nLab
duality

From a fundamental category theoretic perspective, there are two fundamental kinds of dualities:

• abstract duality: formally nothing but the reversal of arrows in a category $C$, i.e. the passage from $C$ to the opposite category $C^{\mathrm{op}}$;

• concrete duality: the oppositely-oriented image of some diagram under a contravariant internal hom functor $\mathrm{hom}(-,V)$.

(See Chapter 7 of Lawvere and Rosebrugh’s Sets for Mathematics.)

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)

Examples

• the duality between space and quantity
• Pontryagin duality?
• Stone duality?
• Eckmann-Hilton duality

Definition

A duality or dual equivalence is an equivalence between a category $C$ and the abstract dual (i.e. opposite) of a category $D$.

Remarks

• Often two categories are not related by a dual equivalence, but just by a dual adjunction: an adjunction between $C^{\mathrm{op}}$ and $C$. But every dual adjunction induces a maximal dual equivalence between subcategories of $C$ and $D$, as described below.

Maximal dualities inside dual adjunctions

…

Dualizing objects

Of particular interest are concrete dualities between concrete categories $C,D$, i.e. categories equipped with a faithful functor $f:C\to \mathrm{Set}$, $\hat{f}:D\to \mathrm{Set}$ 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

• H.-E. Porst, W. Tholen, Concrete Dualities in Category Theory at Work, Herrlich, Porst (eds.) (pdf)

Blog discussion

• David Corfield: More on duality