Dual properties

Functors

Given a property $P$ of functors, its dual property $P^{\op}$ is defined as follows: a functor $F : \C \to \D$ satisfies $P^{\op}$ if and only if its dual functor $F^{\op} : \C^{\op} \to \D^{\op}$ satisfies $P$ . Notice that taking the dual does not reverse the direction of the functor.

For example, since a functor is essentially injective if and only if its dual is essentially injective, the property "is essentially injective" is self-dual. In particular, it is not dual to the property "is essentially surjective", which is itself also self-dual.

Again, notice that $(P^{\op})^{\op} = P$ , and that $F : \C \to \D$ satisfies $P$ if and only if $F^{\op} : \C^{\op} \to \D^{\op}$ satisfies $P^{\op}$ .

Morphisms

Given a property $P$ of morphisms, its dual property $P^{\op}$ is defined as follows: a morphism $f : X \to Y$ in a category $\C$ satisfies $P^{\op}$ if and only if its dual morphism $f^{\op} : Y \to X$ in the dual category $\C^{\op}$ satisfies $P$ .

For example, the property monomorphism is dual to epimorphism since $f$ is an epimorphism if and only if $f^{\op}$ is a monomorphism.

Notice that $(P^{\op})^{\op} = P$ , and that $f$ satisfies $P$ if and only if $f^{\op}$ satisfies $P^{\op}$ .

Symmetric Monoidal Categories

The dual of a symmetric monoidal category $(\C,\otimes,1)$ is defined by $(\C^{\op},\otimes,1)$ (and the obvious coherence isomorphisms). Given a property $P$ of symmetric monoidal categories, its dual property $P^{\op}$ is defined as follows: a symmetric monoidal category satisfies $P^{\op}$ if and only if its dual satisfies $P$ .

For example, the properties closed and coclosed are dual to each other.

A monoidal category has another dual, namely $(\C,\otimes^{\op},1)$ , but for a symmetric monoidal category this is isomorphic to $(\C,\otimes,1)$ .