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\begin{document}

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\section*{adjunction}

\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{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{in_terms_of_string_diagrams}{In terms of string diagrams}\dotfill \pageref*{in_terms_of_string_diagrams} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_monads}{Relation to monads}\dotfill \pageref*{relation_to_monads} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{AdjointFunctors}{Adjoint functors}\dotfill \pageref*{AdjointFunctors} \linebreak
\noindent\hyperlink{more_examples}{More examples}\dotfill \pageref*{more_examples} \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}

Two [[morphisms]] $C \stackrel{L}{\to} D$ and $D \stackrel{R}{\to} C$ in a [[2-category]] $\mathcal{C}$ form an \textbf{adjunction} if they are \textbf{dual} to each other (\hyperlink{Lambek82}{Lambek 82}).

There are two archetypical examples:

\begin{itemize}%
\item If $A$ is a [[monoidal category]] and $\mathcal{C} = \mathbf{B}A$ is the one-object 2-category incarnation of $A$ (the [[delooping]] of $A$), so that the morphisms in $\mathcal{C}$ correspond to the objects of $A$, then the notion of adjoint morphisms in $\mathcal{C}$ coincides precisely with the notion of [[dualizable object|dual objects]] in $A$.


\item If $\mathcal{C}$ is the $2$-category [[Cat]], so that the morphisms in $\mathcal{C}$ are [[functors]], then the notion of adjoint morphisms in $\mathcal{C}$ coincides precisely with the notion of \emph{[[adjoint functors]]}.



\end{itemize}
The notion of adjunction may usefully be thought of as a weakened version of the notion of [[equivalence]] in a [[2-category]]: a morphism in an adjunction need not be invertible, but it has in some sense \emph{a left inverse from below} and \emph{a right inverse from above}. If the morphism in an adjunction does happen to be a genuine equivalence, then we speak of the adjunction being an [[adjoint equivalence]].

Essentially everything that makes category theory nontrivial and interesting beyond [[groupoid]] theory can be derived from the concept of adjoint functors. In particular [[universal constructions]] such as [[limit|limits and colimits]] are examples of certain adjunctions. Adjunctions are already interesting (but simpler) in [[2-posets]], such as the $2$-poset [[Pos]] of [[posets]].

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

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

An \textbf{adjunction} in a [[2-category]] is a pair of [[objects]] $C,D$ together with [[morphisms]] $L: C \to D$, $R : D \to C$ and [[2-morphisms]] $\eta: 1_C \to R \circ L$, $\epsilon: L \circ R \to 1_D$ satisfying the equations

\begin{displaymath}
(R \epsilon) \cdot (\eta R) = 1_R
  \qquad \text{i.e.} \qquad
  R \stackrel{\eta \circ 1_R}{\to} R \circ L \circ R \stackrel{1_R \circ \epsilon}{\to} R = R \stackrel{1_R}{\to} R
  \qquad \text{i.e.} \qquad
  \itexarray{\arrayopts{ \padding{0} }
    &&&&1_C& 
    \\
    &&\cellopts{\colspan{5}} \\
    D
    & \stackrel{R}{\to}&
    C
    & \stackrel{L}{\to}&
    D
    & \stackrel{R}{\to}&
    C
    \\
    \cellopts{\colspan{4}} 
    \\
    &&1_D&
  }
  \quad = \quad D \stackrel{R}{\to} C
\end{displaymath}
and

\begin{displaymath}
(\epsilon L) \cdot (L \eta) = 1_L
  \qquad \text{i.e.} \qquad
  L \stackrel{1_L \circ \eta}{\to} L \circ R \circ L \stackrel{\epsilon \circ 1_L}{\to} L = L \stackrel{1_L}{\to} L
  \qquad \text{i.e.} \qquad
  \itexarray{\arrayopts{ \padding{0} }
    &&1_C& 
    \\
    \cellopts{\colspan{5}} \\
    C
    & \stackrel{L}{\to}&
    D
    & \stackrel{R}{\to}&
    C
    & \stackrel{L}{\to}&
    D
    \\
    &&\cellopts{\colspan{4}} 
    \\
    &&&&1_D&
  }
  \quad = \quad C \stackrel{L}{\to} D
\end{displaymath}
variously called the \emph{[[triangle identities]]} or the \emph{zig-zag identities}. We call $L$ the \textbf{[[left adjoint]]} (of $R$) and $R$ the \textbf{[[right adjoint]]} (of $L$). We call $\eta$ the [[unit of the adjunction]] and $\epsilon$ the [[counit of the adjunction]].

