nLab
dinatural transformation

Contents

Idea

Dinatural transformations are a generalization of ordinary natural transformations and also of extranatural transformations. The differences can be summarized thusly:

  • In an ordinary natural transformation FGF\to G, both FF and GG involved depend on some variable xx with the same variance (covariant or contravariant).
  • In an extranatural transformation FGF\to G, either FF depends on xx both covariantly and contravariantly and GG does not depend on xx at all, or vice versa.
  • In a dinatural transformation FGF\to G, both FF and GG can depend on xx both covariantly and contravariantly.

If the dependence of FF or GG on xx in a dinatural transformation is trivial, it reduces to an extranatural one. Similarly, if the contravariant (or, dually, the covariant) dependence of FF and GG on xx are trivial, it reduces to an ordinary natural one.

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

Formalization

Let F,G:C op×CDF, G: C^{op} \times C \to D be functors. A dinatural transformation from FF to GG, sometimes written

α:FG,\alpha: F \stackrel{\bullet}{\to} G,

consists of a collection of morphisms

α c:F(c,c)G(c,c)\alpha_{c}: F(c, c) \to G(c, c)

such that for every morphism f:ccf: c \to c' in CC,

(1)G(c,f)α cF(f,c)=G(f,c)α cF(c,f):F(c,c)G(c,c) G(c, f)\alpha_c F(f, c) = G(f, c')\alpha_{c'}F(c', f): F(c', c) \to G(c, c')

This “hexagon identity” gives the “domain” version of extranaturality when GG is constant, and the “codomain” version when FF is constant. The domain version involves a commutative square of the form

(2)α cF(f,c)=α cF(c,f):F(c,c)G \alpha_c F(f, c) = \alpha_{c'} F(c', f): F(c', c) \to G

where GG is constant with respect to the argument cc, and the codomain version a commutative square of the form

(3)G(c,f)α c=G(f,c)α c:FG(c,c) G(c, f) \alpha_c = G(f, c')\alpha_{c'}: F \to G(c, c')

when FF is constant with respect to the argument cc.

Dinaturality versus extranaturality

Many people who encounter the notion of dinaturality through the general definition (as in equation (1) 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 (2) and (3) – 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. Here perhaps the most well known example is where F=hom:Set op×SetSetF = \hom: Set^{op} \times Set \to Set, where we have a class of dinatural transformations

hom(x,x)α nhom(x,x)\hom(x, x) \stackrel{\alpha_n}{\to} \hom(x, x)

defined by the rule alpha n(f)=f (n)alpha_n(f) = f^{(n)} (“Church numeral?s”). But these examples can be “bent” into domain extranaturality by defining

Set op×SetGSet:(x,y)hom(x,y) hom(y,x)Set^{op} \times Set \stackrel{G}{\to} Set: (x, y) \mapsto \hom(x, y)^{\hom(y, x)}

and considering extranatural transformations from the constant 11 (the terminal set) to GG. Such tricks support the counterargument that the extra generality of the traditional definition is largely spurious, and not particularly helpful in terms of comprehension.

Composition

Dinatural transformations cannot, in general, be composed with each other, although there are certain circumstances when they can be (such as when certain squares are pushouts or pullbacks, or when they are in fact ordinary natural transformations). In general, what we can say is that dinatural transformations with fixed source and target form a paracategory.

References

Here is a blog post inspired by the above discussion:

It discusses these concepts in the context of the programming language Haskell.

Revised on June 11, 2011 15:11:03 by Urs Schreiber (89.204.153.109)