More on Duality
Posted by David Corfield
Continuing our earlier discussion about duality, it’s worth noting a distinction that Lawvere and Rosebrugh introduce in chapter 7 of their Sets for Mathematics between ‘formal’ and ‘concrete’ duality. Formal duality concerns mere arrow reversal in the relevant diagrams, so
of course if the original diagrams had been given specific interpretation in terms of specific sets and mappings, such interpretation is lost when we pass to this formal dual in that the formal dualization process in itself does not determine specific sets and specific mappings that interpret the dualized statement. (p. 121)
Concrete duality, on the other hand, occurs in situations where a new diagram is formed from an old one by exponentiating each object with respect to a given dualizing object, e.g., becomes , with the dualizing object. The arrows are naturally reversed in the new diagram.
Now,
Not every statement will be taken into its formal dual by the process of dualizing with respect to , and indeed a large part of the study of mathematics
space vs. quantity
and of logic
theory vs. example
may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite correspond or fail to correspond. (p. 122)
Very relevant for concrete dualities is Peter Johnstone’s Stone Spaces, especially chapter 6 and its discussion of schizophrenic objects. For a hard-hitting review by Johnstone of some Universal algebraists’ attempt to treat duality while minimizing contact with category theory, see this.
Schizophrenic objects have found a role in clarifying -categorical issues. See Makkai and Zawadowski’s Duality for Simple -Categories and Disks.
Re: More on Duality
In connection with these formal and concrete dualities, I can’t resist mentioning the Chu construction and especially Chu spaces: Vaughn Pratt has emphasized that many familiar concrete dualities are embodied in the category of Chu spaces.
In a nutshell, the Chu construction is a way of constructing a *-autonomous category from the data of a symmetric monoidal closed category M with pullbacks and an object D of M. An object of Chu(M, D) is a triple (A, B, p) where p is a D-valued pairing between A and B, and a morphism (A, B, p) –> (A’, B’, p’) is a pair of morphisms f: A –> A’, g: B’ –> B which are transposes of one another with respect to the pairings p and p’. If you write down a formal description of these hom-sets in terms of pullback diagrams involving hom-sets of M, then you can more or less guess what the closed structure of Chu(M, D) looks like, by internalizing these homs. The dual of (A, B, p) is obtained by switching A and B, and the dualizing object is the pair (D, I) with the obvious pairing.
The category of Chu spaces is Chu(Set, 2). A nice point emphasized by Pratt is that for many instances of concrete dualities where the underlying set of the schizophrenic object is 2, the category and its opposite embed as dual subcategories in the self-dual category of Chu spaces. This applies in particular to Stone duality, the duality between ordinals and intervals [emphasized by Joyal in connection with his category of disks], the self-duality of sup-lattices, and many others that you can think of.