Definition:Dual Statement (Category Theory)

Morphisms-Only Category Theory
Let $\Sigma$ be a statement in the language of category theory.

The dual statement $\Sigma^*$ of $\Sigma$ is the statement obtained from substituting:

Object Category Theory
In the more convenient description of metacategories by using objects, the dual statement $\Sigma^*$ of $\Sigma$ then becomes the statement obtained from substituting:

Example
For example, if $\Sigma$ is the statement:


 * $\exists g: g \circ f = \operatorname{id}_{\operatorname{dom} f}$

describing that $f$ is a split mono, then $\Sigma^*$ becomes:


 * $\exists g: f \circ g = \operatorname{id}_{\operatorname{cod} f}$

which precisely expresses $f$ to be a split epi.

For a set $\mathcal E$ of statements, write:


 * $\mathcal E^* := \left\{{\Sigma^*: \Sigma \in \mathcal E}\right\}$

for the set comprising of the dual statement of those in $\mathcal E$.

Also see

 * Category Axioms are Self-Dual
 * Duality Principle (Category Theory)