valued function $\sigma_i : U_i \to V$. Then on double overlaps it is a smooth [[natural transformation]] $\sigma_i|_{U_{i j}} \to \sigma_j|_{U_i j}$ whose components in $G$ are required to be those of the given cocycle $g$. That means exactly that the functions $(\sigma_i)$ are glued on double overlaps precisely as the local trivializations of a global section $\sigma : X \to P \times_G V$ would.

Hence we find the $p$-twisted cohomology is

$$
  H_{[g]}(X,V) =
  \{sections\; of\; P \times_G V\}
  \,.
$$

In this sense **a section is a twisted function**.

Notice that $V//G \stackrel{p}{\to} \mathbf{B}G$ is not itself a homotopy fiber, but is a _lax_ fiber, in that we have a [[2-limit|lax pullback]] (really a _comma object_ )

$$
  \array{
    V//G &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\to& Vect
  }
  \,,
$$

where in the bottom right corner we have [[Vect]] (regarded as a [[stack]] on $CartSp$ in the evident way) and where the right vertical morphism sends the point to the ground field vector space $k$ (or rather sends $U \in CartSp$ to the trivial vector bundle $U \times k$ ).

We may paste to this the homotopy pullback along the cocycle $g : X \to \mathbf{B}G$ to obtain

$$
  \array{
    P\times_G V &\to& V//G &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\stackrel{g}{\to}& \mathbf{B}G &\to& Vect
  }
  \,.
$$

This makes is manifest that a section $\sigma : X \to P \times_G V$ is also the same as a natural transformation from $const_k : X \to Vect$ to $X \stackrel{g}{\to} \mathbf{B}G \to Vect$.

Notice moreover that in the special case that $G = U(1)$ and for ground field $k = \mathbb{C}$ we may think of $\mathbf{B}U(1)$ as the category $\mathbb{C} Line \hookrightarrow \mathbb{C} Mod = Vect$ and think of the twisting cocycle $g$ as

$$
  X \stackrel{g}{\to} \mathbb{C}Line \hookrightarrow \mathbb{C}Mod
  \,.
$$

#### ... and $\infty$-sections as twisted $\infty$-functions
 {#InfSections}

Regarded this way, the above discussion has a generalization to the case where the [[monoid]] $\mathbb{C}$ is replaced with any [[ring spectrum]] $R$ and we consider

$$
  X \stackrel{\tau}{\to} R Line \hookrightarrow R Mod
  \,.
$$

Twisted cohomology in terms of such morphisms $\tau$ is effectively considered in

* {#AndoBlumbergGepner10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in [[Jonathan Rosenberg]] et al. (eds.), _Superstrings, Geometry, Topology, and $C^\ast$-algebras_, volume 81 of _Proceedings of Symposia in Pure Mathematics_, 2009 ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

and in unpublished work of [[Ulrich Bunke]] et al. For more on this see the discussion at _[[(∞,1)-vector bundle]]_.

More generally one can hence twist with maps

$$
  X \to Pic(R) \hookrightarrow R Mod
$$

into the [[Picard ∞-group]] of $R Mod$.

See also at _[∞-group of units -- augmented definition](infinity-group+of+units#AugmentedDefinition)_.

### twisted K-theory

In the context of [[generalized (Eilenberg-Steenrod) cohomology]] a coefficient object for [[cohomology]] is a [[spectrum]] $A$: the $A$-cohomology of a [[topological space]] $X$ with coefficients in $A$ is the set of homotopy classes of maps $X \to A$. For instance, as a model of the degree-$0$ space in the [[K-theory spectrum]] one can take $A = Fred(H)$, the space of Fredholm operators on a separable Hilbert space $H$. There is a canonical action on this space of the projective unitary group $G = P U(H)$ of $H$. Since $P U(H)$ has the homotopy type of an [[Eilenberg-Mac Lane space]] $K(\mathbb{Z},2)$, a $P U(H)$-[[principal bundle]] $P \to X$ defines a class $c \in H^3(X,\mathbb{Z})$ in [[Eilenberg-MacLane spectrum|ordinary integral cohomology]] (this may also be thought of as the class of a twisting [[bundle gerbe]]). The twisted K-theory (in degree $0$) of $X$ with that class as its twist is the set of homotopy classes of sections $X \to P \times_{P U(H)} Fred(H)$ of the associated bundle.

