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\begin{document}

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\section*{(infinity,n)-category of correspondences}

[[!redirects (infinity,n)-category of spans]]

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{direct_definition}{Direct definition}\dotfill \pageref*{direct_definition} \linebreak
\noindent\hyperlink{DefinitionViaCoalgebras}{Definition via coalgebras}\dotfill \pageref*{DefinitionViaCoalgebras} \linebreak
\noindent\hyperlink{PhasedTensorProduct}{With the phased tensor product}\dotfill \pageref*{PhasedTensorProduct} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{FullDualizability}{Full dualizability}\dotfill \pageref*{FullDualizability} \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}

The generalization of the [[bicategory]] \emph{[[Span]]} to [[(∞,n)-categories]]:

An \emph{$(\infty,n)$-category of correspondences} in [[∞-groupoid]] is an [[(∞,n)-category]] whose

\begin{itemize}%
\item [[objects]] are [[∞-groupoids]];


\item [[morphism]]s $X \to Y$ are [[correspondences]]

\begin{displaymath}
\itexarray{
    && Z
    \\
    & \swarrow && \searrow
    \\
    X &&&& Y
  }
\end{displaymath}
in [[∞Grpd]]


\item [[2-morphisms]] are correspondences of correspondences

\begin{displaymath}
\itexarray{
    && Z
    \\
    & \swarrow &\uparrow& \searrow
    \\
    X &&Q&& Y
    \\
    & \nwarrow &\downarrow& \nearrow
    \\
    && Z'
  }
\end{displaymath}
(where the triangular sub-[[diagram]]s are filled with [[2-morphism]]s in [[∞Grpd]] which we do not display here)


\item and so on up to [[k-morphism|n-morphism]]s


\item $k \gt n$-morphisms are equivalences of order $(k-n)$ of higher correspondences.



\end{itemize}
Using the symmetric monoidal structure on [[∞Grpd]] this becomes a [[symmetric monoidal (∞,n)-category]].

More generally, for $C$ some [[symmetric monoidal (∞,n)-category]], there is a symmetric monoidal $(\infty,n)$-category of correspondences over $C$, whose

\begin{itemize}%
\item [[object]]s are [[∞-groupoid]]s $X$ equipped with an [[(∞,n)-functor]] $X \to C$;


\item [[morphism]]s $X \to Y$ are [[correspondences]] in [[(∞,1)Cat]] over $C$

\begin{displaymath}
\itexarray{
    && Z
    \\
    & \swarrow && \searrow
    \\
    X &&\swArrow&& Y
    \\
    & \searrow && \swarrow
    \\
    && C
  }
\end{displaymath}

\item and so on.



\end{itemize}
Even more generally one can allow the [[∞-groupoid]]s $X, Y, \cdots$ to be [[(∞,n)-categories]] themselves.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{direct_definition}{}\subsubsection*{{Direct definition}}\label{direct_definition}

The [[(∞,2)-category]] of correspondences in [[∞Grpd]] is discussed in some detail in (\hyperlink{DyckerhoffKapranov12}{Dyckerhoff-Kapranov 12, section 10}). A sketch of the definition for all $n$ was given in (\hyperlink{Lurie}{Lurie, page 57}). A fully detailed version of this definition is in (\hyperlink{Haugseng14}{Haugseng 14}).

\hypertarget{DefinitionViaCoalgebras}{}\subsubsection*{{Definition via coalgebras}}\label{DefinitionViaCoalgebras}

In (\hyperlink{BenZviNadler13}{BenZvi-Nadler 13, remark 1.17}) it is observed that

\begin{displaymath}
Corr_n(\mathbf{H}) \simeq E_n Alg_b(\mathbf{H}^{op})
\end{displaymath}
is equivalently the [[(∞,n)-category]] of [[En-algebras]] and [[(∞,1)-bimodules]] between them in the [[opposite (∞,1)-category]] of $\mathbf{H}$ (since every object in a cartesian category is uniquely a [[coalgebra]] by its [[diagonal]] map).

(This immediately implies that every object in $Corr_n(\mathbf{H})$ is a self-[[fully dualizable object]].)

To see how this works, consider $X \in \mathbf{H}$ any object regarded as a coalgebra in $\mathbf{H}$ via its [[diagonal map]] (\href{cartesian+monoidal+infinity%2C1-category#CoalgebraObjects}{here}). Then a [[comodule]] $E$ over it is a [[co-action]]

\begin{displaymath}
E \to E \times X
\end{displaymath}
and hence is canonically given by just a map $E \to X$.

