group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 delooped groupoid $\mathbf{B} \hat G$.
Given a realization of $\hat G$ as an abelian extension
of groups, i.e. given a fibration sequence
of groupoids such that $\mathbf{B}A$ is once deloopable so that the fibration sequence continues to the right at least one step as
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.
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 topological and smooth twisted bundles, respectively, whose twists are correspondingly topological or smooth bundle gerbes/circle n-bundles.
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
representing a characteristic class
Notice that if $G$ is a 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
where on the right we have the ordinary integral cohomology of the classifying space $B G \in$ Top of $G$.
Let $G$ and $\mathbf{c}$ be as above.
Write
for the homotopy fiber of $\mathbf{c}$.
This identifies $\hat G$ as the group extension of $G$ by the 2-cocycle $\mathbf{c}$.
Equivalently this means that
is the smooth circle 2-bundle/bundle gerbe classified by $\mathbf{c}$; and its loop space object
the corresponding circle group principal bundle on $G$.
Let $X \in \mathbf{H}$ be any object. From twisted cohomology we have the following notion.
The degree-1 total twisted cohomology $H_{tw}^1(X, \hat G)$ of $X$ with coefficients in $\hat G$, def. 1, relative to the characteristic class $[\mathbf{c}]$ is the set
of connected components of the (∞,1)-pullback
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
is the $[\alpha]$-twisted cohomology of $X$ with coefficients in $\hat G$ relative to $\mathbf{c}$.
For $[\alpha] = 0$ the trivial twist, $[\alpha]$-twisted cohomology coincides with ordinary cohomology:
By the discussion at principal ∞-bundle we may identify the elements of $H^1_{Smooth}(X, \hat G)$ with $\hat G$-principal ∞-bundles $P \to X$. In particular if $\hat G$ is an ordinary Lie group and $X$ is an ordinary smooth manifold, then these are ordinary $\hat G$-principal bundles over $X$. This justifies equivalently calling the elements of $H^1_{tw}(X,\hat G)$ twisted principal $\infty$-bundles; and we shall write
where throughout we leave the characteristic class $[\mathbf{c}]$ with respect to which the twisting is defined implcitly understood.
We unwind the abstract definition, def. 2, to obtain the explicit definition of twisted bundles by Cech cocycles the way they appear in the traditional literature (see the General References below).
Let $U(1) \to \hat G \to G$ be a group extension of topological groups.
Let $X \in$ Mfd $\hookrightarrow$ ETop∞Grpd $=: \mathbf{H}$ be a paracompact topological manifold with good open cover $\{U_i \to X\}$.
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
satisfying on every quadruple intersection the equation
I terms of this cocycle data the twisted cohomology $H^1_{[\alpha]}(X, \hat G)$ is given by equivalence classes of cocycles consisting of
collections of functions
subject to the condition that on each triple overlap the equation
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;
subject to the equivalence relation that identifies two such collections of cocycle data $\{g_{i j}\}$ and $\{g'_{i j}\}$ if there exists functions
and
such that
and
We pass to the standard 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
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
and so $\alpha$ is presented by an ∞-anafunctor
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 sets
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
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.
(…)
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
and a tensor product operation
(…)
Equivalence classes of twisted $U(n)$-bundles for fixed $\mathbf{B}U(1)$-twist $\alpha$ form a model for topological $\alpha$-twisted K-theory. See there for details.
The notion and term twisted bundle (with finite rank) apparently first appears in
The equivalent notion of gerbe module apparently appears first in
there explicitly in terms of Cech cocycles relative to an open cover. The generalization to infinite rank and arbitrary covering morphisms was amplified in (CBMMS) below.
Just as vector bundles model cocycles in K-theory, twisted vector bundles model cocycles in twisted K-theory.
For twists $c$ that are torsion class (i.e. have finite order as group elements in the cohomology group $H(X,\mathbf{B}^2 A)$ ) this was realized in
which also, apparently, is the source where gerbe modules as such were first introduced.
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
There the twisted cocycle equation discussed above appears on the bottom of page 7.
Then there is
The observation that twisted vector bundles may be understood as higher-order sections of 2-vector bundles associated with circle 2-bundles/bundle gerbes appears in
A discussion of this with 2-connections taken into account is in section 4.4.3 of
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”
The observation then re-appears independently in