When interpreted in the prototypical 2-category [[Cat]], $C$ and $D$ are [[categories]], $L$ and $R$ are [[functors]], and $\eta$ and $\epsilon$ are [[natural transformations]]. In this case (which was of course the first to be defined) there are a number of equivalent definitions of an adjunction, which can be found on the page [[adjoint functor]]. Conversely, the definition in any 2-category can be obtained by [[internalization]] from the definition in $\Cat$.

\hypertarget{in_terms_of_string_diagrams}{}\subsubsection*{{In terms of string diagrams}}\label{in_terms_of_string_diagrams}

The definition of an adjunction may be nicely expressed using [[string diagrams]]. The data $L: C \to D$, $R : D \to C$ and 2-cells $\eta: 1_C \to R \circ L$, $\epsilon: L \circ R \to 1_D$ are depicted as

[[adjunction-L.png:pic]] ~~~~~~~~ [[adjunction-R.png:pic]] ~~~~~~~~ [[adjunction-unit.png:pic]] ~~~~~~~~ [[adjunction-co-unit.png:pic]]

(where 1-cells read from right to left and 2-cells from bottom to top), and the zigzag identities are expressed as ``pulling zigzags straight'' (hence the name):

[[adjunction-up-string.png:pic]] ~~~~~~~~ [[adjunction-down-string.png:pic]]

Often, arrows on strings are used to distinguish $L$ and $R$, and most or all other labels are left implicit; so the zigzag identities, for instance, become:

[[adjunction-up-string-minimal.png:pic]] ~~~~~~~~ [[adjunction-down-string-minimal.png:pic]]

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

\hypertarget{relation_to_monads}{}\subsubsection*{{Relation to monads}}\label{relation_to_monads}

See at \emph{\href{monad#RelationToAdjunctionsAndMonadicity}{monad -- Relation to adjunctions}}.

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

\hypertarget{AdjointFunctors}{}\subsubsection*{{Adjoint functors}}\label{AdjointFunctors}

An adjunction in [[Cat]] is a pair of [[adjoint functors]].

At the cost of some repetition (compare [[adjoint functor]]), we outline how one gets from the [[hom-functor]] formulation of adjunction in [[Cat]] to the elementary definition in terms of [[unit of an adjunction|units and counits]]. This will motivate the definition in the section that follows, which is elementary (definable in the first-order theory of categories) and portable to any 2-category.

We start from a familiar example. Let $U: Grp \to Set$ from groups to sets be the usual forgetful functor. When we say ``$F(X)$ is the free group generated by a set $X$'', we mean there is a function $\eta_X: X \to U(F(X))$ which is universal among functions from $X$ to the underlying set of a group, which means in turn that given a function $f: X \to U(G)$, there is a unique group homomorphism $g: F(X) \to G$ such that

\begin{displaymath}
f = (X \stackrel{\eta_X}{\to} U(F(X)) \stackrel{U(g)}{\to} U(G))
\end{displaymath}
Here $\eta_X$ is a component of what we call the [[unit of an adjunction|unit of the adjunction]] $F \dashv U$, and the equation above is a recipe for the relationship between the map $g: F(X) \to G$ and the map $f: X \to U(G)$ in terms of the unit.

