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\begin{document}

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\section*{mate}

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

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{naturality}{Naturality}\dotfill \pageref*{naturality} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{multivariable_mates}{Multi-variable mates}\dotfill \pageref*{multivariable_mates} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{prop}
\label{MateBijection}\hypertarget{MateBijection}{}
Given a [[2-category]] $K$, [[adjunction|adjoint pairs]] $(\eta,\epsilon) \colon f \dashv u \colon b \to a$ and $(\eta',\epsilon') \colon f' \dashv u' \colon b' \to a'$ , and [[1-morphisms|1-cells]] $x \colon a \to a'$ and $y \colon b \to b'$, there is a [[bijection]]

\begin{displaymath}
K(a,b')(f' x,y f) \cong K(b,a')(x u,u' y)
\end{displaymath}
given by [[pasting]] with the [[unit of an adjunction|unit]] of one adjunction and the [[counit of an adjunction|counit]] of the other, i.e.

\begin{displaymath}
\itexarray{
    a & \overset{x}{\to} & a' 
    \\
    \mathllap{f} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow     \mathrlap{f'} 
    \\
    b & \underset{y}{\to} & b'
  }
  \;\;\;\;\;
    \mapsto
  \;\;\;\;\;
  \itexarray{
    b & \overset{u}{\to} & a & \overset{x}{\to} 
    & a' &   \overset{1}{\to} &     a' 
   \\
    \mathllap{1} \downarrow & \mathllap{\epsilon} \Downarrow & \mathllap{f} \downarrow &  
    \mathllap{\lambda}      \Downarrow     & \downarrow \mathrlap{f'} & \Downarrow \mathrlap{\eta'} 
    & \downarrow \mathrlap{1} \\
    b & \underset{1}{\to} & b & \underset{y}{\to} 
    & b' & \underset{u'}{\to}    & a'
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    b & \overset{y}{\to} & b' \\ 
    \mathllap{u} \downarrow & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} \\
    a & \underset{x}{\to} & a'
  }
  \;\;\;\;\;
    \mapsto 
  \;\;\;\;\;
  \itexarray{
    a & \overset{f}{\to} & b & \overset{y}{\to} & b' 
    & \overset{1}{\to} &     b' \\
    \mathllap{1} \downarrow & \mathllap{\eta} \Uparrow & \mathllap{u} \downarrow 
    & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} & \Uparrow \mathrlap{\epsilon'} & \downarrow \mathrlap{1} \\
    a & \underset{1}{\to} & a & \underset{x}{\to} & a' & \underset{f'}{\to} &     b'
  }
\end{displaymath}
\end{prop}
\begin{proof}
That this is a bijection follows easily from the [[triangle identities]].

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
The 2-cells $\lambda$ and $\mu$ in prop. \ref{MateBijection} are called \textbf{mates} (or sometimes \textbf{conjugates}) with respect to the adjunctions $f \dashv u$ and $f' \dashv u'$ (and to the 1-cells $x$ and $y$).

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{prop}
\label{}\hypertarget{}{}
Strict [[2-functors]] preserve adjunctions and [[pasting diagrams]], so that if $F \colon K \to J$ is a 2-functor and if $\lambda$ and $\mu$ are mates wrt $f \dashv u$ and $f' \dashv u'$ in $K$, then $F \lambda$ and $F \mu$ are mates wrt $F f \dashv F u$ and $F f' \dashv F u'$ in $J$.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
If $\alpha \colon F \Rightarrow G$ is a [[2-natural transformation]], then the naturality identities $\alpha_b \circ F f = G f \circ \alpha_a$ and $\alpha_a \circ F u = G u \circ \alpha_b$ are mates wrt $F f \dashv F u$ and $G f \dashv G u$.

\end{prop}
\hypertarget{naturality}{}\subsubsection*{{Naturality}}\label{naturality}

There are two [[double categories]] with objects those of $K$, vertical arrows [[adjoint pairs]] in $K$ and horizontal arrows 1-cells of $K$. In one the 2-cells are those of the form $\lambda$ above, while in the other they are those of the form $\mu$. It is easily shown, as in Kelly--Street, that the triangle identities and the definition of composition of adjoints make these two double categories isomorphic. So for any $K$ there is a double category $Adj(K)$, defined up to isomorphism as above but with mate-pairs in $K$ as 2-cells.

What this means is that, for example, the mate of a square coming from a [[pasting diagram]] is given by pasting the mates of the individual 2-cells (whenever this makes sense).

