Duality Principle (Category Theory)

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This proof is about Duality Principle in the context of Category Theory. For other uses, see Duality Principle.

Theorem

In the study of metacategories and categories, the following two duality principles are very useful.


Formal Duality

Morphisms-Only Category Theory

Let $\Sigma$ be a statement in the language of category theory.

Suppose $\Sigma$ is provable from the axioms for morphisms-only category theory $\mathrm{MOCT}$:

$\mathrm{MOCT} \vdash \Sigma$


Then the dual statement $\Sigma^*$ is also provable from these axioms, i.e.:

$\mathrm{MOCT} \vdash \Sigma^*$


Object Category Theory

Let $\mathrm{CT}$ be the collection of seven axioms on Characterization of Metacategory via Equations.

Suppose a statement $\Sigma$ about metacategories follows from the axioms $\mathrm{CT}$.


Then so does its dual statement $\Sigma^*$.


Conceptual Duality

Let $\Sigma$ be a statement about metacategories, be it in natural language or otherwise.

Suppose that $\Sigma$ holds for all metacategories.


Then so does its dual statement $\Sigma^*$.