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\begin{document}

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\section*{dinatural transformation}

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

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

[[!include 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{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{dinaturality_versus_extranaturality}{Dinaturality versus extranaturality}\dotfill \pageref*{dinaturality_versus_extranaturality} \linebreak
\noindent\hyperlink{composition}{Composition}\dotfill \pageref*{composition} \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}

\textbf{Dinatural transformations} are a generalization of ordinary [[natural transformations]] and also of [[extranatural transformations]]. The differences can be summarized thus:

\begin{itemize}%
\item In an ordinary \emph{natural transformation} $F\to G$, both $F$ and $G$ involved depend on some variable $x$ with the same variance (covariant or contravariant).
\item In an \emph{extranatural transformation} $F\to G$, either $F$ depends on $x$ both covariantly \emph{and} contravariantly and $G$ does not depend on $x$ at all, or vice versa.
\item In a \textbf{dinatural transformation} $F\to G$, both $F$ and $G$ can depend on $x$ both covariantly and contravariantly.

\end{itemize}
If the dependence of $F$ or $G$ on $x$ in a dinatural transformation is trivial, it reduces to an extranatural one. Similarly, if the contravariant (or, dually, the covariant) dependence of $F$ and $G$ on $x$ are trivial, it reduces to an ordinary natural one.

Arguably, most dinatural transformations which arise in practice are ordinary or extranatural.

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

Let $F, G: C^{op} \times C \to D$ be functors. A \textbf{dinatural transformation} from $F$ to $G$, sometimes written

\begin{displaymath}
\alpha: F \stackrel{\bullet}{\to} G,
\end{displaymath}
consists of a collection of morphisms

\begin{displaymath}
\alpha_{c}: F(c, c) \to G(c, c)
\end{displaymath}
such that for every morphism $f: c \to c'$ in $C$,

\begin{equation}
G(c, f)\alpha_c F(f, c) = G(f, c')\alpha_{c'}F(c', f): F(c', c) \to G(c, c')
\label{hexagon}\end{equation}
If drawn out as a commutative diagram, this becomes a ``hexagon identity'':

\begin{displaymath}
\begin{array}{ccccccc}
 &  & F(c,c) & \overset{\alpha_{c}}{\to} & G(c,c)\\
 & \overset{\mathclap{F(f,c)}}{\nearrow} &  &  &  & \overset{\mathclap{G(c,f)}}{\searrow}\\
F(c',c) &  &  &  &  &  & G(c,c')\\
 & \underset{\mathclap{F(c',f)}}{\searrow} &  &  &  & \underset{\mathclap{G(f,c')}}{\nearrow}\\
 &  & F(c',c') & \underset{\alpha_{c'}}{\to} & G(c',c')
\end{array}\,.
\end{displaymath}
\hypertarget{special_cases}{}\subsubsection*{{Special cases}}\label{special_cases}

If $F$ and $G$ both factor through the projection $C^{op}\times C \to C$, then the notion reduces to an ordinary natural transformation, and similarly if they both factor through $C^{op}$.

If $F$ factors through $C$ while $G$ factors through $C^{op}$, then we obtain a notion of natural transformation from a covariant functor to a contravariant one, and dually.

If $G$ is constant, the hexagon identity reduces to the ``domain'' version of extranaturality, involving a commutative square of the form

\begin{equation}
\alpha_c F(f, c) = \alpha_{c'} F(c', f): F(c', c) \to G
\label{domain}\end{equation}
where $G$ is constant with respect to the argument $c$. Similarly, if $F$ is constant, it yields the ``codomain'' version with a commutative square of the form

\begin{equation}
G(c, f) \alpha_c = G(f, c')\alpha_{c'}: F \to G(c, c')
\label{codomain}\end{equation}
when $F$ is constant with respect to the argument $c$.

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

Of course, all ordinary natural transformations, and also all [[extranatural transformations]], are also dinatural ones. Here we will confine ourselves to examples that do not reduce to either of these.

\begin{itemize}%
\item Perhaps the most well known example is the [[Church numerals]]: for any category $C$, we have a class of dinatural transformations from the hom-functor $hom: C^{op}\times C\to Set$

\begin{displaymath}
\hom(x, x) \stackrel{\alpha_n}{\to} \hom(x, x)
\end{displaymath}
defined by the rule $\alpha_n(f) = f^{(n)}$.