Now we work more generally. Suppose given functors $L: C \to D$, $R: D \to C$ and the structure of an adjunction in the form of a natural isomorphism

\begin{displaymath}
\Psi_{c, d}: \hom_D(L(c), d) \cong \hom_C(c, R(d))
\end{displaymath}
Now the idea is that, in the spirit of the (proof of the) [[Yoneda lemma]], we would like $\Psi$ to be determined by what it does to identity maps. With that in mind, define the unit $\eta : 1_C \to R L$ by the formula $\eta_c = \Psi_{c, L(c)}(1_{L(c)})$. Dually, define the counit $\varepsilon : L R \to 1_D$ by the formula $\varepsilon_d = \Psi^{-1}_{R(d), d}(1_{R(d)})$. Then given $g: L(c) \to d$, the claim is that

\begin{displaymath}
\Psi_{c, d}(g) = (c \stackrel{\eta_c}{\to} R(L(c)) \stackrel{R(g)}{\to} R(d)).
\end{displaymath}
This may be left as an exercise in the yoga of the Yoneda lemma, applied to $\hom_D(L(c), -) \to \hom_C(c, R(-))$. By duality, given $f: c \to R(d)$,

\begin{displaymath}
\Psi^{-1}_{c, d}(f) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d).
\end{displaymath}
(In fact, we spell out the Yoneda-lemma proof of this dual form below.)

Finally, these operations should obviously be mutually inverse, but that can again be entirely encapsulated Yoneda-wise in terms of the effect on identity maps. Thus, if $\eta_c \coloneqq \Psi_{c, L(c)}(1_{L(c)})$, via the recipe just given for $\Psi^{-1}$ we recover

\begin{displaymath}
1_{L(c)} = (L(c) \stackrel{L(\eta_c)}{\to} L R L(c) \stackrel{\varepsilon_{L(c)}}{\to} L(c))
\end{displaymath}
and this is one of the famous [[triangle identities]]: $1_L = (L \stackrel{L \eta}{\to} L R L \stackrel{\varepsilon L}{\to} L)$. Here, juxtaposition of functors and natural transformations denotes neither functor application, nor vertical composition, nor horizontal composition, but [[whiskering]]. By duality, we have the other [[triangle identity]] $1_R = (R \stackrel{\eta R}{\to} R L R \stackrel{R \varepsilon}{\to} R)$. These two triangular equations are enough to guarantee that the recipes for $\Psi$ and $\Psi^{-1}$ indeed yield mutual inverses.

Thus, it is perfectly sufficient to define an adjoint pair of functors in $Cat$ as given by unit and counit transformations $\eta: 1_C \to R L$, $\varepsilon: L R \to 1_D$, satisfying the triangle identities above.

The definition of adjunctions via units and counits is an ``elementary'' definition (so that by implication, the formulation in terms of hom-functors is not elementary) in the sense that while the hom-functor formulation relies some notion of hom-\_set\_, the formulation in terms of units and counits is purely in the first-order language of categories and makes no reference to a model of set theory. (Cf. [[first-order logic]] and [[second-order logic]].) The definition via (co)units therefore gives us a definition of adjunctions even if an assumption of [[locally small|local smallness]] is not made.

\begin{proof}
We claim that $\Psi^{-1}_{c, d}: \hom_C(c, R(d)) \to \hom_D(L(c), d)$ can be defined by the formula

\begin{displaymath}
\Psi^{-1}_{c, d}(f: c \to R(d)) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d)
\end{displaymath}
where $\varepsilon_d \coloneqq \Psi^{-1}_{R(d), d}(1_{R(d)})$. This is by appeal to the proof of the Yoneda lemma applied to the transformation

\begin{displaymath}
\Psi^{-1}_{-, d}: \hom_C(-, R(d)) \to \hom_D(L(-), d)
\end{displaymath}
For the naturality of $\Psi^{-1}$ in the argument $(-)$ would imply that given $f: c \to R(d)$, we have a commutative square

\begin{displaymath}
\itexarray{
\hom_C(R(d), R(d)) & \stackrel{\Psi^{-1}_{R(d), d}}{\to} & \hom_D(L(R(d)), d) \\
\hom_C(f, R(d)) \downarrow & & \downarrow \hom_D(L(f), d) \\
\hom_C(c, R(d)) & \underset{\Psi^{-1}_{c, d}}{\to} & \hom_D(L(c), d)
}
\end{displaymath}
Chasing the element $1_{R(d)}$ down and then across, we get $f: c \to R(d)$ and then $\Psi^{-1}_{c, d}(f)$. Chasing across and then down, we get $\varepsilon_d$ and then $\varepsilon_d \circ L(f)$. This completes the verification of the claim.

