# Definition:Dual Statement (Category Theory)

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## Contents

## 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:

\(\displaystyle R_\circ \left({y, x, z}\right)\) | \(\) for \(\) | \(\displaystyle R_\circ \left({x, y, z}\right)\) | |||||||||||

\(\displaystyle \operatorname{dom}\) | \(\) for \(\) | \(\displaystyle \operatorname{cod}\) | |||||||||||

\(\displaystyle \operatorname{cod}\) | \(\) for \(\) | \(\displaystyle \operatorname{dom}\) |

### 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:

\(\displaystyle f \circ g\) | \(\) for \(\) | \(\displaystyle g \circ f\) | |||||||||||

\(\displaystyle \operatorname{cod}\) | \(\) for \(\) | \(\displaystyle \operatorname{dom}\) | |||||||||||

\(\displaystyle \operatorname{dom}\) | \(\) for \(\) | \(\displaystyle \operatorname{cod}\) |

### 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

## Sources

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