\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{span}

\begin{quote}%
This entry is about the notion of spans/correspondences which generalizes that of \emph{[[relations]]}. For spans in [[vector spaces]] or [[modules]], see \emph{[[linear span]]}.


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

\hypertarget{relations}{}\paragraph*{{Relations}}\label{relations}

[[!include relations - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{correspondences}{Correspondences}\dotfill \pageref*{correspondences} \linebreak
\noindent\hyperlink{categories_of_spans}{Categories of spans}\dotfill \pageref*{categories_of_spans} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_1category_of_spans}{The 1-category of spans}\dotfill \pageref*{the_1category_of_spans} \linebreak
\noindent\hyperlink{universal_property_of_the_2category_of_spans}{Universal property of the 2-category of spans}\dotfill \pageref*{universal_property_of_the_2category_of_spans} \linebreak
\noindent\hyperlink{LimitsAndColimits}{Limits and colimits}\dotfill \pageref*{LimitsAndColimits} \linebreak
\noindent\hyperlink{RelationToRelations}{Relation to relations}\dotfill \pageref*{RelationToRelations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{correspondences}{}\subsubsection*{{Correspondences}}\label{correspondences}

In any [[category]] $C$, a \textbf{span}, or \textbf{roof}, or \textbf{correspondence}, from an [[object]] $x$ to an object $y$ is a [[diagram]] of the form

\begin{displaymath}
\itexarray{
     && s
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
     \\
     x
     &&&&
     y
  }
\end{displaymath}
where $s$ is some other object of the category. (The word ``correspondence'' is also sometimes used for a [[profunctor]].)

This [[diagram]] is also called a `span' because it looks like a little bridge; `roof' is similar. The term `correspondence' is prevalent in geometry and related areas; it comes about because a correspondence is a generalisation of a binary [[relation]].

Note that a span with $f = 1$ is just a morphism from $x$ to $y$, while a span with $g = 1$ is a morphism from $y$ to $x$. So, a span can be thought of as a generalization of a morphism in which there is no longer any asymmetry between source and target.

A span in the [[opposite category]] $C^op$ is called a [[co-span]] in $C$.

A span that has a [[cocone]] is called a [[coquadrable span]].

\hypertarget{categories_of_spans}{}\subsubsection*{{Categories of spans}}\label{categories_of_spans}

If the category $C$ has [[pullback|pullbacks]], we can compose spans. Namely, given a span from $x$ to $y$ and a span from $y$ to $z$:

\begin{displaymath}
\itexarray{
     && s &&&& t
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
       &
       & {}^{h}\swarrow
      && \searrow^{i}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
\end{displaymath}
we can take a pullback in the middle:

\begin{displaymath}
\itexarray{
     &&&& s \times_y t 
     \\& 
    &&
      {}^{p_s}\swarrow
      && \searrow^{p_t}
    \\
     && s &&&& t
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
       &
       & {}^{h}\swarrow
      && \searrow^{i}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
\end{displaymath}
and obtain a span from $x$ to $z$:

\begin{displaymath}
\itexarray{
     && s \times_y t
      \\
      & {}^{f p_s}\swarrow
      && \searrow^{i p_t}
     \\
     x
     &&&&
     z
  }
\end{displaymath}
This way of composing spans lets us define a [[2-category]] [[Span]]$(C)$ with:

\begin{itemize}%
\item objects of $C$ as objects
\item spans as morphisms
\item maps between spans as 2-morphisms

\end{itemize}
This is a weak 2-category: it has a nontrivial [[associator]]: composition of spans is not strictly associative, because pullbacks are defined only up to canonical isomorphism. A [[Lack's coherence theorem|naturally defined]] [[strict 2-category]] which is equivalent to $Span(C)$ is the strict 2-category of [[linear polynomial functors]] between [[slice categories]] of $C$.