\end{proof}
\hypertarget{more_examples}{}\subsubsection*{{More examples}}\label{more_examples}

\begin{itemize}%
\item For $A$ a [[monoidal category]] and $\mathbf{B}A$ its incarnation as a one-object [[2-category]] (its [[delooping]]), an adjunction in $\mathbf{B}A$ is a [[dualizable object|pair of dual objects]].


\item For $\mathcal{C}$ a [[2-groupoid]], an adjunction in $\mathcal{C}$ is an [[adjoint equivalence]].



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

\begin{itemize}%
\item [[left adjoint]], [[right adjoint]]


\item [[duality]]


\item [[adjunct]]


\item [[dual adjunction]], [[Galois connection]]


\item [[monoidal adjunction]]


\item [[enriched adjunction]]


\item [[adjoint monad]]


\item [[heteromorphism]]


\item [[adjoint string]]

\begin{itemize}%
\item [[adjoint triple]], [[adjoint quadruple]]

\end{itemize}

\item [[ambidextrous adjunction]]


\item [[adjoint cylinder]]


\item [[adjoint logic]]



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

For the basics, see any text on [[category theory]] (and see the references at \emph{[[adjoint functor]]}), for instance:

\begin{itemize}%
\item [[Francis Borceux]], Vol 1, Section 3 of \emph{[[Handbook of Categorical Algebra]]}


\item \emph{[[geometry of physics -- categories and toposes]] -- \href{https://ncatlab.org/nlab/show/geometry+of+physics+--+categories+and+toposes#Adjunctions}{Adjunctions}}



\end{itemize}
For some early history and illustrative examples see

\begin{itemize}%
\item [[Joachim Lambek]], \emph{The Influence of Heraclitus on Modern Mathematics}, In \emph{Scientific Philosophy Today: Essays in Honor of Mario Bunge}, edited by Joseph Agassi and Robert S Cohen, 111--21. Boston: D. Reidel Publishing Co. (1982) (\href{https://link.springer.com/chapter/10.1007/978-94-009-8462-2_6}{doi:10.1007/978-94-009-8462-2\_6})

(more along these lines at \emph{[[objective logic]]}).



\end{itemize}
The fundamental role of adjunctions in [[logic]]/[[type theory]] originates with the observaiton that [[substitution]] forms an [[adjoint triple]] with [[existential quantification]] and [[universal quantification]]:

\begin{itemize}%
\item [[William Lawvere]], \emph{Adjointness in Foundations}, (\href{http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html}{tac:16}), Dialectica 23 (1969), 281-296


\item [[William Lawvere]], \emph{Quantifiers and sheaves}, Actes, Congr\`e{}s intern, math., 1970. Tome 1, p. 329 \`a{} 334 (\href{http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf}{pdf})



\end{itemize}
Adjunctions in [[programming languages]]:

\begin{itemize}%
\item Ralf Hinze, \emph{Generic Programming with Adjunctions}, In: J. Gibbons (ed.) \emph{Generic and Indexed Programming} Lecture Notes in Computer Science, vol 7470. Springer 2012 (\href{http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf}{pdf}, \href{http://www.cs.ox.ac.uk/ralf.hinze/SSGIP10/Slides.pdf}{slides} \href{https://doi.org/10.1007/978-3-642-32202-0_2}{doi:10.1007/978-3-642-32202-0\_2})


\item Jeremy Gibbons, Fritz Henglein, Ralf Hinze, Nicolas Wu, \emph{Relational Algebra by Way of Adjunctions}, Proceedings of the ACM on Programming Languages archive Volume 2 Issue ICFP, September 2018 Article No. 86 (\href{https://www.cs.ox.ac.uk/jeremy.gibbons/publications/reladj.pdf}{pdf}, \href{https://dl.acm.org/citation.cfm?doid=3243631.3236781}{doi:10.1145/3236781})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \href{http://en.wikipedia.org/wiki/Adjoint_functors}{Adjoint Functors}


\item Catsters, \emph{Adjunctions} (\href{http://www.youtube.com/watch?v=loOJxIOmShE&feature=channel_page}{YouTube})



\end{itemize}

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[[!redirects adjoint pair]] [[!redirects adjoint pairs]] [[!redirects adjunctions]]



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