Duality Principle (Category Theory)/Conceptual Duality

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

Theorem

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 $\Sigma^*$.

Thus it is sufficient to verify $\Sigma^*$ holds in all dual categories.

In any dual category $\mathbf C^{\mathrm {op}}$, we see that:

$\operatorname{dom} f^{\mathrm {op}} = \paren {\operatorname{cod} f}^{\mathrm {op}}$

$\operatorname{cod} f^{\mathrm {op}} = \paren {\operatorname{dom} f}^{\mathrm {op}}$

$f^\mathrm {op} \circ g^\mathrm {op} = \paren {g \circ f}^{\mathrm {op}}$

Thus, the interpretation of $\Sigma^*$ in $\mathbf C^{\mathrm {op}}$ comes down to the interpretation of $\Sigma$ in $\mathbf C$.

That is, except for amending every symbol $\star$ of $\mathbf C$ with a superscript "$\mathrm {op}$", yielding $\star^\mathrm {op}$.

This operation does not change the validity of $\Sigma$ in any possible way, and $\Sigma$ was assumed to hold in $\mathbf C$.

Hence we conclude that $\Sigma^*$ must hold in $\mathbf C^{\mathrm {op}}$ as well.

The result follows.

$\blacksquare$