(Note that we must choose a specific pullback when defining the composite of a pair of morphisms in $Span(C)$, if we want to obtain a [[bicategory]] as traditionally defined; this requires the [[axiom of choice]]. Otherwise we obtain a bicategory with `composites of morphisms defined only up to canonical iso-2-morphism'; such a structure can be modeled by an [[anabicategory]] or an [[opetopic bicategory]].)

By including functions as well, instead of a 2-category we obtain a [[double category]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{the_1category_of_spans}{}\subsubsection*{{The 1-category of spans}}\label{the_1category_of_spans}

Let $C$ be a category with [[pullback]]s and let $Span_1(C) := (Span(C))_{\sim 1}$ be the 1-category of objects of $C$ and isomorphism class of spans between them as morphisms.

Then

\begin{itemize}%
\item $Span_1(C)$ is a [[dagger category]].

\end{itemize}
Next assume that $C$ is a [[cartesian monoidal category]]. Then clearly $Span_1(C)$ naturally becomes a [[monoidal category]] itself, but more: then

\begin{itemize}%
\item $Span_1(C)$ is a [[dagger compact category]].

\end{itemize}
\hypertarget{universal_property_of_the_2category_of_spans}{}\subsubsection*{{Universal property of the 2-category of spans}}\label{universal_property_of_the_2category_of_spans}

(\hyperlink{DawsonParePronk04}{Dawson-Par\'e{}-Pronk 04}) (\ldots{})

\hypertarget{LimitsAndColimits}{}\subsubsection*{{Limits and colimits}}\label{LimitsAndColimits}

Since a category of spans/correspondences $Corr(\mathcal{C})$ is evidently [[equivalence of categories|equivalent]] to its [[opposite category]], it follows that to the extent that [[limits]] exists they are also [[colimits]] and vice versa.

If the underlying category $\mathcal{C}$ is an [[extensive category]], then the [[coproduct]]/[[product]] in $Corr(\mathcal{C})$ is given by the [[disjoint union]] in $\mathcal{C}$. (See also \href{http://mathoverflow.net/questions/32526/what-is-the-product-in-the-2-category-of-spans}{this} MO discussion).

More generally, every [[van Kampen colimit]] in $\mathcal{C}$ is a (co)limit in $Corr(\mathcal{C})$ --- and conversely, this property characterizes van Kampen colimits. (\href{SobocinskiHeindel11}{Sobocinski-Heindel 11}).

\hypertarget{RelationToRelations}{}\subsubsection*{{Relation to relations}}\label{RelationToRelations}

Correspondences may be seen as generalizations of [[relations]]. A relation is a [[correspondence]] which is [[(-1)-truncated]] as a [[morphism]] into the [[cartesian product]]. See at \emph{[[relation]]} and at \emph{[[Rel]]} for more on this.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Spans in [[FinSet]] behave like the [[categorification]] of [[matrices]] with entries in the [[natural number]]s: for $X_1 \leftarrow N \to X_2$ a span of finite sets, the [[cardinality]] of the [[fiber]] $X_{x_1, x_2}$ over any two elements $x_1 \in X_1$ and $x_2 \in X_2$ plays the role of the corresponding matrix entry. Under this identification composition of spans indeed corresponds to matrix multiplication. This implies that the category of spans of finite sets is equivalent to the [[Lawvere theory]] of commutative monoids, that is, to the category of finitely generated free commutative monoids.


\item The [[Burnside category]] is essentially the category of correspondences in [[G-sets]] for $G$ a [[finite group]].


\item A [[cobordism]] $\Sigma$ from $\Sigma_{in}$ to $\Sigma_{out}$ is an example of a cospan $\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out}$ in the category of [[smooth manifold]]s. However, composition of cobordisms is not quite the pushpout-composition of these cospans: to make the composition be a [[smooth manifold]] again some extra technical aspects must be added (``[[collars]]'').


\item In [[prequantum field theory]] (see there for details), spans of [[stacks]] model [[trajectories]] of [[field (physics)|fields]].


\item The [[category of Chow motives]] has as [[morphisms]] [[equivalence classes]] of [[linear combinations]] of spans of [[smooth variety|smooth]] [[projective varieties]].


