group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Traditionally, in the strict sense of the term, the Chern character is a universal characteristic class of vector bundles or equivalently of their topological K-theory classes, which is a rational combination of all Chern classes.
This is a special case of the following more general construction (Hopkins-Singer 02, section 4.8):
for $E$ a spectrum representing a generalized (Eilenberg-Steenrod) cohomology theory there is a canonical localization map
to the smash product with the Eilenberg-MacLane spectrum over the real numbers. This represents the $E$-Chern character (see also Bunke-Gepner 13, around def. 2.1).
In the case that $E =$ KU this reproduces the traditional Chern character. (In which case this is a map from a complex oriented cohomology theory of chromatic level 1 to chromatic level 0. More generally one can also consider higher chromatic Chern characters that take values not in ordinary cohomology but in some cohomology theory of higher chromatic level. See at higher chromatic Chern character for more on this.)
The Chern character $ch_E$ may be used to define differential cohomology refinements $\hat E$ of the cohomology theory $E$ by choosing a differential form-model for $E \wedge H\mathbb{R}$ (Hopkins-Singer 02, see also at differential function complex). In that case $ch_E$ is the real cohomology class associated to a chern character differential form $CH_E$ via the de Rham theorem. Here $CH_E$ has the interpretation of being the curvature forms of the differential cohomology cocycles thought of as ∞-connections.
This may be turned around (Bunke-Nikolaus-Völkl 13, prop. .3.5): given any refinement $\hat E$ of $E$ in a tangent cohesive (∞,1)-topos $T \mathbf{H}$, then it is induced from homotopy pullback of its de Rham coefficients along a Chern character map
where $\Pi$ is the shape modality and $\theta_E$ the Maurer-Cartan form of $E$. This reproduces the above definition for ordinary differential form models, see at differential cohomology diagram – Hopkins-Singer coefficients.
But more generally, given for instance a K(n)-localization $E \longrightarrow L_{K(n)} E$ then any choice of cohesive refinement of $L_{K(n)} E$ (i.e. lift through the unit of the shape modality $\Pi$) which is in the kernel of $\flat$ yields a generalized differential cohomology theory $\hat E$ whose intrinsic Chern-character $\Pi \theta_{\hat E}$ is the $K(n)$-localization. See at differential cohomology diagram – Chern character and differential fracture.
In words this is summarized succintly as: The Chern character is the shape of the Maurer-Cartan form.
In the context of algebraic K-theory Chern characters appear at Beilinson regulators. There are analogues in algebraic geometry (e.g. a Chern character between Chow groups and algebraic K-theory) and in noncommutative geometry (Chern-Connes character) where the role of usual cohomology is taken by some variant of cyclic cohomology.
The classical theory of the Chern character applies to the spectrum of complex K-theory, $E = KU$. In this case, the Chern character is made up from Chern classes: each characteristic class is by Chern-Weil theory in the image of a certain element in the Weil algebra via taking the class of evaluation at the curvature operator for some choice of a connection. Consider the symmetric functions in $n$ variables $t_1,\ldots, t_n$ and let the Chern classes of a complex vector bundle $\xi$ (representing a complex K-theory class) be $c_1,\ldots, c_n$. Define the formal power series
Then $ch(\chi) = \phi(c_1,\ldots,c_n)$.
Let us describe this a bit differently. The cocycle $H^0(X,KU)$ may be represented by a complex vector bundle, and the image of this cocycle under the Chern-character is the class in even-graded real cohomology that is represented (under the deRham theorem isomorphism of deRham cohomology with real cohomology) by the even graded closed differential form
where
$\nabla$ is any chosen connection on the vector bundle;
$F = F_\nabla \in \Omega^2(X,End(V))$ is the curvature of this connection;
$k_j \in \mathbb{R}$ are normalization constants, $k_j = \frac{1}{j!} \left( \frac{1}{2\pi i}\right)^j$;
the trace of the wedge products produces the curvature characteristic forms.
The Chern character applied to the Whitney sum of two vector bundles is a sum of the Chern characters for the two: $ch(\xi\oplus \eta) = ch(\chi)+ch(\eta)$ and it is multiplicative under the tensor product of vector bundles: $ch(\xi\otimes\eta)=ch(\chi)ch(\eta)$. Therefore we get a ring homomorphism.
Let $E$ be a spectrum The isomorphism
that defines the Chern-character map is induced by a canonical cocycle on the spectrum $E$ that is called the fundamental cocycle.
This is described for instance in section 4.8, page 47 of Hopkins-Singer Quadratic Functions in Geometry, Topology,and M-Theory.
More generally, for $\hat E$ a stable homotopy type in a cohesive (∞,1)-topos, then the underlying bare homotopy type is $E \coloneqq \Pi(\hat E)$ and the corresponding Chern character is
For more on this see at differential cohomology diagram.
Generalizing in another direction, generalized Chern characters are given by passage to derived loop spaces and their cyclic homology or, more generally, topological cyclic homology (Toen-Vezzosi 08, Hoyois-Scherotzke-Sibillia 15).
The behaviour of the Chern-character under fiber integration in generalized cohomology along proper maps is described by the Grothendieck-Riemann-Roch theorem.
for algebraic K-theory:
The universal Chern character for generalized (Eilenberg-Steenrod) cohomology theory is discussed in section 4.8, page 47 of
in the context of differential cohomology via differential function complexes.
The observation putting this into the general context of differential cohomology diagrams (see there) of stable homotopy types in cohesion is due to
based on
A characterization of Chern-character maps for K-theory is in
A discussion of Chern characters in terms of free loop space objects in derived geometry is in
which conjectures a construction that is fully developed in
See also
Marc Hoyois, Chern character and derived algebraic geometry (2009) (pdf)
Marc Hoyois, Sarah Scherotzke, Nicolò Sibilla, Higher traces, noncommutative motives, and the categorified Chern character (arXiv:1511.03589)