duality

# Duality

## Idea

Instances of “dualities” relating two different, maybe opposing, but to some extent equivalent concepts or phenomena are ubiquitous in mathematics (and in mathematical physics, see at dualities in physics).

In terms of general abstract concepts in category theory instances of dualities might be (and have been) organized as follows:

1. involution – any automorphism which is an involution, hence which squares to the identity? may be though of as exhibiting two dual perspectives on the objects that it acts on;

• abstract duality – the operation of sending a category to its opposite category is such an involution on Cat itself (and in fact this is the only non-trivial automorphism of Cat, see here). This has been called abstract duality. While the construction is a priori tautologous, any given opposite category often is equivalent to a category known by other means, which makes abstract duality interesting.

• concrete duality – given a closed category $\mathcal{C}$ and any object $D$ of it, then the operation $[-,D] : \mathcal{C} \to \mathcal{C}^{op}$ obtained by forming the internal hom into $D$ sends each object to something like a $D$-dual object. This is particularly so if $D$ is indeed a dualizing object in a closed category in that applying this operation twice yields an equivalence of categories $[[-,D],D] : \mathcal{C} \stackrel{\simeq}{\to} \mathcal{C}$ (so that $[-,D]$ is a (contravariant) involution on $\mathcal{C}$). If $\mathcal{C}$ is in addition a closed monoidal category then under some conditions on $D$ (but not in general) this kind of concrete dualization coincides with the concept of forming dual objects in monoidal categories.

From (Lawvere-Rosebrugh, chapter 7):

Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, 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 $V$ correspond or fail to correspond. (p. 122)

2. adjunction – another categorical concept of duality is that of adjunction, as in pairs of adjoint functors. Via the many incarnations of universal constructions in category theory this accounts for all dualities that arise as instances as the dual pairs

(Given that the saying has it that “Everything in mathematics is a Kan extension”, this goes some way in explaining the ubiquity of duality in mathematics.)

When the adjoint functors are monads and hence modalities, then adjointness between them has been argued to specifically express the concept of duality of opposites.

Adjunctions and specifically dual adjunctions (“Galois connections”) may be thought of as a generalized version of the above abstract duality: every dual adjunction induces a maximal dual equivalence between subcategories.

## Dualizing objects

Of particular interest are concrete dualities between concrete categories $C, D$, i.e. categories equipped with faithful functors

$f : C \to Set$

to Set, which are represented by objects $a \in C$, $\hat a \in D$ with the same underlying set $f(a) = \hat f(\hat a)$. Such objects are known as dualizing objects.

## References

Discussion of duality specifically in homological algebra and stable homotopy theory with emphasis on the concept of dualizing object in a closed category (and the induced Umkehr maps etc.) is in

Revised on February 27, 2014 10:13:10 by Urs Schreiber (89.204.137.80)