### $G$-Actions on spectra

The above example generalizes straightforwardly to the case that

* $A$ is a [[spectrum|connective spectrum]], i.e. [[topological space]] that is an infinite [[loop space]]. (We need to assume a connective spectrum given by an infinite loop space so that $A$ can be regarded as living in the category of topologicall spaces along with the other objects, such as classifying spaces $\mathbf{B}G$ of nonabelian groups);

* with a (topological) [[group]] $G$ acting on $A$ by automorphisms and

* a $G$-[[principal bundle]] $P \to X.$

In this case there is an established definition of [[generalized (Eilenberg-Steenrod) cohomology]] with coefficients $A$ _twisted_ by a $G$-[[principal bundle]] as follows.

From the $G$-[[principal bundle]] $P \to X$ we obtain the associated $A$-bundle $P \times_G A \to X$. Then a twisted $A$-cocycle on $X$ is defined to be a [[section]] $X \to P \times_G A$ of this associated bundle. The collection of homotopy classes of these sections is the _twisted $A$-cohomology of $X$_ with the twist specified by the class of $P$.

This is the definition of twisted cohomology as it appears for instance essentially as definition 22.1.1 of the May–Sigursson reference below (when comparing with their definition take their $G$ to be the trivial group and identify their $\Gamma$ and $\Pi$ with our $G$).

It is clearly a particular case of the general definition of twisted cohomology given above:

* the $(\infty,1)$-topos $\mathbf{H}$ is the $(\infty,1)$-category of [[Top]] of topological spaces

* the object $B$ is the [[delooping]] $\mathbf{B}G$ of the [[group]] $G$.

* the object $\hat{B}$ is the homotopy quotient $A//G\simeq \mathbf{E}G\times_G A$.

* the twisting cocycle $c$ is the element in $\mathbf{Top}(X,\mathbf{B}G)$ defining the principal $G$-bundle $P\to X$.

Indeed, $B$ is pointed, we have a fibration sequence
$$
A \to A//G \to \mathbf{B}G
$$
and the homotopy pullback
$$
  \array{
    P_A &\to& {A//G}
     \\
     \downarrow && \downarrow{f}
     \\
     X  &\stackrel{c}{\to}& \mathbf{B}G
  }\,
$$
is the $A$-bundle $P\times_G A\to X$.

The obstruction problem described by this example reads as folllows:

*  the obstruction to lifting a ("nonabelian" or "twisted") $A//G$-cocycle $g : X \to A//G$ to an $A$-cocycle $\hat g : X \to A$ is its image $\delta g $ in first $G$-cohomology  $\delta g \in H^1(X,G) :=\pi_0 \mathbf{Top}(X, \mathbf{B} G)$.

Read the other way round it says:

* $A$-cocycles are precisely those $G$-twisted $A$-cocycles whose twist vanishes.

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

Since the associated bundle $P \times_G A$ is in general no longer itself a spectrum, twisted cohomology is not an example of [[generalized (Eilenberg-Steenrod) cohomology|generalized Eilenberg-Steenrod cohomology]].

To stay within the spectrum point of view, May–Sigurdsson suggested that twisted cohomology should instead be formalized in terms of _[[parameterized homotopy theory]]_, where one thinks of $P \times_G A$ as a [[parameterized spectrum|parameterized family of spectra]].

=--

### Group cohomology with coefficients in a module

Some somewhat trivial examples of this appear in various context. For instance [[group cohomology]] on a group with coefficients in a nontrivial module can be regarded as an example of twisted cohomology. See there for more details.

Compare this to the example below of cohomology "with local coefficients". It is the same principle in both cases.

### Twisted bundles

To get a feeling for how the definition does, it is
instructive to see how for the fibration sequence coming from an ordinary central extension $K \to \hat G \to G$ of ordinary groups as

$$
  \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\omega}{\to}
  \mathbf{B}^2 K
$$

classified by a [[group cohomology|group 2-cocycle]] $\omega$, $c$-twisted $\hat G$-cohomology produces precisely the familiar notion of [[twisted bundles]], with the twist being the lifting [[gerbe]] that obstructs the lift of a $G$-bundle to a $\hat G$-bundle.

This is also the first example in the list
in the last section of

* [[schreiber:Background fields in twisted differential nonabelian cohomology|Background fields in twisted differential nonabelian cohomology]]

and contains examples that are of interest in the wider context of [[string theory]].

