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\begin{document}

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\section*{bivariant cohomology theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{AxiomatizazionInHomotopyTheory}{Axiomatization in homotopy theory}\dotfill \pageref*{AxiomatizazionInHomotopyTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Where a [[homology theory]] is a [[covariant functor]] and a [[cohomology theory]] is a [[contravariant functor]] on some [[category]] of [[spaces]], a \emph{bivariant cohomology theory} is a [[bifunctor]], hence a functor of two variables, contravariant in the first, and covariant in the second.

Examples:

\begin{itemize}%
\item In [[noncommutative topology]]: [[KK-theory]].


\item In [[noncommutative algebraic geometry]]: [[bivariant algebraic K-theory]], [[noncommutative motives]].



\end{itemize}
\hypertarget{AxiomatizazionInHomotopyTheory}{}\subsection*{{Axiomatization in homotopy theory}}\label{AxiomatizazionInHomotopyTheory}

Here are some notes on a proposal for how to usefully formalize bivariant cohomology theory in [[stable homotopy theory|stable]] [[homotopy theory]]. (This is in generalization of the structure of [[KK-theory]], while the original axioms of (\hyperlink{FultonMacPherson81}{Fulton-MacPherson 81}) are a little different\footnote{Thanks to [[Thomas Nikolaus]] for patiently emphasizing this.} . Aspects of the following appear in (\hyperlink{Nuiten13}{Nuiten 13}, \hyperlink{Schreiber14}{Schreiber 14}). See also at \emph{[[dependent linear type theory]]} the section on \emph{\href{dependent+linear+type+theory#SecondaryIntegralTransforms}{secondary integral transforms}}).

$\,$

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)-module bundle|(∞,1)-line bundles]] over $E$. Consider an [[(∞,1)-functor]]

\begin{displaymath}
\Gamma^\ast \;\colon \; \mathbf{H}_{/\mathbf{B}GL_1(E)} \to E Mod
\end{displaymath}
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]] $\mathbf{H}$.

\begin{defn}
\label{BivariantCohomologyByHomsOfCoSections}\hypertarget{BivariantCohomologyByHomsOfCoSections}{}
For $\chi_i \colon X_i \to \mathbf{B}GL_1(E)$ two [[objects]] of $\mathbf{H}_{/\mathbf{B}GL_1(E)}$, the \emph{$(\chi_1,\chi_2)$-[[twisted cohomology|twisted]] bivariant $E$-[[cohomology theory|cohomology]]} on $(X_1,X_2)$ is

\begin{displaymath}
E^{\bullet + \chi_2 - \chi_1}(X_1,X_2)
  \;\coloneqq\;
  Hom_{E Mod}\left(\Gamma^\ast_{X_1}\left(\chi_1\right), \Gamma^\ast_{X_2}\left(\chi_2\right)\right)
  \in 
  E Mod
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
By the general discussion at [[twisted cohomology]], following (\hyperlink{ABG}{ABG, def. 5.1}) we have

\begin{itemize}%
\item for $X_2 = \ast$ the point, the above bivariant cohomology is the $\chi_1$-twisted $E$-cohomology of $X_1$;

\begin{displaymath}
E^{\bullet + \chi_1}(X_1, \ast) \simeq E^{\bullet + \chi_1}(X_1)
  \,.
\end{displaymath}

\item for $X_1 = \ast$ the point, the above bivariant cohomology is the $\chi_2$-twisted $E$-[[generalized homology|homology]] of $X_2$;

\begin{displaymath}
E^{\bullet + \chi_2}(\ast, X_2) \simeq E_{\bullet + \chi_2}(X_2)
  \,.
\end{displaymath}


\end{itemize}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
[[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 \hyperlink{TuXuLG03}{Tu-Xu-LG 03}). According to (\hyperlink{JoachimStolz09}{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 functor|monoidal]] [[enriched functor]]

\begin{displaymath}
\mathbb{KK} \to KU Mod
  \,.
\end{displaymath}
This sends an object to its [[operator K-theory]] spectrum, hence to the $E$-[[dual object|dual]] of the $E$-module of co-sections.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
Generally, one may want to consider in def. \ref{BivariantCohomologyByHomsOfCoSections} the dualized co-section functor

