nLab strong dinatural transformation

Idea

The notion of strong dinatural transformation is a notion of natural transformation between pairs of functors $C^{op}\times C\to D$ that is stronger than that of dinatural transformations.

Unlike dinatural transformations, strong dinatural transformations can always be composed. They have close connections to parametricity in computer science.

Definition

Let $F,G:C^{op}\times C\to D$ be functors. A strong dinatural transformation $\alpha :F\to G$ consists of, for each $c\in C$, a component $\alpha_c : F(c,c) \to G(c,c)$, such that for any morphism $f:c\to c'$ in $C$ and any span $F(c',c') \leftarrow W \to F(c,c)$ in $D$, if the square on the left commutes, then the outer hexagon also commutes:

$$\begin{array}{ccccccc} & & F(c,c) & \overset{\alpha_{c}}{\to} & G(c,c)\\ & \nearrow & & \overset{\mathclap{F(c,f)}}{\searrow} & & \overset{\mathclap{G(c,f)}}{\searrow}\\ W & & & & F(c,c') & & G(c,c')\\ & \searrow & & \underset{\mathclap{F(f,c')}}{\nearrow} & & \underset{\mathclap{G(f,c')}}{\nearrow}\\ & & F(c',c') & \underset{\alpha_{c'}}{\to} & G(c',c') \end{array}\,.$$

If the pullback $F(c,c) \times_{F(c,c')} F(c',c')$ exists in $D$, it suffices to assert this when the square on the left is the defining one of that pullback. On the other hand, if $D$ has a separator (such as $1\in Set$), it suffices to assert this when $W$ is the separator.

By comparison, a dinatural transformation asserts this condition only when $W = F(c',c)$ with the span consisting of $F(c',f)$ and $F(f,c)$.

Connection to relational model

We can regard strong dinatural transformations as a variation on the relational approach to parametric polymorphism, as follows.

If $F:C^{op}\times C\to D$ and $D$ has finite limits then for every $f:c\to c'$ in $C$ we can regard the pullback $F(c,c) \times_{F(c,c')} F(c',c')$ as a relation

$$R_{F,f} \subseteq F(c,c) \times F(c',c').$$

From this point of view, the family $\{\alpha_c:F(c,c)\to G(c,c)\}_c$ is strong dinatural if and only if it preserves these relations, i.e.

$$(x,y) \in R_{F,f} \;implies\; \big( \alpha_{c}(x), \alpha_{c'}(y) \big) \in R_{G,f} \,.$$

As shown by Vene (2006), for a class of types, the usual relational interpretation corresponds to a strong dinaturality interpretation.

However, the strong dinatural transformations don’t form a Cartesian closed category in general (e.g. Uustalu 2010) so might not serve to interpret all types, including function types.

References

Originally introduced (as “strong dinatural transformations”) in Definition 2.7 of:

• Philip S. Mulry, Strong Monads, Algebras and Fixed Points, in: Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series 177 (1991) 202–216 [doi:10.1017/CBO9780511525902.012]

Further developments:

• Philip S. Mulry: Strong monads, algebras and fixed points, Applications of Categories in Computer Science 177 (1992) 202–216 [doi:10.1017/CBO9780511525902.012]

• Robert Paré, Leopoldo Román: Dinatural numbers. Journal of Pure and Applied Algebra 128 1 (1998) 33–92 [doi:10.1016/S0022-4049(97)00036-4]

• A. Eppendahl, Parametricity and Mulry’s Strong Dinaturality, Department of Computer Science Technical Report No 768, Queen Mary (1999) [pdf, ISSN:1369-1961]

• Varmo Vene: Parametricity and Strong Dinaturality, talk notes (2006) [pdf, archive:pdf]

• Jennifer Hackett, Graham Hutton: Programs for cheap!, In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE, 115–126 [doi:10.1109/LICS.2015.21]

• Tarmo Uustalu: Strong dinaturality and initial algebras 12th Nordic Wksh. on Programming Theory, NWPT 2 (2000).

• Tarmo Uustalu: A Note on Strong Dinaturality, Initial Algebras and Uniform Parameterized Fixpoint Operators, In: FICS 2010. 77–82 [pdf, pdf]

Strong dinatural transformations are called paranatural transformations in:

• Jacob Neumann, Paranatural Category Theory, 2023, [arxiv:2307.09289]