See also [Twisted Differential String- and Fivebrane-Structures](http://golem.ph.utexas.edu/category/2009/03/twisted_differential_string_an.html).

### Cohomology with local coefficients

What is called **cohomology with local coefficients** is twisted cohomology with the twist given by the class represented by the universal cover space of the base space, which is to say: by the action of the fundamental group of the base space.

In the classical case of ordinary cohomology, C. A. Robinson in 1972 constructed a _twisted_ $K(\pi,n)$,
denoted $\tilde K(\pi,n)$, so that, for nice spaces,
the cohomology with local coefficients $\tilde H^n(X,\pi)$
with respect to a homomorphism $\varepsilon:\pi_1(X)\to Aut(\pi)$ is
given by homotopy classes of maps $X\to \tilde K(\pi,n)$ compatible with $\varepsilon.$

More generally, for any space $X$, let $A$ be a coefficient object that is equipped with an action of the first [[fundamental group]] $\pi_1(X)$ of $X$. (Such an action is also called an $A$-valued [[local system]] on $X$).

Then there is the [[fibration sequence]]

$$
  A \to A//\pi_1(X) \to \mathbf{B} \pi_1(X)
$$

of this action.

Notice that there is a canonical map $c : X \to \mathbf{B} \pi_1(X)$, the one that classifies the universal cover of $X$.

Then **$A$-cohomology with local coefficients** on $X$ is nothing but the $c$-twisted $A$-cohomology of $X$ in the above sense.

#### Effective computation of cohomology with local coefficients

By _effective_, we mean involving as much as possible only calculations within finite dimensional linear algebra. For definiteness, we work in the smooth context and require the locally constant sheaf $\mathcal{A}$ to have stalks of finite dimensional vector spaces over a field $k$ ($\mathbb{R}$ or $\mathbb{C}$). Let $X$ be a connected $n$-dimensional manifold. The sheaf $\mathcal{A}$ can then be seen as the sheaf of germs of locally constant sections of a vector bundle $A\to X$ endowed with a flat connection $\nabla$. The fibers of $A$ are isomorphic with the stalks of $\mathcal{A}$, all of which are isomorphic to some finite dimensional vector space $\bar{A}$. Let $\tilde{X} \to X$ denote the universal cover of $X$ and $\pi = \pi_1(X)$ its fundamental group. It is well known that $\pi$ acts by [[deck transformation]] diffeomorphisms on $\tilde{X}$ and also induces a [[holonomy]] representation $\rho\colon \pi \to \mathbf{Aut}(\bar{A})$ on $\bar{A}$.

Consider the vector bundles $\Lambda^p X \otimes_X A$, where $\Lambda^p X$ is the bundle of differential $p$-forms, with $\Omega^p_A(-)$ denoting the sheaf of its sections (differential forms _twisted_ by $A$). Let $d_\nabla \colon \Omega^p_A \to \Omega^{p+1}_A$ denote the correspondingly twisted de Rham differential, defined by the property that
\[ d_\nabla(\omega \otimes a) = d\omega \otimes a + (-1)^{|\omega|} \omega \wedge \nabla a , \]
where $d$ is the ordinary de Rham differential, $\nabla a$ is seen as a section of $\Lambda^1 X \otimes_X A$, and with the wedge operation acting in the obvious way. The complex of sheaves $(\Omega^\bullet_A(-),d_\nabla)$ is then a [[soft|soft sheaf]] resolution of the sheaf $\mathcal{A}$ of locally constant sections of $A$. Its cohomology $H^p(X;A,\nabla) = H^p(\Omega^\bullet_A(X),d_\nabla)$ is then isomorphic to the sheaf cohomology of $X$ with coefficients in the locally constant sheaf $\mathcal{A}$, $H^p(X,\mathcal{A}) \simeq H^p(X;A,\nabla)$.

Now, the bundle $A\to X$ and the connection $\nabla$ both pull back to the universal covering space $\tilde{X}$, that is to $\tilde{A} \to \tilde{X}$ and $\tilde{\nabla}$. Since now $\tilde{X}$ is simply connected, we can globally trivialize this bundle as $\tilde{A} \simeq \tilde{X} \times \bar{A}$ and $\tilde{\nabla}$ to the trivial connection thereon. Similarly, the structure of the sheaf of twisted differential forms can be simplified to $\Omega^p_{\tilde{A}}(-) \simeq \Omega^p(-) \otimes \bar{A}$, with the action of the twisted de Rham differential given by $d_{\tilde{\nabla}} (\omega \otimes \bar{a}) = d\omega \otimes \bar{a}$. This observation allows us to conclude that, on the universal covering space, we have the isomorphism $H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}$, where $H^p(\tilde{X})$ denotes the ordinary de Rham cohomology.