\begin{displaymath}
\Gamma = [\Gamma^\ast(-), E] \;\colon\; \left(\mathbf{H}_{/\mathbf{B}GL_1(E)}\right)^{op} \to E Mod
  \,.
\end{displaymath}
\end{remark}
\begin{example}
\label{}\hypertarget{}{}
A [[correspondence]] in $\mathbf{H}_{/\mathbf{B}GL_1(E)}$

\begin{displaymath}
\itexarray{
    && Q
    \\
    & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
    \\
    X_1 && \swArrow_{\xi} && X_2
    \\
    & {}_{\mathllap{\chi_1}}\searrow && \swarrow_{\mathrlap{\chi_2}}
    \\
    && \mathbf{B}GL_1(E)
  }
\end{displaymath}
is a morphism of ``twisted $E$-[[pure motive|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]]

\begin{displaymath}
\Gamma_{X_1}(\chi_1)
  \stackrel{\xi}{\rightarrow}
  \Gamma_Q(i_2^\ast \chi_2)
  \stackrel{i_2^\ast}{\leftarrow}
  \Gamma_{X_2}(\chi_2)
\end{displaymath}
in $E Mod$. If the wrong-way map of this is [[orientation in generalized cohomology|orientable]] in $E$-cohomology then we may form its [[dual morphism]]/[[Umkehr map]] to obtain the corresponding ``[[index]]''

\begin{displaymath}
\Gamma_{X_1}(\chi_1)
   \stackrel{(i_2)_! \xi}{\to}
  \Gamma_{X_2}(\chi_2)
\end{displaymath}
in $E Mod$. Identifying correspondences that yield the same ``[[index]]'' this way yields a presentation of bivariant cohomology by [[pure motive|motive]]-like structures. This is how (equivariant) [[bivariant K-theory]] is presented, at least over manifolds, see at \emph{\href{KK-theory#ReferencesInTermsOfCorrespondences}{KK-theory -- References -- In terms of correspondences}}.

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[noncommutative motive]]


\item [[motivic quantization]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A general introduction to bivariant cohomology theories is in

\begin{itemize}%
\item [[William Fulton]], [[Robert MacPherson]], \emph{Categorical framework for the study of singular spaces}, Memoirs of the AMS, 243, 1981

\end{itemize}
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

\begin{itemize}%
\item Martin Jakob, \emph{Bivariant theories for smooth manifolds}, Applied Categorical Structures 10 no. 3 (2002)

\end{itemize}
A similar construction for PL manifolds is in

\begin{itemize}%
\item S. Buoncristiano, C. P. Rourke and B. J. Sanderson, \emph{A geometric approach to homology theory}, Cambridge Univ. Press, Cambridge, Mass. (1976)

\end{itemize}
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

\begin{itemize}%
\item F. Déglise, \emph{Bivariant theories in motivic stable homotopy}, (\href{https://arxiv.org/abs/1705.01528}{arXiv:1705.01528})

\end{itemize}
References related to the discussion in \emph{\hyperlink{AxiomatizazionInHomotopyTheory}{Axiomatization in homotopy theory}} above include the following

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Twists of K-theory and TMF}, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], \emph{Superstrings, Geometry, Topology, and $C^*$-algebras}, Proceedings of Symposia in Pure Mathematics \href{http://www.ams.org/bookstore-getitem/item=PSPUM-81}{vol 81}, American Mathematical Society (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004})


\item [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], \emph{Twisted K-theory of differentiable stacks} (\href{http://arxiv.org/abs/math/0306138}{arXiv:math/0306138})


\item [[Michael Joachim]], [[Stephan Stolz]], \emph{An enrichment of $KK$-theory over the category of symmetric spectra} M\"u{}nster J. of Math. 2 (2009), 143--182 (\href{http://www3.nd.edu/~stolz/KKenrich.pdf}{pdf})


\item [[Joost Nuiten]], \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]}, master thesis, August 2013


\item [[Urs Schreiber]], \emph{[[schreiber:Quantization via Linear homotopy types]]} (\href{http://arxiv.org/abs/1402.7041}{arXiv:1402.7041})



\end{itemize}
[[!redirects bivariant cohomology theories]]

[[!redirects bivariant homology theory]] [[!redirects bivariant homology theories]]

[[!redirects bivariant cohomology]]

[[!redirects twisted bivariant cohomology]] [[!redirects bivariant twisted cohomology]]



\end{document}