Then for

\begin{displaymath}
\itexarray{
    && E_1 &&&& E_2
    \\
    & \swarrow && \searrow && \swarrow && \searrow
    \\
    X && && Y && && Z
  }
\end{displaymath}
two consecutive correspondences, now interpreted as two bi-comodules, their [[tensor product]] of comodules over $Y$ as a coalgebra is the limit over

\begin{displaymath}
E_1 \times E_2
  \stackrel{\to}{\to}
  E_1 \times Y \times E_2
  \stackrel{\to}{\stackrel{\to}{\to}}
  ...
\end{displaymath}
This is indeed the fiber product

\begin{displaymath}
E_1 \underset{Y}{\times} E_2 \stackrel{(p_1, p_2)}{\to} E_1 \times E_2
\end{displaymath}
as it should be for the composition of [[correspondences]].

\hypertarget{PhasedTensorProduct}{}\subsubsection*{{With the phased tensor product}}\label{PhasedTensorProduct}

\begin{defn}
\label{InSliceWithPhasedTensorProduct}\hypertarget{InSliceWithPhasedTensorProduct}{}
For $\mathbf{H}$ an [[(∞,1)-topos]] and $\mathcal{C} \in Cat_{(\infty,n)}(\mathbf{H})$ a [[symmetric monoidal (∞,n)-category|symmetric monoidal]] [[internal (∞,n)-category]] then there is a [[symmetric monoidal (∞,n)-category]]

\begin{displaymath}
Corr_n(\mathbf{H}_{/\mathcal{C}})^\otimes \in SymmMon (\infty,n)Cat
\end{displaymath}
whose [[k-morphisms]] are $k$-fold correspondence in $\mathbf{H}$ over $k$-fold correspondences in $\mathcal{C}$, and whose monoidal structure is given by

\begin{displaymath}
\left[
    \itexarray{
      X_1
      \\
      \downarrow^{\mathrlap{\mathbf{L}_1}}
      \\
      \mathcal{C}_0
    }
  \right]
  \otimes
  \left[
    \itexarray{
      X_2
      \\
      \downarrow^{\mathrlap{\mathbf{L}_2}}
      \\
      \mathcal{C}_0
    }
  \right]
  \coloneqq
  \left[
    \itexarray{
      X_1 \times X_2
      \\
      \downarrow^{\mathrlap{(\mathbf{L}_1, \mathbf{L}_2)}}
      \\
      \mathcal{C}_0 \times \mathcal{C}_0
      \\
      \downarrow^{\mathrlap{\otimes_{\mathcal{C}}}}
      \\
      \mathcal{C}_0
    }
  \right]  
  \,.
\end{displaymath}
\end{defn}
This is (\hyperlink{Haugseng14}{Haugseng 14, def. 4.6, corollary 7.5})

\begin{remark}
\label{}\hypertarget{}{}
If $\mathcal{C}_0$ is (or is regarded as) a [[moduli stack]] for some kind of bundles forming a [[linear homotopy type theory]] over $\mathbf{H}$, then the phased tensor product is what is also called the \emph{[[external tensor product]]}.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Examples of phased tensor products include

\begin{itemize}%
\item the \href{species#HoTTCauchyProduct}{Cauchy product of species};


\item some [[external tensor products]] in [[indexed monoidal categories]];



\end{itemize}
\end{example}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{FullDualizability}{}\subsubsection*{{Full dualizability}}\label{FullDualizability}

\begin{prop}
\label{CorrnGrpdHasDuals}\hypertarget{CorrnGrpdHasDuals}{}
$Corr_n(\infty Grpd)$ is a [[symmetric monoidal (∞,n)-category|symmetric monoidal]] [[(∞,n)-category with duals]].

More generally, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-category with duals, then so is $Corr_n(\infty Grpd,\mathcal{C})^\otimes$ equipped with the phased tensor product of prop. \ref{InSliceWithPhasedTensorProduct}.

In particular every object in these is a [[fully dualizable object]].

\end{prop}
This appears as (\hyperlink{Lurie}{Lurie, remark 3.2.3}). A proof is written down in (\hyperlink{Haugseng14}{Haugseng 14, corollary 6.6}).

\begin{conjecture}
\label{}\hypertarget{}{}
The canonical $O(n)$-[[∞-action]] on $Corr_n(\infty Grpd)$ induced via prop. \ref{CorrnGrpdHasDuals} by the [[cobordism hypothesis]] (see there at \emph{\href{cobordism+hypothesis#TheCanonicalOnAction}{the canonical O(n)-action}}) is trivial.

\end{conjecture}
This statement appears in (\hyperlink{Lurie}{Lurie, below remark 3.2.3}) without formal proof. For more see (\hyperlink{Haugseng14}{Haugseng 14, remark 9.7}).

More generally:

\begin{prop}
\label{}\hypertarget{}{}
For $\mathbf{H}$ an [[(∞,1)-topos]], then $Corr_n(\mathbf{H})$ is an [[(∞,n)-category with duals]].