The pull back along a deck transformation diffeomorphism, induces a linear action of $\pi$ on forms $\Omega^p(\tilde{X})$. Combined with the holonomy representation of $\pi$ on $\bar{A}$, this defines a representation of $\pi$ on $\Omega^p_{\tilde{A}}(X)$. Let $\Omega^p_{\tilde{A}}(\tilde{X})^\pi \subseteq \Omega^p_{\tilde{A}}$ denote the subspace of twisted forms that is invariant under the action of $\pi$. It is not hard to notice the isomorphism of complexes $(\Omega^\bullet_\tilde{A}(\tilde{X})^\pi, d_{\tilde{\nabla}}) \simeq (\Omega^\bullet_A(X), d_\nabla)$ and hence of their cohomologies, $H^p(X; A,\nabla) \simeq H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^\pi, d_{\tilde{\nabla}})$. Furthermore, the action of $\pi$ on $\Omega^p_{\tilde{A}}(\tilde{X})$ commutes with the differential $d_{\tilde{\nabla}}$ and hence induces an action on $H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}$, where $\pi$ also acts in the obvious and compatible way on each tensor factor. Let $(H^p(\tilde{X})\otimes \bar{A})^\pi$ denote the corresponding $\pi$-invariant subspace.

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

Suppose that there exists a decomposition $\Omega^p_{\tilde{A}}(\tilde{X}) \simeq \Omega^p_{\tilde{A}}(\tilde{X})^\pi \oplus \Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}}$ as representations of $\pi$, with $\Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}}$ having no $\pi$-invariant subspace. Then we have the following isomorphism for each $p$: $H^p(X,\mathcal{A}) \simeq (H^p(\tilde{X})\otimes \bar{A})^\pi$.

=--

Whether the decomposition hypothesis actually holds may depend on the properties of the group $\pi$. For instance, it does hold if $\pi$ is compact (finite, in particular). Other cases, have to be examined individually.

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

Start with the short exact sequence of complexes
\[ 0 \to \Omega^\bullet_{\tilde{A}}(\tilde{X})^\pi
  \to \Omega^\bullet_{\tilde{A}}(\tilde{X})
  \to \Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}
  \to 0 . \]
The corresponding long exact sequence in cohomology is equivalent to the short exact sequences
\[ 0 \to H^p(X,\mathcal{A})
  \to H^p(\tilde{X}) \otimes \bar{A}
  \to H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}, d_{\tilde{\nabla}})
  \to 0 \]
for each value of $p$. The reason that all the connecting maps in the long exact sequence are zero is representation theoretic, since all the relevant maps are $\pi$-equivariant. Since, $H^{p+1}(X,\mathcal{A})$ carries a trivial representation by $\pi$, while the $H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}, d_{\tilde{\nabla}})$ representation has no $\pi$-invariant subspace, by [[Schur's lemma]], the only equivariant map from the latter to the former is zero. From the same observation, we easily see that the inclusion of $H^p(X,\mathcal{A})$ must coincide with the $\pi$-invariant subspace $(H^p(\tilde{X}) \otimes \bar{A})^\pi$.

=--

The presentation of cohomology of $X$ with local coefficients $\mathcal{A}$ as $\pi$-invariant de Rham cohomology of the universal covering space $\tilde{X}$ twisted by the holonomy representation on the stalk $\bar{A}$ is originally due to ([Eilenberg 47](#Eil47)). It is also discussed in Chapter VI of ([Whitehead 78](#Whit78)). The idea to look at the $\pi$-invariant subspace of the twisted de Rham cohomology of the universal covering space scan be found in an answer by [Peter Michor](#MichorMO) on [MathOverlflow](http://mathoverflow.net/).

