# Definition:Dual Statement (Category Theory)

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## 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 \(\) | \(\ds \operatorname {Cdm}\) |

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### 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 $\EE$.

## Also see

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 3.1$