nLab
duality

Idea

From the point of view of category theory, 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

• opposite category
• dual space, dual vector space
• the duality between space and quantity
• Poincaré duality for finite dimensional (oriented) closed manifolds
• Pontryagin duality for commutative (Hausdorff) topological groups
• Cartier duality of a finite flat commutative group scheme
• Serre duality? on nonsingular projective algebraic varieties which has as a special case the statement of the Riemann-Roch theorem
• Grothendieck duality, coherent duality? for coherent sheaves?
• Verdier duality? for abelian categories of sheaves; e.g. for a category of sheaves of abelian groups.
• Artin-Verdier duality? generalizing Tate duality? for constructible sheaves over the spectrum of a ring of algebraic numbers
• Koszul duality
• Stone duality
• syntax - semantics duality
• Eckmann-Hilton duality
• Spanier-Whitehead duality
• Tannaka 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.

Dualizing objects

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

$$f:C\to \mathrm{Set}$$f : C \to Set$, $\hat f : D \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.

Related concepts

• duality in physics

References

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

• David Corfield: More on duality (blog)

• wikipedia duality (mathematics)

• MathOverflow: the-concept-of-duality

• W. G. Dwyer, J. P. C. Greenlees, S. Iyengar, Duality in algebra and topology, Hopf archive