The above result can be seen as an effective way to compute the sheaf cohomology groups $H^p(X,\mathcal{A})$ since all it requires is the knowledge of the following finite dimensional representations of the fundamental group $\pi = \pi_1(X)$: the deck transformations on the de Rham cohomology $H^k(\tilde{X})$ of the covering space, and the holonomy representation on a typical stalk $\bar{A}$ of the locally constant sheaf $\mathcal{A}$. Obtaining the invariant subspace of their tensor product can then be done using usual representation theory methods, which involve only finite dimensional linear algebra. Unfortunately, it appears that the requirement that $\pi$ is finite is rather important for the argument. It is not entirely clearly how to proceed if, for instance $\pi = \mathbb{Z}$ or is non-abelian and infinite.

## Related entries

* [[local coefficient bundle]]

* [[twisted Umkehr map]]

* [[twisted differential cohomology]]

* [[relative cohomology]]

* [[twisted equivariant cohomology]]

* <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos+--+structures#TwistedCohomology">twisted cohomology</a> in [[cohesive (∞,1)-topos -- structures]]

* [[twisted ordinary cohomology]], [[twisted K-theory]], [[twisted tmf]]

**Examples**

* [[twisted ordinary cohomology]], [[twisted Bredon cohomology]]

* [[twisted K-theory]], [[twisted equivariant K-theory]]

* [[twisted Cohomotopy]]

* [[twisted equivariant cohomology]]

## References

### General

Original articles on [[twisted ordinary homology]]:

* [[Kurt Reidemeister]], _Überdeckungen von Komplexen_, Crelle **173** (1935), 164-173 [[doi:10.1515/crll.1935.173.164](http://dx.doi.org/10.1515/crll.1935.173.164)]

and independently:

* [[Norman E. Steenrod]], _Homology with local coefficients_, Annals of Mathematics (Second Series) **44** 4 (1943) 610-627 [[doi:10.2307/1969099](https://doi.org/10.2307/1969099)]

On representing twisted [[Eilenberg–MacLane spaces]] and [[cohomology operations]] for twisted ordinary cohomology:

* [[Samuel Gitler]], _Cohomology operations with local coefficients_, American Journal of Mathematics **85** 2 (1963) 156–188  [[doi:10.2307/2373208](http://dx.doi.org/10.2307/2373208)]

* M. Bullejos, E. Faro, M. A. García-Muñoz, _Homotopy colimits and cohomology with local coefficients_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 ([numdam:CTGDC_2003__44_1_63_0](http://www.numdam.org/item?id=CTGDC_2003__44_1_63_0))

The case of twisted [[generalized (Eilenberg-Steenrod) cohomology]] twisted by a $G$-[[principal bundle]]:

* [[Peter May]], [[Johann Sigurdsson]], Section 22.1 of: _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006 ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))

* [[Chris Douglas]], _Twisted stable homotopy theory_ PhD thesis 2005 ([dspace:1721.1/7582](http://dspace.mit.edu/handle/1721.1/7582))

This in turn is based on the definition of [[twisted K-theory]] given in

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory_, Ukrainian Math. Bull. **1**, 3 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

Review:

* [[Jonathan Rosenberg]], *Twisted cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.03966](https://arxiv.org/abs/2401.03966)]

Details on Larmore's work on twisted cohomology are at

* [[Larmore twisted cohomology]].

The abstract discussion of twisted nonabelian cohomology in $\infty$-toposes is in

* {#NikolausSchreiberStevenson12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]],  _[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]_

The abstract discussion of twisted ES-type cohomology in the [[stable (infinity,1)-category of spectra]] is in

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics 81, American Mathematical Society 2010 ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004), [ISBN:978-0-8218-4887-6](https://bookstore.ams.org/pspum-81))

* {#ABGHR14} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], *An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology*, Journal of Topology **7** 3 (2014) 869893  [[arXiv:1403.4325](http://arxiv.org/abs/1403.4325), [doi:10.1112/jtopol/jtt035](https://doi.org/10.1112/jtopol/jtt035)]

The presentation of cohomology with local coefficients in terms $\pi_1$-equivariant de Rham cohomology on the universal covering space is discussed in

* {#Eil47} [[Samuel Eilenberg]], _Homology of spaces with operators I_, Trans. Amer. Math. Soc. 6 (1947), 378-417. ([doi](http://dx.doi.org/10.1090/S0002-9947-1947-0021313-4))

* {#Whit78} [[George Whitehead]], *Elements of homotopy theory*, Springer-Veriag, 1978.

