# Duality Principle (Category Theory)/Formal Duality

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

## Contents

## Theorem

### 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 $\Sigma^*$.

## Proof

### Proof for Morphisms-Only Category Theory

From a formal perspective, if one would have derived (for some collection of statements $\Delta$):

- $\Delta \vdash \Sigma$

without using the axioms, then because $R_\circ$, $\operatorname{dom}$ and $\operatorname{cod}$ would still be undefined, it follows that necessarily also:

- $\Delta^* \vdash \Sigma^*$

In this correspondence, taking $\Delta$ to be $\mathrm{CT}$, it follows that:

- $\mathrm{CT} \vdash \Sigma$

implies:

- $\mathrm{CT}^* \vdash \Sigma^*$

As the Category Axioms are Self-Dual, in that $\mathrm{CT}^* = \mathrm{CT}$, we obtain the result.

$\blacksquare$

### Proof for Object Category Theory

## Sources

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