# Definition:Dual Statement (Category Theory)

## Definition

### 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:

 $\ds R_\circ \tuple {y, x, z}$  for  $\ds R_\circ \tuple {x, y, z}$ $\ds \operatorname {Dom}$  for  $\ds \operatorname {Cdm}$ $\ds \operatorname {Cdm}$  for  $\ds \operatorname {Dom}$

### 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:

 $\ds f \circ g$  for  $\ds g \circ f$ $\ds \operatorname {Cdm}$  for  $\ds \operatorname {Dom}$ $\ds \operatorname {Dom}$  for  $\ds \operatorname {Cdm}$

### Example

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

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

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

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

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

For a set $\EE$ of statements, write:

$\EE^* := \set {\Sigma^*: \Sigma \in \EE}$

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