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): $\S 3.1$