And generally, for $\mathcal{C} \in SymmMon (\infty,n)Cat(\mathbf{H})$ a [[symmetric monoidal (∞,n)-category]] [[internal (∞,n)-category|internal]] to $\mathbf{C}$, then $Corr_n(\mathbf{H}_{/\mathbf{C}})$ equipped with the phased tensor product of prop. \ref{InSliceWithPhasedTensorProduct} is an [[(∞,n)-category with duals]]

\end{prop}
(\hyperlink{Haugseng14}{Haugseng 14, cor. 7.8})

Let $Bord_n$ be the [[(∞,n)-category of cobordisms]].

\begin{prop}
\label{HomsFromBordIntoSpan}\hypertarget{HomsFromBordIntoSpan}{}
The following data are equivalent

\begin{enumerate}%
\item Symmetric monoidal $(\infty,n)$-functors

\begin{displaymath}
Bord_n \to Corr_n(\infty Grpd)
\end{displaymath}

\item Pairs $(X,V)$, where $X$ is a [[topological space]] and $V \to X$ a [[vector bundle]] of [[rank]] $n$.



\end{enumerate}
\end{prop}
This appears as (\hyperlink{Lurie}{Lurie, claim 3.2.4}).

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

\begin{itemize}%
\item [[symplectic category]]


\item [[span trace]]


\item [[relations]], [[bicategory of relations]]


\item [[sheaf with transfer]], [[Mackey functor]]



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

For references on 1- and 2-categories of spans see at \emph{[[correspondences]]}.

An explicit definition of the [[(∞,2)-category]] of spans in [[∞Grpd]] is in section 10 of

\begin{itemize}%
\item Tobias Dyckerhoff, [[Mikhail Kapranov]], \emph{Higher Segal spaces I}, (\href{http://arxiv.org/abs/1212.3563}{arxiv:1212.3563})

\end{itemize}
An inductive definition of the [[symmetric monoidal (∞,n)-category]] $Span_n(\infty Grpd)/C$ of spans of [[∞-groupoid]] over a symmetric monoidal $(\infty,n)$-category $C$ is sketched in section 3.2 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}

\end{itemize}
there denoted $Fam_n(C)$. Notice the heuristic discussion on page 59.

More detailed discussion is given in

\begin{itemize}%
\item [[Rune Haugseng]], \emph{Iterated spans and ``classical'' topological field theories} (\href{http://arxiv.org/abs/1409.0837}{arXiv:1409.0837})


\item [[Yonatan Harpaz]], \emph{Ambidexterity and the universality of finite spans} (\href{https://arxiv.org/abs/1703.09764}{arXiv:1703.09764})



\end{itemize}
Both articles comment on the relation to [[schreiber:Local prequantum field theory]].

The generalization to an $(\infty,n)$-category $Span_n((\infty,1)Cat^Adj)$ of spans between [[cobordism hypothesis|(∞,n)-categories with duals]] is discussed on p. 107 and 108.

The extension to the case when the ambient $\infty$-topos is varied is in

\begin{itemize}%
\item [[David Li-Bland]], \emph{The stack of higher internal categories and stacks of iterated spans}, (\href{http://arxiv.org/abs/1506.08870}{arXiv:1506.08870})

\end{itemize}
The application of $Span_n(\infty Grpd/C)$ to the construction of [[FQFT]]s is further discussed in section 3 of

\begin{itemize}%
\item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]}

\end{itemize}
Discussion of $Span_n(\mathbf{H}) \simeq Alg_{E_n}(\mathbf{H}^{op})$ is around remark 1.17 of

\begin{itemize}%
\item [[David Ben-Zvi]], [[David Nadler]], \emph{Nonlinear traces} (\href{http://arxiv.org/abs/1305.7175}{arXiv:1305.7175})

\end{itemize}
A discussion of a version $Span(B)$for $B$ a [[2-category]] with $Span(B)$ regarded as a [[tricategory]] and then as a 1-object [[tetracategory]] is in

\begin{itemize}%
\item [[Alex Hoffnung]], \emph{Spans in 2-Categories: A monoidal tricategory} (\href{http://arxiv.org/abs/1112.0560}{arXiv:1112.0560})

\end{itemize}
A discussion that $Span_2(-)$ in a [[2-category]] with weak [[finite limits]] is a [[compact closed 2-category]]:

\begin{itemize}%
\item [[Mike Stay]], \emph{Compact Closed Bicategories} (\href{http://arxiv.org/abs/1301.1053}{arXiv:1301.1053})

\end{itemize}
See also

\begin{itemize}%
\item [[David Ayala]], [[John Francis]], \emph{Fibrations of $\infty$-Categories} (\href{https://arxiv.org/abs/1702.02681}{arXiv:1702.02681})

\end{itemize}
Coisotropic orrespondences for derived Poisson stacks:

\begin{itemize}%
\item [[Rune Haugseng]], Valerio Melani, [[Pavel Safronov]], \emph{Shifted Coisotropic Correspondences} (\href{https://arxiv.org/abs/1904.11312}{arXiv:1904.11312})

\end{itemize}
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