In the double category $Adj(K)$, every vertical arrow has both a [[companion]] (the left adjoint) and a [[conjoint]] (the right adjoint). (In fact, in some sense it is the universal double category cosntructed from $K$ with this property.) Therefore, it is equivalent to a [[2-category equipped with proarrows]]. More explicitly, there is a forgetful functor $L \colon Adj_V(K) \to K$ from the 2-category of objects, adjunctions and mate-pairs in $K$ to $K$ that sends an adjunction $f \dashv u$ to $f$. It is [[locally fully faithful 2-functor|locally fully faithful]], and moreover every $L f$ has a right adjoint in $K$ by definition; this gives the more traditional definition of a proarrow equipment.

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

\begin{example}
\label{}\hypertarget{}{}
Let $F \dashv U \colon D \to C$ be an [[adjunction]] in the [[2-category]] [[Cat]], i.e. a pair of [[adjoint functors]], and $A \colon * \to C$ and $X \colon * \to D$ be objects of $C$ and $D$ considered as [[functors]] out of the [[terminal category]] $*$. Then taking mates with respect to $1 \dashv 1 \colon * \to *$ and $F \dashv U$ yields the familiar [[bijection]]

\begin{displaymath}
D(F A,X) \cong C(A,U X)
\end{displaymath}
and the pasting operations as above yield the usual definition of the isomorphism of adjunction by means of [[unit of an adjunction|unit]] and [[counit of an adjunction|counit]]. Moreover, the naturality of the mate correspondence yields [[natural isomorphism|naturality]] of the bijection.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
If the ambient [[2-category]] $K$ is the [[delooping]] of a [[monoidal category]] $(\mathcal{C}, \otimes)$ in that

\begin{displaymath}
K \simeq \mathbf{B}_\otimes \mathcal{C}
\end{displaymath}
then an [[adjunction]] in $K$ is a pair of [[dual objects]] and the mate-construction is the construction of [[dual morphisms]] between [[dualizable objects]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Suppose given a [[commutative square]] (up to [[isomorphism]]) of [[functor]]s:

\begin{displaymath}
\itexarray{ & \overset{f^*}{\to} & \\
  ^{g^*}\downarrow && \downarrow^{k^*}\\
  & \underset{h^*}{\to} & }
\end{displaymath}
in which $f^*$ and $h^*$ have [[left adjoint]]s $f_!$ and $h_!$, respectively. (The classical example is a [[Wirthmüller context]].) Then the [[natural isomorphism]] that makes the square commute

\begin{displaymath}
k^* f^* \to h^* g^*
\end{displaymath}
has a mate

\begin{displaymath}
h_! k^* \to g^* f_!
\end{displaymath}
defined as the composite

\begin{displaymath}
h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_!   
  \,.
\end{displaymath}
Ones says that the original square satisfies the \textbf{[[Beck-Chevalley condition]]} if this mate is an [[equivalence]].

\end{example}
\hypertarget{multivariable_mates}{}\subsection*{{Multi-variable mates}}\label{multivariable_mates}

There is a version of the mate correspondence that applies to [[two-variable adjunctions]] and $n$-variable adjunctions; see \hyperlink{CGR}{Cheng-Gurski-Riehl}.

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

\begin{itemize}%
\item [[Max Kelly]], [[Ross Street]], \emph{Review of the elements of 2-categories}, in Kelly (ed.), Category Seminar, LNM 420.


\item [[Tom Leinster]], \emph{Higher operads, higher categories}, \href{http://arxiv.org/abs/math/0305049}{math.CT/0305049}, Section 6.1


\item [[Eugenia Cheng]], [[Nick Gurski]], [[Emily Riehl]], ``Multivariable adjunctions and mates'', \href{http://arxiv.org/abs/1208.4520}{arXiv:1208.4520}, (to appear: Journal of K-Theory).



\end{itemize}
 Generalization to [[bicategories]] is discussed in

\begin{itemize}%
\item [[Aaron Lauda]], \S{}3 of \emph{Frobenius algebras and ambidextrous adjunctions}, Theory and Applications of Categories, 16:84--122, 2006. 52


\item [[Richard Garner]], [[Michael Shulman]], around 13.7 of \emph{Enriched categories as a free cocompletion} (\href{https://arxiv.org/abs/1301.3191}{arXiv:1301.3191})



\end{itemize}
[[!redirects mates]] [[!redirects mate correspondence]] [[!redirects mates correspondence]]



\end{document}