Extraordinary natural transformations, or extranatural transformations, are what you get when you “bend the rules” for natural transformations. One intuitive approach to them is through string diagrams: every time you bend a string (that represents a component of a natural transformation) into a U-shape or upside-down U-shape, the U-shape represents a component of an extranatural transformation. Thus, the rules for extranatural transformations mirror rules for ordinary natural transformations, except they are bent into shapes with a covariant part and a contravariant part. (Cf. interactions between particles and their corresponding antiparticles.)
A transformation can also be ordinary-natural in some variables and extraordinary-natural in other variables. Sometimes this sort of transformation is called a generalized natural transformation. The late Max Kelly was fond of saying that really it’s all the same basic concept, however, so why proliferate terminology needlessly? So he would simply say a transformation was “natural” in all its arguments, both the “ordinary” and the “extraordinary” ones.
The calculus of natural and extranatural transformations is a very simple string diagram calculus; perhaps the most basic one. It was first introduced by Eilenberg, Kelly, and Mac Lane in the mid 60’s.
There is also a yet more general notion of dinatural transformation. However, there are few examples of dinatural transformations which are not extranatural. Also, unlike extranatural transformations, dinatural transformations cannot be generalized to all enriched categories and do not admit a natural string diagram calculus.
Consider the function set functor $\hom: Set^{op} \times Set \to Set$ (or more generally, the internal hom functor $\hom: V^{op} \times V \to V$ where $V$ is symmetric monoidal closed). The identity transformation $1: \hom \to \hom$ has components of the form
and this of course is natural in each of the separate arguments $x, y$. String diagrammatically, this naturality would be represented by placing the domain over the codomain and linking the two instances of $x$ with a straight line and the two instances of $y$ with a straight line. Although it’s trivial, let’s at least record what naturality in say $y$ would mean: it means that for any morphism $g: y \to y'$ we have an equation of the form
Now, the adjunction between tensor product and internal hom allow us to “bend” the transformation into another:
in which the two instances of $y$ are linked by a U-shape. This gives a transformation which is natural in $x$ but not of course in $y$; rather, in $y$ we have an equation which is companion to (1):
and we say in this case that $eval_{x, y}$ is extranatural in $y$. Notice how the extranatural variable $y$ in $eval_{x, y}$ appears once covariantly [in the tensor factor] and once contravariantly [in the exponent], but together on the same side of the arrow [here the domain]. (There is a nice string diagram picture for (2) which the reader might like to draw at this point.)
Thus we already see that $eval_{x, y}$ is a “generalized natural” transformation, involving a mixture of naturality (in $x$) and extranaturality (in $y$).
The basic idea should now be clear, but let’s give a few more examples. Starting with the identity transformation
we can again bend it using the tensor-hom adjunction to form an arrow
where again $y$ appears once covariantly and once contravariantly, this time on the codomain side. The extranaturality in $y$ is the condition
for every arrow $g: y \to y'$.
As these examples indicate, instances of ordinary naturality are typically transferred into instances of extranaturality by means of adjunctions. Indeed, basic instances of U-shapes or upside-down U-shapes in string diagrams come about through counits and units of adjunctions,
Let $F\colon A\times B\times B^{op} \to D$ and $G\colon A\times C\times C^{op}\to D$ be functors. A family of morphisms
for $a\in A$, $b\in B$, and $c\in C$ is said to be natural, or more precisely ordinary-natural in $a$ and extranatural in $b$ and $c$, if the following hold.
It is convenient to draw “string diagrams” specifying in what variables a transformation is natural and extranatural. For instance, the above transformation can be notated in this way:
Similarly, a transformation from $f\colon (A,B,A^{op},C^{op}) \to X$ to $g\colon (C^{op},B,D^{op},D) \to X$ which is natural in $B$ and $C^{op}$ and extranatural in $A$ and $D$ would be notated in this way:
We set down a few basic lemmas which describe how extranatural transformations compose. These lemmas become very intuitive once one draws string diagrams to accompany them. (Cf. “yanking moves” in the string diagram calculus of adjunctions.)
Let $F \colon C^{op} \times C \to D$ and $G \colon \underbrace{C^{op} \times C \times \cdots \times C^{op} \times C}_{2n} \to D$ be functors. If $\alpha_{x, y_1, \ldots, y_{n-1}, z} \colon F(x, z) \to G(x, y_1, y_1, \ldots, y_{n-1}, y_{n-1}, z)$ is natural in $x,z$ and extranatural in $y_1, \ldots, y_{n-1}$, and $\beta_{y_1, \ldots, y_n} \colon G(y_1, y_1, \ldots, y_n, y_n) \to H$ (for some object $H$ of $D$) is extranatural in $y_1, \ldots, y_n$, then
is extranatural in $x$.
Let $G \colon \underbrace{C^{op} \times C \times \cdots \times C^{op} \times C}_{2n} \to D$ and $H \colon C^{op} \times C \to D$ be functors. If $\alpha_{y_1, \ldots, y_n} \colon F \to G(y_1, y_1, \ldots, y_n, y_n)$ (for some object $F$ of $D$) is extranatural in $y_1, \ldots, y_n$, and $\beta_{x, y_1, \ldots, y_{n-1}, z} \colon G(x, y_1, y_1, \ldots, y_{n-1}, y_{n-1} ,z) \to H(x,z)$ is natural in $x, z$ and extranatural in $y_1, \ldots, y_{n-1}$, then
is extranatural in $x$.
Let $F, H$ be functors of the form $C \to D$, and let $G: \underbrace{C \times C^{op} \times C \times \ldots \times C^{op} \times C}_{2n+1} \to D$ be a functor. If $\alpha_{x, y_1, \ldots, y_n} \colon F(x) \to G(x, y_1, y_1, \ldots, y_n, y_n)$ is natural in $x$ and extranatural in $y_1, \ldots, y_n$, and if $\beta_{y_1, \ldots, y_n, z} \colon G(y_1, y_1, \ldots, y_n, y_n ,z) \to H(z)$, is natural in $z$ and extranatural in $y_1, \ldots, y_n$, then
is natural in $x$.
In fact, these lemmas essentially capture “all possible” ways in which extranatural transformations can be composed. The general statement, which is obtained by combining these, is that if the graphs representing the two transformations can be composed without creating any cycles, then the transformations can be composed. In some detail, if there are no cycles in the composed graph, then it can be shown that each connected components has one of the forms of the compositions in Lemma 1-3. It then follows from the lemmas (and from the fact that extranaturality can be determined by separation of variables) that each connected component can be replaced by a single edge, and the resulting graph defines the type of the composed transformation.
This can be found in the original paper about extranatural transformations:
With the operation of “loop-free composition,” extranatural transformations with a given target form a paracategory. And as we vary the source and target categories, they assemble into an extraordinary 2-multicategory?.
One abstract way to describe the structure of extranatural transformations is as an extraordinary 2-multicategory?. Another abstract structure, which arguably arises more naturally in practice (but also includes more data than necessary), is an compact closed monoidal bicategory, double category, or proarrow equipment.
More should go here, but for now see compact closed double category.
Samuel Eilenberg and G. M. Kelly, A generalization of the functorial calculus, J. Algebra 3:3, 366–375 (1966). (doi)
Max Kelly, Saunders MacLane, Coherence in closed categories, JPAA 1 (1971), 97-140. (doi)