3. Natural duality

One advantage of the category-theoretic point of view is that it becomes possible to define concretely what it means for isomorphisms to be ``natural''. The idea is to look at isomorphisms of whole categories at a time, instead of individual objects; the naturalness consists of compatibility with morphisms between objects.

There are two concepts: natural isomorphism of two categories, defined with covariant functors, and natural duality, defined with contravariant functors. Anticipating Boolean duality, let's concentrate on the second kind.

Definition. A natural duality between categories ${\cal C}$ and ${\cal D}$ consists of

(i) a contravariant functor $F$ from ${\cal C}$ to ${\cal D}$ and a contravariant functor $G$ from ${\cal D}$ to ${\cal C}$,

(ii) for each $A \in {\cal C}$, an isomorphism $\alpha _ A$ of $A$ with $G(F(A))$ and for each $B \in {\cal D}$, an isomorphism $\beta _ B$ of $B$ with $F(G(B))$, such that

(a) for each $A _ 1, A _ 2 \in {\cal C}$ and morphism $\phi \in$   Morph$(A _ 1, A _ 2)$, the following diagram is commutative, i.e., the same morphism is obtained by taking the composition along either route from the upper left corner to the lower right:

and (b) for each $B _ 1, B _ 2 \in {\cal D}$, a similar diagram is commutative.