Duality Principle (Category Theory)/Formal Duality

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This proof is about Formal Duality in Category Theory. For other uses, see Duality Principle.

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