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\begin{document}

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\section*{Weinstein symplectic category}

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

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{refinements}{Refinements}\dotfill \pageref*{refinements} \linebreak
\noindent\hyperlink{PrequantumCorrespondences}{Prequantum correspondences}\dotfill \pageref*{PrequantumCorrespondences} \linebreak
\noindent\hyperlink{motivic_stabilization}{Motivic stabilization}\dotfill \pageref*{motivic_stabilization} \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}

When [[symplectic geometry]] is used to model [[mechanics]] in [[physics]], then a [[symplectic manifold]] $(X,\omega)$ encodes the [[phase space]] of a [[mechanical system]] and a [[symplectomorphism]]

\begin{displaymath}
\phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)
\end{displaymath}
encodes a process undergone by this system, for instance the time evolution induced by a [[Hamiltonian vector field]].

Now the [[graph]] of a [[symplectomorphism]] $\phi$ is a [[Lagrangian submanifold]] of the [[Cartesian product]] space $X_1 \times X_2$ regarded as a [[symplectic manifold]] with symplectic form $p_1^\ast \omega_1 - p_2^\ast \omega_2$. In other words, a symplectomorphism $\phi$ as above constitutes a [[Lagrangian correspondence]] between $(X_1,\omega_1)$ and $(X_2, \omega_2)$. See for instance (\hyperlink{CattaneoMnevReshetikhin12}{Cattaneo-Mnev-Reshetikhin 12}) for a review.

This suggests that instead of the [[category]] whose [[objects]] are [[symplectic manifolds]] and whose [[morphisms]] are [[symplectomorphisms]], one might consider a kind of [[category of correspondences]] whose objects are symplectic manifolds, and whose morphisms include [[Lagrangian correspondences]], so that [[composition]] is given by forming the [[fiber product]] along adjacent legs of [[correspondences]].

[[Alan Weinstein]] called this would-be category the \emph{symplectic category} and suggested that it is the natural [[domain]] for [[geometric quantization]].

However, take at face value, symplectic manifolds with [[Lagrangian correspondences]] between them do not quite form a [[category]], since the usual [[composition]] is only well-defined when the intersection of $L_1 \times L_2 \cap X_1 \times \Delta(X_2) \times X_3$ is \href{http://ncatlab.org/nlab/show/transversal+maps}{transverse}.

Proposals for how to rectify this are in (\hyperlink{WehrheimWoodward}{Wehrheim-Woodward}) and in (\hyperlink{Kitchloo}{Kitchloo}) (by turning this into an [[(infinity,1)-category]]).

\hypertarget{refinements}{}\subsection*{{Refinements}}\label{refinements}

\hypertarget{PrequantumCorrespondences}{}\subsubsection*{{Prequantum correspondences}}\label{PrequantumCorrespondences}

A refinement of the symplectic category to [[prequantum geometry]] is the following (see \hyperlink{SyntheticQFT}{S 13}).

Write $\mathbf{B}U(1)_{conn}$ for the [[moduli stack]] of smooth [[circle group]]-[[principal connections]]. Write [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]], and $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$ for the corresponding [[slice (∞,1)-topos]]. Finally write

\begin{displaymath}
Corr_1(Smooth\infty Grpd,\mathbf{B}U(1)_{conn})
\end{displaymath}
for the [[(∞,1)-category of correspondences]] in $Smooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}$.

An object in here is a [[prequantum geometry]] $(X,\nabla)$ given by a map

\begin{displaymath}
\itexarray{
    X
    \\
    \downarrow^{\mathrlap{\nabla}}

    \\
    \mathbf{B}U(1)_{conn}
  }
  \,.
\end{displaymath}
Under the [[curvature]] map $F_{(-)} \colon \mathbf{B}U(1)_{conn} \to \Omega^2_{cl}$ this maps to a [[presymplectic structure]]

\begin{displaymath}
(X,\omega) = (X, F_{\nabla})
  \,.
\end{displaymath}
If here $\omega$ is non-degenerate, this is a symplectic structure as in Weinstein's symplectic category.

Moreover, a [[morphism]] $(X_1,\nabla_1) \to (X_2,\nabla_2)$ is a [[diagram]] of the form

\begin{displaymath}
\itexarray{
    && Z
    \\
    & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
    \\
    X_1
    && \swArrow_{\eta} &&
    X_2
    \\
    & {}_{\mathllap{\nabla_1}}\searrow && \swarrow_{\mathrlap{\nabla_2}}
    \\
    && \mathbf{B}U(1)_{conn}
  }
  \,,
\end{displaymath}
hence a [[correspondence]] space (smooth $\infty$-groupoid) $Z$ over $X$ and $Y$ together with an [[equivalence in an (∞,1)-category]]

\begin{displaymath}
\eta \colon i_2^\ast \nabla_2 \stackrel{\simeq}{\to} i_1^\ast \nabla_1
  \,.
\end{displaymath}
On the underlying [[curvatures]] this implies that

\begin{displaymath}
i_2^\ast \omega_2 = i_1^\ast \omega_1
  \,.
\end{displaymath}
Hence if $Z \to X \times Y$ is a maximal inclusion with this property, the above diagram is a prequantization of a morphism in the Weinstein symplectic category.

\hypertarget{motivic_stabilization}{}\subsubsection*{{Motivic stabilization}}\label{motivic_stabilization}

[[Nitu Kitchloo]] defines the [[stable (infinity,1)-category|stable]] symplectic category $\mathbb{S}$, which has as [[objects]] [[symplectic manifolds]], and [[morphisms]] are certain [[Thom spectra]] associated to [[Lagrangian correspondences]] $\overline{M} \times N$, where $\overline{M}$ denotes the conjugate with symplectic form $-\omega$. One can view this as a category of symplectic [[motives]].

