# Duality Principle (Category Theory)

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.

### 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 $\Sigma^*$.