CatDat

Dual properties

Categories

Given a property $P$ of categories, its dual property $P^{\op}$ is defined as follows: a category $\C$ satisfies $P^{\op}$ if and only if its dual category $\C^{\op}$ satisfies $P$.

For example, since a category has an initial object if and only if its dual category has a terminal object, the property "has an initial object" is dual to the property "has a terminal object". In practice, dual properties can be obtained by reversing all arrows.

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

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 $P^{\op}$.