\item Not every endo-dinatural-transformation of a hom-functor is of this form; several other examples are given in \hyperlink{PareRoman}{Pare-Roman}. For instance, if $C=FinSet$ then the operation sending each endomorphism to its [[eventual image]] is dinatural.


\item The following example of a dinatural transformation from a covariant functor to a contravariant one is found in \hyperlink{DubucStreet}{Dubuc-Street} and attributed to [[Mac Lane]]. Let $C$ be the category of [[inner product spaces]] and isometries, and $D$ the category of [[vector spaces]] and linear transformations (all over a fixed [[field]]). Let $F:C\to D$ be the forgetful functor and let $G:C^{op}\to D$ be the functor sending each inner product space to its dual. Define $\alpha_V : F(V) \to G(V)$ to send $v\in V$ to the operation $\langle v,-\rangle$; then $\alpha$ is dinatural.



\end{itemize}
\hypertarget{dinaturality_versus_extranaturality}{}\subsection*{{Dinaturality versus extranaturality}}\label{dinaturality_versus_extranaturality}

Many people who encounter the notion of dinaturality through the general definition (as in equation \eqref{hexagon}) have subsequent difficulty grokking it. It is the opinion of at least one author of this article ([[Todd Trimble]]), and it was certainly the opinion of Max Kelly, that this ``efficient'' definition is not the most useful or intuitive one. Rather, one may be better off grokking the separate squares \eqref{domain} and \eqref{codomain} -- that is, the notion of [[extranaturality]] -- and how they arise in practice.

One could try to argue against that by pointing to dinatural transformations which do not reduce to extranatural ones.

A counterargument, however, is that \emph{any} dinatural transformation between functors $F,G:C^{op}\times C\to D$ can be ``bent'' into domain extranaturality by defining

\begin{displaymath}
C^{op} \times C \stackrel{[F,G]}{\to} Set: (x, y) \mapsto G(x, y)^{F(y, x)}.
\end{displaymath}
Then a dinatural transformation $F\to G$ can be identified with an extranatural transformation from the constant $1$ (the terminal set) to $[F,G]$. Such tricks support the counterargument that the extra generality of the traditional definition is largely spurious, and not particularly helpful in terms of comprehension.

A further argument for the relative importance of extranatural transformations over dinatural ones is that extranatural transformations can be defined for any sort of [[enriched categories]], whereas dinatural ones (including the other special case of natural transformations from covariant functors to contravariant ones) only make sense when the enriching category is [[cartesian monoidal category|cartesian]].

\hypertarget{composition}{}\subsection*{{Composition}}\label{composition}

Dinatural transformations cannot, in general, be composed with each other, although there are certain circumstances when they can be.

\begin{itemize}%
\item One such case is when certain squares are pushouts or pullbacks.


\item Of course, ordinary natural transformations can be composed.


\item In fact, there is a category whose objects consist of both covariant functors $C\to D$ and contravariant ones $C^{op}\to D$, and whose morphisms are dinatural transformations.



\end{itemize}
In general, what we can say is that for two fixed categories $C$ and $D$, the functors $C^{op}\times C^ \to D$ and the dinatural transformations between them form a [[paracategory]].

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

\begin{itemize}%
\item [[homotopy]]


\item [[contravariant functor]], [[2-category with contravariance]]


\item [[transfor]]

\begin{itemize}%
\item [[natural transformation]]

\begin{itemize}%
\item [[extranatural transformation]], \textbf{dinatural transformation}

\end{itemize}

\item [[pseudonatural transformation]]


\item [[lax natural transformation]]


\item [[2-dinatural transformation]]



\end{itemize}


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

\begin{itemize}%
\item [[Eduardo Dubuc]] and [[Ross Street]], \emph{Dinatural transformations}, Reports of the Midwest Category Seminar IV, Lecture Notes in Math., vol. 137, Springer, Berlin (1970), pp. 126--137

\end{itemize}
\begin{itemize}%
\item [[Robert Paré]] and Leopoldo Rom\'a{}n, \emph{Dinatural numbers}, \href{http://www.sciencedirect.com/science/article/pii/S0022404997000364}{JPAA}

\end{itemize}
Here is a blog post inspired by the above discussion that discusses these concepts in the context of the programming language Haskell:

\begin{itemize}%
\item Dan Piponi, \href{http://blog.sigfpe.com/2009/03/dinatural-transformations-and-coends.html}{Dinatural transformations and coends}

\end{itemize}
[[!redirects dinaturality]] [[!redirects dinatural transformations]]



\end{document}