\item The [[Weinstein symplectic category]] has as morphisms [[Lagrangian correspondences]] between [[symplectic manifolds]].

More generally [[symplectic dual pairs]] are correspondences between [[Poisson manifolds]].


\item Cospans of homomorphisms of [[C\emph{-algebras]] represent morphisms in [[KK-theory]] (by Cuntz' result).}


\item Correspondences of [[flag manifolds]] play a role as [[twistor correspondences]], see at \emph{\href{Schubert+calculus#Correspondences}{Schubert calculus -- Correspondences}} and at \emph{[[horocycle correspondence]]}


\item The [[Fourier-Mukai transform]] is a pull-push operation through correspondences equipped with objects in a cocycle given by an object in a [[derived category of quasi-coherent sheaves]].


\item [[Hecke correspondence]]


\item A [[hypergraph]] is a span from a set of vertices to a set of (hyper)edges.



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

\begin{itemize}%
\item [[(∞,n)-category of spans]].


\item [[multispan]]



\end{itemize}
A category of correspondences is a refinement of a category [[Rel]] of relations. See there for more.

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

The $Span(C)$ construction was introduced by Jean B\'e{}nabou (as an example of a bicategory) in

\begin{itemize}%
\item [[Jean Bénabou]], Introduction to Bicategories, \emph{Lecture Notes in Mathematics 47}, Springer (1967), pp.1-77. (\href{http://dx.doi.org/10.1007/BFb0074299}{doi})

\end{itemize}
B\'e{}nabou cites an article by Yoneda (1954) for introducing the concept of span (in the category of categories).

An exposition discussing the role of spans in [[quantum theory]]:

\begin{itemize}%
\item [[John Baez]], \emph{Spans in quantum Theory} (\href{http://math.ucr.edu/home/baez/span/}{web}, \href{http://math.ucr.edu/home/baez/span/span.pdf}{pdf}, \href{http://golem.ph.utexas.edu/category/2007/10/spans_in_quantum_theory.html}{blog})

\end{itemize}
The relationship between spans and [[bimodules]] is briefly discussed in

\begin{itemize}%
\item [[John Baez]], \emph{Bimodules versus spans} (\href{http://golem.ph.utexas.edu/category/2008/08/bimodules_versus_spans.html}{blog})

\end{itemize}
The relation to [[van Kampen colimits]] is discussed in

\begin{itemize}%
\item [[Pawel Sobocinski]], Tobias Heindel, \emph{Being Van Kampen is a universal property}, (\href{http://arxiv.org/abs/1101.4594}{arXiv:1101.4594})

\end{itemize}
The [[universal property]] of categories of spans is discussed in

\begin{itemize}%
\item R. Dawson, [[Robert Paré]], [[Dorette Pronk]], \emph{Universal properties of Span}, Theory and Appl. of Categories \textbf{13}, 2004, No. 4, 61-85, \href{http://www.tac.mta.ca/tac/volumes/13/4/13-04abs.html}{TAC}, \href{http://www.ams.org/mathscinet-getitem?mr=2116323}{MR2005m:18002}


\item R. Dawson, [[Robert Paré]], [[Dorette Pronk]], \emph{The span construction}, Theory Appl. Categ. \textbf{24} (2010), No. 13, 302--377, \href{http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html}{TAC} \href{http://www.ams.org/mathscinet-getitem?mr=2720187}{MR2720187}



\end{itemize}
The structure of a [[k-tuply monoidal (n,r)-category|monoidal]] [[tricategory]] on spans in [[2-categories]] is discussed 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}
Generally, an [[(∞,n)-category of spans]] is indicated in section 3.2 of

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

\end{itemize}
[[!redirects spans]]

[[!redirects correspondence]] [[!redirects correspondences]]

[[!redirects category of spans]] [[!redirects categories of spans]]

[[!redirects category of correspondences]] [[!redirects categories of correspondences]]

[[!redirects correspondence space]] [[!redirects correspondence spaces]]



\end{document}