group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Where a homology theory is a covariant functor and a cohomology theory is a contravariant functor on some category of spaces, a bivariant cohomology theory is a bifunctor, hence a functor of two variables, contravariant in the first, and covariant in the second.
Examples:
Here are some notes on a proposal for how to usefully formalize bivariant cohomology theory in stable homotopy theory. (This is in generalization of the structure of KK-theory, while the original axioms of (Fulton-MacPherson 81) are a little different^{1}. Aspects of the following appear in (Nuiten 13, Schreiber 14). See also at dependent linear type theory the section on secondary integral transforms).
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Let $E$ be an E-∞ ring, write $GL_1(E)$ for its ∞-group of units. With $\mathbf{H}$ the ambient (∞,1)-topos, write $\mathbf{H}_{/\mathbf{B}GL_1(E)}$ for the slice (∞,1)-topos over the delooping of this abelian ∞-group. This is the (∞,1)-category of spaces equipped with (∞,1)-line bundles over $E$. Consider an (∞,1)-functor
to the (∞,1)-category of (∞,1)-modules over $E$, which form $E$-modules of co-sections of $E$-(∞,1)-module bundles (generalized Thom spectra).
This is well understood for $\mathbf{H} =$ ∞Grpd in which case $\Gamma \simeq \underset{\to}{\lim} \circ i$ is the (∞,1)-functor homotopy colimits in $E Mod$ under the canonical embedding $\mathbf{B} GL_1(E) \simeq E Line \hookrightarrow E Mod$. But one can consider similar constructions $\Gamma$ for more general ambient (∞,1)-toposes $\mathbf{H}$.
For $\chi_i \colon X_i \to \mathbf{B}GL_1(E)$ two objects of $\mathbf{H}_{/\mathbf{B}GL_1(E)}$, the $(\chi_1,\chi_2)$-twisted bivariant $E$-cohomology on $(X_1,X_2)$ is
By the general discussion at twisted cohomology, following (ABG, def. 5.1) we have
for $X_2 = \ast$ the point, the above bivariant cohomology is the $\chi_1$-twisted $E$-cohomology of $X_1$;
for $X_1 = \ast$ the point, the above bivariant cohomology is the $\chi_2$-twisted $E$-homology of $X_2$;
KK-theory is a model for bivariant twisted topological K-theory over differentiable stacks (hence 1-truncated suitably representable objects in $\mathbf{H} =$ Smooth∞Grpd, see Tu-Xu-LG 03). According to (Joachim-Stolz 09, around p. 4) the category $KK$ first of all is naturally an enriched category $\mathbb{KK}$ over the category $\mathcal{S}$ of symmetric spectra and as such comes with a symmetric monoidal enriched functor
This sends an object to its operator K-theory spectrum, hence to the $E$-dual of the $E$-module of co-sections.
Generally, one may want to consider in def. the dualized co-section functor
A correspondence in $\mathbf{H}_{/\mathbf{B}GL_1(E)}$
is a morphism of “twisted $E$-motives” in that it is a correspondence in $\mathbf{H}$ between the spaces $X_1$ and $X_2$ equipped with an $(i_1^\ast \chi_1, i_2^\ast \chi_2)$-twisted bivariant $E$-cohomology cocycle $\xi$ on the correspondence space $Q$. Under the co-sections / Thom spectrum functor this is sent to a correspondence
in $E Mod$. If the wrong-way map of this is orientable in $E$-cohomology then we may form its dual morphism/Umkehr map to obtain the corresponding “index”
in $E Mod$. Identifying correspondences that yield the same “index” this way yields a presentation of bivariant cohomology by motive-like structures. This is how (equivariant) bivariant K-theory is presented, at least over manifolds, see at KK-theory – References – In terms of correspondences.
A general introduction to bivariant cohomology theories is in
A general construction of bivariant theories on smooth manifolds from cohomology theories by geometric cycles, generalizing the construction of K-homology by Baum-Douglas geometric cycles, is in
A similar construction for PL manifolds is in
A study of bivariant theories in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism is in
References related to the discussion in Axiomatization in homotopy theory above include the following
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 vol 81, American Mathematical Society (arXiv:1002.3004)
Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux, Twisted K-theory of differentiable stacks (arXiv:math/0306138)
Michael Joachim, Stephan Stolz, An enrichment of $KK$-theory over the category of symmetric spectra Münster J. of Math. 2 (2009), 143–182 (pdf)
Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, master thesis, August 2013
Urs Schreiber, Quantization via Linear homotopy types (arXiv:1402.7041)
Thanks to Thomas Nikolaus for patiently emphasizing this. ↩