* [Peter Michor](http://mathoverflow.net/users/26935/peter-michor), [answer](http://mathoverflow.net/a/129246/2622) to MathOverflow question [_de Rham cohomology and flat vector bundles_](http://mathoverflow.net/q/129246), (version: 2013-04-30).
 {#MichorMO}

### Chronology of literature on twisted cohomology

The oldest meaning of twisted cohomology is that of **cohomology with local coefficients** (see above).

For more on the history of that notion see

* [[History of cohomology with local coefficients]]

In the following we shall abbreviate

* tc = twisted cohomology

Searching MathSciNet for _twisted cohomology_ led to the following chronology: It missed older references in which the phrase was not used but the concept was in the sense of local coefficient systems – ancient and honorable.

Most notably missing are

* [[Kurt Reidemeister]] (1938) Topologie der Polyeder und kombinatorische

Topologie der Komplexe_, Mathematik und ihre Anwendungen in
Physik und Technik,_(But note that reprints appear, sans reviews. There is a short English and longer German review on Zentralblatt)

* [[Norman Steenrod]] (1942,1943)

* Olum (thesis 1947, published 1950)

Next come several that involve twisted differentials more generally.

Few are in terms of homotopy of spaces

tc ops should be treated as a single phrase – it may be that the ops are twisted, not the cohomology

* 1966 McClendon thesis – summarized in

* 1967  Emery Thomas  tc ops

* 1967 Larmore tc ops

* 1969 McClendon tc ops

* 1969 Larmore tc

* 1970 Peterson tc ops

* 1971 McClendon tc ops

* 1972 Deligne Weil conjecture for K3 tc – meaning?

* 1972 Larmore tc

* 1973 Larmore and Thomas tc

* 1973 Larmore tc

* gap

* 1980 Coelho & Pesennec tc

* 1980 Tsukiyama sequel to McClendon

* 1983 Coelho & Pesennec tc

* 1985 Morava but getsted at 1975 ??

* 1986 Fried tc

* 1988 Baum & Connes ??

* 1989 Lott torsion

* 1990 Dwork ??

* 1993 Gomez–Tato tc minimal models

* 1993 Duflo & Vergne diff tc

* 1993 Vaisman tc and connections

* 1993 Mimachi tc and holomorphic

* 1994 Kita tc and intersection

* 1995 Cho, Mimachi and Yoshida tc and configs

* 1995 Cho, Mimachi tc and intersection

* 1996 Iwaski and Kita tc de rham

* 1996 Asada nc geom and strings

* 1997 H Kimura tc de Rham and hypergeom

* 1998 [[Michael Farber]], [[Gabriel Katz]], [[Jerome Levine]], [Morse theory of harmonic forms](http://www.sciencedirect.com/science/article/pii/S0040938397827309), Topology, (Volume 37, Issue 3, May 1998, Pages 469–483)

* 1998 Knudson tc SL_n

* 1998 Morita  tc de Rham

* 1999 Kachi, Mtsumoto, Mihara tc and intersection

* 1999 Hanamura & Yoshida Hodge tc

* 1999 Felshtyn & Sanchez–Morgado Reidemeister torsion

* 1999 Haraoka hypergeom

* 2000 Tsou & Zois tc de rham

* 2000 Manea tc Czech

* 2001 Royo Prieto tc Euler

* 2001 Takeyama q-twisted

* 2001 Gaberdiel &Schaefr–Nameki tc of Klein bottle

* 2001 Iwaskai  tc deRham

* 2001 Proc Rims tc and DEs and several papers in this book

* 2001 Knudson tc SL_n

* 2001 Royo Prieto tc as $d+k\wedge$

* 2001 Barlewtta & Dragomir tc and integrability

* 2002 Lueck $L^2$

* 2002 Verbitsky HyperKahler, torsion, etc

* 2003 Etingof & Grana tc of Carter, Elhamdadi and Saito

* 2003 Cruikshank tc of Eilenberg

* 2003 various in Proc NATO workshop

* 2003 Dimca tc of hyperplanes

* 2004 Kirk & Lesch tc and index

* 2004 Bouwknegt, Evslin, Mathai tc and tK

* 2004 Bouwknegt, Hannbuss, Mathai tc in re: T-duality

* 2005 Bouwknegt, Hannbuss, Mathai tc in re: T-duality

* 2005 Bunke & Schick tc in re: T-duality

* 2006 Dubois tc and Reidemeister (elsewhere he considers twisted Reidemesiter)

