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:

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