Considering an oriented version of the category $\mathbb{S}$, there is a canonical [[fiber functor]] $F : M \mapsto \mathbb{S}(pt, M)$, and one may consider the [[motivic Galois group]] $G$ of monoidal [[automorphisms]] of $F$. (\hyperlink{Kitchloo}{Kitchloo 12, question 8.6, p. 19}).

It turns out to have a natural subgroup which is isomorphic to the quotient of the [[Grothendieck-Teichmüller group]].

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

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


\item [[Lagrangian correspondences and category-valued TFT]]



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

The way that [[Lagrangian correspondences]] encode [[symplectomorphisms]] in [[symplectic geometry]] and hence evolution in [[mechanics]] is reviewed (and put in the broader context of [[BV-BRST formalism]]) in

\begin{itemize}%
\item [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], \emph{Classical and quantum Lagrangian field theories with boundary} (\href{http://arxiv.org/abs/1207.0239}{arXiv:1207.0239})

\end{itemize}
In his work on Fourier integral operators,

\begin{itemize}%
\item [[Lars Hörmander]], \emph{Fourier Integral Operators I.}, Acta Math. 127 (1971) 79--183. 14

\end{itemize}
following

\begin{itemize}%
\item [[Victor Maslov]], \emph{Theory of Perturbations and Asymptotic Methods (in Russian)}, Moskov. Gos. Univ., Moscow, (1965).

\end{itemize}
observed that, under a transversality assumption, the set-theoretic composition of two [[Lagrangian submanifolds]] is again a Lagrangian submanifold, and that this composition is a ``[[classical limit]]'' of the composition of certain [[linear operators]].

Shortly thereafter,

\begin{itemize}%
\item J. Sniatycki, W.M. Tulczyjew, \emph{Generating forms of Lagrangian submanifolds}, Indiana Univ. Math. J. 22(1972), 267--275.

\end{itemize}
defined symplectic relations as [[isotropic submanifolds]] of products and showed that this class of relations was closed under ``clean'' composition. Following in part some (unpublished) ideas of [[Alan Weinstein]],

\begin{itemize}%
\item [[Victor Guillemin]] [[Shlomo Sternberg]], \emph{Some problems in integral geometry and some related problems in microlocal analysis}, Amer. J. Math. 101 (1979), 915--955 (\href{http://www.jstor.org/stable/2373923}{JSTOR})

\end{itemize}
observed that the linear canonical relations (i.e., lagrangian subspaces of products of [[symplectic vector spaces]]) could be considered as the morphisms of a category, and they constructed a partial [[quantization]] of this category (in which lagrangian subspaces are enhanced by halfdensities.) The quantization of the linear symplectic category was part of a larger project of quantizing canonical relations (enhanced with extra structure, such as half-densities) in a functorial way, and this program was set out more formally

\begin{itemize}%
\item [[Alan Weinstein]], \emph{Symplectic manifolds and their lagrangian submanifolds}, Advances in Math. 6 (1971), 329--346.


\item [[Alan Weinstein]], \emph{Symplectic geometry}, Bulletin Amer. Math. Soc. (new series) 5 (1981), 1--13.



\end{itemize}
Lecture notes reviewing these developments include

\begin{itemize}%
\item [[Alan Weinstein]], \emph{Symplectic Categories}, proceedings of Geometry Summer School, Lisbon, July 2009 (\href{http://arxiv.org/abs/0911.4133}{arXiv:0911.4133})

\end{itemize}
from the introduction of which parts of the commented list of references above is taken. Further review includes

\begin{itemize}%
\item Santiago Canez, \emph{Double Groupoids, Orbifolds, and the Symplectic Category} (\href{http://arxiv.org/abs/1105.2592}{arXiv:1105.2592})

\end{itemize}
Further refinements in [[higher category theory]] are discussed in

\begin{itemize}%
\item [[Katrin Wehrheim]] and [[Chris Woodward]], \emph{Functoriality for Lagrangian correspondences in Floer theory}, \href{http://arxiv.org/abs/0708.2851}{arXiv:0708.2851}

\end{itemize}
\begin{itemize}%
\item [[Nitu Kitchloo]], \emph{The Stable Symplectic Category and Geometric Quantization} (\href{http://arxiv.org/abs/1204.5720v1}{arXiv:1204.5720})

\end{itemize}
\begin{itemize}%
\item [[Nitu Kitchloo]], [[Jack Morava]], \emph{The Grothendieck--Teichm\"u{}ller group and the stable symplectic category}, 2012. (\href{http://arxiv.org/abs/1212.6905}{arxiv:1212.6905})

\end{itemize}
A closed [[symmetric monoidal category]] version of the symplectic category and the observation that this hence is a [[categorical semantics]] for [[quantum logic]] qua [[linear logic]] is in

\begin{itemize}%
\item [[Sergey Slavnov]], \emph{From proof-nets to bordisms: the geometric meaning of multiplicative connectives}, Mathematical Structures in Computer Science / Volume 15 / Issue 06 / December 2005, pp 1151 - 1178

\end{itemize}
Remarks about refinements to correspondences of smooth $\infty$-groupoids in the slice over prequantum moduli is in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Synthetic Quantum Field Theory]]}, Talk at \href{http://cms.math.ca/Events/summer13/}{CMS Summer Meeting 2013}

\end{itemize}
[[!redirects stable symplectic category]]

[[!redirects symplectic category]]

[[!redirects motivic symplectic category]]



\end{document}