* 2006 Bunke & Schick tc in re: T-duality

* 2006 Sati

* 2006 Atiyah & Segal tc and tK

* 2007 Mickelsson & Pellonpaa tc and tK

* 2007 Sugiyama in re: Galois and Reidemeister

* 2007 Bunke, Schick, Spitzweck tc in re: gerbes

* 2008 Kawahara hypersurfaces

[[!redirects twisted generalized cohomology]]

[[!redirects cohomology with local coefficients]]

[[!redirects twisted cohomology theory]]
[[!redirects twisted cohomology theories]]


twisted cohomotopy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Twisted cohomotopy_ is the [[twisted cohomology]]-variant of the
the [[non-abelian cohomology]]-[[generalized cohomology theory|theory]] _[[cohomotopy]]_, [[representable functor|represented]] by [[homotopy types]] of [[n-spheres]].

The [[coefficients]]/[[twisted cohomology|twist]] for twisted cohomotopy are [[spherical fibrations]], and [[cocycles]] are [[sections]] of these. For those [[spherical fibrations]] arising as [[unit sphere bundles]] of [[real vector bundles]] the twist may be understood as given by the [[J-homomorphism]].

## Properties

Various classical theorems of [[differential topology]] are secretly theorems about [[twisted cohomotopy]]

> table grabbed from [FSS 19b](#FSS19b)

### Twisted Pontrjagin theorem
 {#TwistedPontrjaginThomTheorem}

The _[[twisted Pontrjagin theorem]]_ relates twisted cohomotopy with [[cobordism classes]] of [[normally twisted-framed submanifolds]].

### Twisted May-Segal theorem

The [May-Segal theorem](configuration+space+of+points#LoopSpacesOfSuspensions) has a twisted generalization:

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

Let

1. $X^n$ be a [[smooth manifold|smooth]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[scanning map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ J-twisted cohomotopy space}}
  }{
    Maps_{{}_{/B O(n)}}
    \Big(
      X^n
      \;,\;
      S^{
        \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}}
      }
      \!\sslash\!
      O(n)
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{scanning map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. the [[J-homomorphism|J]]-[[twisted cohomotopy|twisted (n+k)-cohomotopy]] space of $X^n$, hence the [[space of sections]] of the $(n + k)$-[[spherical fibration]] over $X$ which is [[associated fiber bundle|associated]] via the [[tangent bundle]] by the [[O(n)]]-[[action]] on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

([Bödigheimer 87, Prop. 2](#Boedigheimer87), following [McDuff 75](#McDuff75))

+-- {: .num_prop #ScanningMapEquivalenceOverClosedFramedManifold}
###### Remark

In the special case that the [[closed manifold]] $X^n$ in Prop. \ref{ScanningMapEquivalenceOverClosedManifold} is [[parallelizable manifold|parallelizable]], hence that its [[tangent bundle]] is [[trivial bundle|trivializable]], the statement of Prop. \ref{ScanningMapEquivalenceOverClosedManifold} reduces to this:

Let

1. $X^n$ be a [[parallelizable manifold|parallelizable]] [[closed manifold]] of [[dimension]] $n$;

1. $1 \leq k \in \mathbb{N}$ a [[positive number|positive]] [[natural number]].

Then the [[scanning map]] constitutes a [[weak homotopy equivalence]]

$$
  \underset{
    \color{blue}
    { \phantom{a} \atop \text{ cohomotopy space}}
  }{
    Maps
    \Big(
      X^n
      \;,\;
      S^{
       n + k
      }
    \Big)
  }
  \underoverset
  {\simeq}
  {
    \color{blue}
    \text{scanning map}
  }
  {\longleftarrow}
  \underset{
    \mathclap{
    \color{blue}
      {
        \phantom{a}
        \atop
        {
          \text{configuration space}
          \atop
          \text{of points}
        }
      }
    }
  }{
    Conf
    \big(
      X^n,
      S^k
    \big)
  }
$$

between

1. $(n+k)$-[[cohomotopy]] space of $X^n$, hence the [[space of maps]] from $X$ to the [[n-sphere|(n+k)-sphere]]

1. the [[configuration space of points]] on $X^n$ with labels in $S^k$.

=--

### Poincaré–Hopf theorem

See at

* [[Poincaré–Hopf theorem]]

### Equivariant Hopf degree theorem

On [[flat orbifolds]], twisted cohomotopy becomes [[equivariant cohomotopy]] and the twisted [[Hopf degree theorem]] becomes the

* [[equivariant Hopf degree theorem]]

## Related concepts

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

* [[twisted Pontrjagin theorem]]

## References

The concept is implicit in classical texts on [[differential topology]], for instance

* {#McDuff75} [[Dusa McDuff]], _Configuration spaces of positive and negative particles_, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], Section 2 of: _Stable splittings of mapping spaces_, Algebraic topology. Springer 1987. 174-187 ([pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf), [[BoedigheimerStableSplittings87.pdf:file]])

Discussion of the [[twisted Pontrjagin theorem]] and of twisted [[stable cohomotopy]] ([[framed manifold|framed]] [[cobordism cohomology theory]]):

* {#Cruickshank03} [[James Cruickshank]], Lemma 5.2 and Section 7 of: _Twisted homotopy theory and the geometric equivariant 1-stem_, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)

Discussion of unstable twisted Cohomotopy, with application to [[flux quantization]] of [[supergravity C-field]] ("[[Hypothesis H]]"):

* {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds]]_, Comm Math. Phys.  **377** (2020) 1961-2025 [[arXiv:1904.10207](https://arxiv.org/abs/1904.10207), [doi:10.1007/s00220-020-03707-2](https://doi.org/10.1007/s00220-020-03707-2)]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane]]_ ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5-brane anomaly cancellation]]_ ([arXiv:2002.07737](https://arxiv.org/abs/2002.07737))

* {#FSS20a} [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies twisted String structure on M5-branes]]_ ([arXiv:2002.11093](https://arxiv.org/abs/2002.11093))

[[!redirects twisted Cohomotopy]]

[[!redirects twisted cohomotopy theory]]
[[!redirects twisted Cohomotopy theory]]

[[!redirects twisted cohomotopy cohomology theory]]
[[!redirects twisted Cohomotopy cohomology theory]]


twisted complex

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

#Contents#
* table of contents
{:toc}

## Definition

Let $C$ be a [[differential graded category]].

A **twisted complex** $E$ in $C$ is

* a [[graded set]] $\{E_i\}_{i \in \mathbb{Z}}$ of [[object]]s of $C$, such that only finitely many $E_i$ are not the [[zero object]];

* a set of [[morphism]]s $\{q_{i j} : E_i \to E_j \}_{i,j \in \mathbb{Z}}$ such that

  * $deg(q_{i j}) = i-j+1$;

  * $\forall i,j : \; d q_{i j} + \sum_{k} q_{k j}\circ q_{i k} = 0$.

The [[differential graded category]] $PreTr(C)$ of twisted complexes in $C$ has as objects twisted complexes and

$$
  PreTr(C)((E_\bullet, q), (E'_\bullet, q'))^k
  =
  \coprod_{l + j - i = k} C(E_i, E'_j)^l
$$

with differential given on $f \in C(E_i, E'_j)^l$ given by

$$
  d f = d_C f + \sum_m (q_{j m}\circ  f + (-1)^{l(i-m+1)} f \circ q_{m i})
  \,.
$$

The construction of categories of twisted complexes is functorial in that for $F : C \to C'$ a dg-functor, there is a dg-functor

$$
  PreTr(F) : PreTr(C) \to PreTr(C')
  \,.
$$

etc.

## Properties

Passing from a [[dg-category]] to its category of twisted complexes is a step towards enhancing it to a [[pretriangulated dg-category]].

## Related concepts

* [[Landau-Ginzburg model]], [[pretriangulated dg-category]]
* {#BondalKapranov90} [[Alexei Bondal]], [[Mikhail Kapranov]], _Enhanced triangulated categories_, Матем. Сборник, Том 181 (1990), No.5, 669–683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93–107, (MR91g:18010) ([[bondalKaprEnhTRiangCat.pdf:file]])
* [[Bernhard Keller]], _A brief introduction to $A_\infty$-algebras_ ([pdf](http://people.math.jussieu.fr/~keller/publ/IntroAinfEdinb.pdf))
* [[Mikhail Kapranov]], Svyatoslav Pimenov, _Derived varieties of complexes and Kostant's theorem for gl(m|n)_,
[arxiv/1504.00339](http://arxiv.org/abs/1504.00339)

[[!redirects twisted complexes]]